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A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean.

When the population standard deviation ? is known, we use a normal distribution to calculate the error bound.

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample.

Write a sentence that interprets the estimate in the context of the situation in the problem. We will first examine each step in more detail, and then illustrate the process with some examples.

When we know the population standard deviation ?, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. The interpretation should clearly state the confidence level (CL), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. Explanation of 90% Confidence Level Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score.

Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user’s body when using the handset.

Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Solution ATo find the confidence interval, start by finding the point estimate: the sample mean. Notice the difference in the confidence intervals calculated in [link] and the following Try It exercise. Explanation of 95% Confidence Level Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Increasing the confidence level increases the error bound, making the confidence interval wider. Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

Suppose we change the original problem in [link] to see what happens to the error bound if the sample size is changed.

Increasing the sample size causes the error bound to decrease, making the confidence interval narrower. Decreasing the sample size causes the error bound to increase, making the confidence interval wider. When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. Use n = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.

Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students. The population standard deviation for the height of high school basketball players is three inches.

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known.

The calculation of EBM depends on the size of the sample and the level of confidence desired. Given a confidence interval, you can work backwards to find the error bound (EBM) or the sample mean. Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size. Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds.

What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? As the sample size increases, there will be less variability in the mean, so the interval size decreases. If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make? If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence?

The level of confidence would decrease because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases. Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Construct a 90% confidence interval for the population mean weight of the heads of lettuce. Construct a 95% confidence interval for the population mean weight of the heads of lettuce. In complete sentences, explain why the confidence interval in [link] is larger than in [link]. What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same? What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

As a result of your answer to [link], state the exact distribution to use when calculating the confidence interval.

Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the next seven exercises.

Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample mean. Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches.

What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. Construct a 95% confidence interval for the population mean length of engineering conferences. Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. Construct a 90% confidence interval for the population mean time to complete the tax forms. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour.

In complete sentences, explain why the confidence interval in part f is larger than the confidence interval in part e. A camp director is interested in the mean number of letters each child sends during his or her camp session.

Construct a 90% confidence interval for the population mean number of letters campers send home. What is meant by the term “90% confident” when constructing a confidence interval for a mean? If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.

If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.

If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle.

Notice the small difference between the two solutions–these differences are simply due to rounding error in the hand calculations.

We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637. The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants.

The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

When you use the appropriate formula for CIs built from that distribution, you will find that you need a slightly different critical value, which will result in a slightly wider CI.

Not the answer you're looking for?Browse other questions tagged self-study confidence-interval or ask your own question.

To maximize the chance of correctly guessing the result of a coin flip, should I always choose the most probable outcome?

DESCRIPTION: Henricus Martellus was the one mapmaker who linked the late medieval cartography, just emerging from social, religious, academic and technological constraints, to mapping that reflected the Renaissance and the new discoveries.

No one at that time had any knowledge of the true position and outlines of East Asia, yet the representation of East Asia is identical on both the Martellus maps and the Behaim globe. Who had most to gain from such a reckless exaggeration of the extent of Eurasia and who was the first to do so?

The Arctic coasts on the Martellus map and those of north and northwest Europe resemble those of the Fra Mauro planisphere of 1459 (#249) rather than those of the Ulm Ptolemy of 1482. In the volume of Imago Mundi, found amongst the possessions of Christopher Columbus after his death, there are numerous notes or postils written in the margins or below the printed matter. Note that in the year a€?88 in the month of December arrived in Lisbon Bartholomew Diaz, Captain of three caravels which the Most Serene King of Portugal had sent to try out the land in Guinea. Another peculiar feature of the Martellus map is the enormous peninsula commencing due south of Aureus Chersonesus [the Malay peninsula] at 28A° south, thereafter widening to reach China. It has become clear to some cartographic scholars that South America was represented as a huge peninsula of southeastern Asia on many world maps of the 16th century, from the Zorzi sketches of 1506 to the Livio Sanuto map of 1574.

It is not so well known that this very same peninsula existed already under the name of India Meridionalis on earlier maps, drawn before the arrival in the western hemisphere of Christopher Columbus. However, the Martellus maps show a very good representation of the South American hydrographic system, including all the great rivers in the sub-continent. A deeper study of the same maps has made possible the identification of several capes on the Atlantic coast, the swamps of the Rio Negro in Brazil, and Lake Titicaca.

There was thus no known pre-Columbian historical exploration of South America by the European nations.

The earlier maps extant include the so-called mappaemundi drawn by medieval churchmen in Western Christendom. In order to detect this peninsula on pre-Martellus maps, we needed an identifying criterion. The following is a theory expressed by Paul Gallez regarding the depiction of South America prior to Columbusa€™ voyages.

In the southernmost part of South America (?), there are the words, next to a strait: Hic sunt gigantes pugnantes cum draconious [Here live some giants who fight against the dragons]. Traditionally, the so-called Legend of the Patagonian Giants is attributed to Antonio Pigafetta, the chronicler of Magellana€™s voyage.

The east-west extent from the Canaries to the coast of China is about 235A°, which agrees with the Toscanelli-Columbus concept and with the 1489 Martellus map now in the British Library. The sheets of paper on which the Yale map is drawn are of different sizes, which excluded the possibility that they were printed map sheets, for they would then have had to be the same size to fit within the map portfolio. Roberto Almagia (1940) stated that he had identified no less than three manuscript maps of the world signed by Martellus, all virtually identical with that in the codex in the British Library. When Columbus left Lisbon in 1485 for Spain, Bartholomew, with his highly trained skills as a cartographer in the Genoese style, stayed on in the map workshop of King John II. These two world maps by Martellus represent, along with Martin Behaima€™s famous globe of 1492, the last view of the old pre-Columbian world as perceived by Western Europe before the great expansion of the world picture during the subsequent twenty years. The Yale example shows more of the Ocean Sea in the Far East than does the British Library manuscript. Very little is known about Henricus Martellus Germanus, a mapmaker working in Florence from 1480 to 1496.

The Indian Ocean is open to the south, but Martellus retains its eastern coast, which Ptolemy had called Sinae, home of the fish-eating Ethiopians. Martellus retains other Ptolemaic features, such as the appearance of the northern coast of the Indian Ocean with a flattened India and a huge Taprobana. Although Martellusa€™s world maps in the Ptolemy manuscripts show an extension of the inhabited world as 180A°, those in the Insularium make it 220A° or more, an attractive feature to those who dreamed of reaching Asia by sailing west. Most of the text still isna€™t legible, but some of the parts that are appear to be drawn from the travels of Marco Polo through east Asia. The work of Henricus Martellus Germanus epitomizes the best of European cartography at the end of the 15th century. Wolff, Hans, a€?The Conception of the World on the Eve of the Discovery of America - Introduction,a€? in Wolff, Hans (ed.), America. AlmagiA , Roberto, a€?I mappamondi di Enrico Martello e alcuni concetti geografici di Cristoforo Colombo,a€? La Bibliofolia, Firenze, XLII (1940), pp.

AlmagiA , Roberto, a€?On the cartographic work of Francesco Roselli,a€? Imago Mundi, VIII (1951), pp. Caraci, Ilaria L., a€?La€™opera cartografica di Enrico Martello e la a€?prescopertoa€™ della€™America,a€? Rivista Geografica Italiana 83 (1976), Florence, pp.

Caraci, Ilaria L., a€?Il planisfero di Enrico Martello della Yale University Library e i fratelli Colombo,a€? Rivista Geografica Italiana 85 (1978), Florence, pp.

Crino, Sebastiano, a€?I Planisferi di Francesco Roselli,a€? LaBibliofilia, Firenze, XLI (1939), pp. Francesco Rosselli was one of the earliest known map stockists and map sellers; in addition he was an important map-maker whose cartographic output spans the decades of the great discoveries.

The mapa€™s most prominent feature is, like the Martellus model, the new outline of Africa, now quite separate from Asia, and reflecting the rounding of the Cape by Bartolomeu Diaz in 1487. First, by 1486 the mathe-matical junta had solved the problem of establishing latitude by measuring the height of the mid-day sun. The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. If we include the central 90%, we leave out a total of ? = 10% in both tails, or 5% in each tail, of the normal distribution. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1). This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.

A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes.

However, sometimes when we read statistical studies, the study may state the confidence interval only.

We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed? If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed? When estimating a population mean, the margin of error is called the error bound for a population mean (EBM).

The confidence level is the percent of all possible samples that can be expected to include the true population parameter. To find the error bound, find the difference of the upper bound of the interval and the mean. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution.

Recall, when all factors remain unchanged, an increase in sample size decreases variability. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. Since we increase the confidence level, we need to increase either our error bound or the sample size.

During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. The most recent survey estimates with 90% confidence that the mean household income in the U.S. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just study for that next big test).

However, even though you report normality, you could still adjust for your smaller sample size by using a different distribution. Unfortunately, using pivotal quantities as a way to construct intervals is usually not taught at this level.

Little is known of this important German cartographer, probably from Nuremberg, who worked in Italy from 1480 to 1496 and produced a number of important manuscript maps. No meridians, parallels or scales of longitude are given, but estimates based on measurements of the map indicate about 230A° from Lisbon to the coast of China, or 240A° from the Canaries. The actual latitude of the Cape of Good Hope is 34A° 22a€™ south based on land measurements by Diaz, who landed three times on the south coast.

He reported to the same Most Serene King that he had sailed beyond Yan 600 leagues, namely 450 to the south and 250 to the north, up to a promontory which he called Cabo de Boa Esperanza [Cape of Good Hope] which we believe to be in Abyssinia. It indicates that he was high in the confidence of the King as an expert cartographer, otherwise he would not have been present at such a secret occasion. It was done probably in Seville after he joined his brother, entering the postil in Imago Mundi and altering his prototype map at the same time. It seems that the prototype originally showed the Cape at 35A° south, well clear of the frame at about 41A° south, as one would expect of a competent cartographer. It is a relic of the continuous coastline that linked Southeast Asia to South Africa in Ptolemaic world maps that displayed a land-locked Indian Ocean, and it needs a name to identify it in argument.

This is the India that Columbus was looking for, because it was marked in the right place on his maps.

In the post-Columbian series, the isthmus of Panama is represented with its true width, because it had been heard of by Columbus and other explorers from the aborigines; in the pre-Columbian series, the union of the peninsula with Asia is much broader, because nobody had exact information about it.

The pre-Magellanic maps have South America extending only to some degrees South; on post-Magellanic maps the land extends to 53 degrees South.

On these pre-Columbian maps, the drainage net is much better drawn than on any other representation made before 1850. So Gallez believes that the deep and sound European knowledge of South America before its exploration by Columbus and his Spanish and Portuguese challengers has been firmly established.

But the detail of its hydrographic features mapped by Martellus in 1489 is a fact, even if this fact remains historically unexplained.

In this way we have identified the Dragona€™s Tail on three maps drawn between 1440 and 1470. As cultured persons, both Pigafetta and Magellan would have seen such maps as Walspergera€™s or others of the same family, and they would surely have taken aboard some copies of them. Long in the possession of a family from Lucca, it went to Austria in the 19th century and was bought for Yale in 1961.

Vietor, then Map Curator of Yale University, reported a gift by an anonymous donor a€?in the form of a magnificent painted world map signed by Henricus Martellus, approximately six feet by four feet (180 x 120 cm). In a private communication of June 1972, Vietor stated that X-ray examination had revealed no evidence of printing on the paper sheets and that it was hand-drawn, lettered and colored.

It was not foretold on Folio 1 and is clearly an unexpected addition to the codex of picture-islands. He was engaged in building up a large map of the world based on Donus Nicolaus and on Portuguese charts.

On his globe, South Africa, between 23A° and 28A° south, extends a horn of land for hundreds of miles due eastward into the Indian Ocean. Among the thousands of islands Marco Polo reported off the coast of Asia, an enormous Sumatra and Java are found in the south, while to the northeast is the huge island of Cipangu [Japan]. The map is remarkable for its exciting new information, although being imperfect because of its acceptance of classical and medieval antecedents.

On the world map in his Insularium at the British Library, he shows the results of Bartholomew Diasa€™ rounding of the Cape of Good Hope in 1488, listing the names of the various ports and landmarks along the way. This was the edge of Ptolemya€™s map - he did not show the full extent of Asia, noting that a€?unknown landsa€? lay beyond. In the north, Greenland is a long, skinny peninsula attached to Europe and north of Scandinavia, a concept derived from the Claudius Clavus map of 1427.

One of the Insularium manuscripts at the Laurentian Library in Florence seems to be a working copy, as it has many cross-outs and corrections on the maps and in the texts. His intelligent effort to reconcile modern discoveries with traditional knowledge, especially Ptolemy, provides a workable model for the world map in a time of rapid change. He worked in Florence prior to 1480, then was away from Italy for about two years before returning to his home town where he was active until his death some time after 1513. The cartography of such maps is very poor: for instance, on the maps of Hieronymo Girava 1556, Johann Honter 1561, Giacomo Gastaldi 1562 and Francesco Basso 1571, the Rio Amazonas has its source in Patagonia and flows from south to north.

However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).

What happens to the error bound and the confidence interval if we increase the sample size and use n = 100 instead of n = 36? If we know the confidence interval, we can work backwards to find both the error bound and the sample mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. With this installment from Internet pedagogical superstar Salman Khan's series of free math tutorials, you'll learn how to calculate a confidence interval in basic statistics.

Ptolemya€™s geographical writings, largely unknown to Western Europeans during the earlier Christian Middle Ages in Europe, became the basis for the Renaissance in geography. His entire hopes of gaining support from King John in 1485 for such an enterprise as sailing westward to Cathay rested on his argument that it lay only 130A° to 140A° to the west. A manuscript copy of Polo and his travels was given by the Doge of Venice to Prince Pedro in 1427 and was thereafter in the Kinga€™s Treasury in Lisbon. Measurements of altitudinal height of the sun by astrolabe or quadrant were accurate on land but could be 2A° or more out on the rolling deck of a ship.

23 is in the handwriting of Bartholomew, and was identified as his by Bishop Bartolomeo de Las Casas, who knew him well. He says that in this place he found by the astrolabe that he was 45A° below the equator and that this place is 3,100 leagues distant from Lisbon [19,850 km].

It was to influence the Catholic Sovereigns who were in the dark owing to the intense secrecy by Portugal regarding discovery. There is no record extant of anyone reporting that they had seen any land or island south of the equator, nor did anybody pretend to have explored the inner part of a trans-Atlantic continent and to have mapped its rivers.

We may thus believe that this knowledge already existed before Martellus, and we should look at older maps in search of the sources that he could have had at his disposal.

They thus knew that, following their maps, they would have to sail to the south along the coast of the Dragona€™s Tail until they reached the Land of Giants, and that at the end of that land, they would find a passage to the West, to the Sinus Magnus, and thus a way to the Moluccas. It shows a great deal more ocean to the east of Asia with several large islands in it, including Japan in the far northeast.

Signora Carla Marzoli of Milan, in a private communication, stated that this large map a€?had left Italy into the possession of family centuries ago and had been lodged in a Swiss bank for safety, for a long time.a€™ In 1959, through trade channels, she learned that this map was for sale. It uses the homeopathic (heart-shaped) projection, as far as can be judged, for it lacks meridians and parallels and scales. They all omitted Cipangu but he found a separate map of Cipangu in a codex which had the same outlines as those in the Behaim globe of 1492. It was, like all important maps at that time, drawn on sheets of parchment which could be joined together almost invisibly, and mounted on linen. In the Indian Ocean, the islands of Madagascar and Zanzibar, rather poorly drawn, add an intriguing aspect. It was the most accurate delineation available to Martin Behaim when he constructed his globe exhibiting the pre-Columbian world. On the Martellus map there is a long peninsula to the east of the Golden Chersonese (Indochina), featuring the mysterious port of Cattigara.

The Mediterranean and Black Seas and the Atlantic coasts are taken from sea charts, while the east coast of Africa, as yet unexplored, also follows Ptolemya€™s design. Here the habitable world is extended to 265A°, with northern Asia coming right up against the right-hand border. According to historian Chet Van Duzer who is participating in a new (2014) digital image scanning of the Yale Martellus, a€?In northern Asia, Martellus talks about this race of wild people who dona€™t have any wine or grain but live off the flesh of deer and ride deer-like horsesa€?.

The role of experience was clearly important to Martellus, for in his copy of the Laurentian Insularium he includes a verse about his own travels, noting that he has been traveling around for many years, and adding that it is worthwhile but difficult to set a white sail upon the stormy sea.

It is a cleanly-engraved copperplate, with finely-stippled sea and six characteristic loose-haired wind-heads grouped round the border of the map.

A legend at the bottom referring to the date 1498 is clearly an error for 1489, and has been copied as such from the 1489 Martellus manuscript world map with obviously similar geographic content to that of Rosselli. These maps depicted graphically the theory that Cipangu [Japan] was but 3,500 miles (5,635 kilometers) westward, and only 1,500 miles (2,415 kilometers) further lay the shores of Cathay [China]. 23 is in the handwriting of Bartholomew, and was identified as his by Bishop Bartolomeo de Las Casas, who knew him well.A Arthur Davies has made a long study of the writing of the Columbus brothers and states, without a shadow of doubt that it is in the hand of Bartholomew.

Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.

To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. Construct a 95% confidence interval estimate for the population mean number of eggs laid per chicken per month. The Martellus delineation included some Ptolemaic dogma in its continental contours and projection but significantly modified and improved upon the ancient model with regards to its contents. The coast of Cathay is approximately 130A° West of Lisbon on the Martellus map, on the Behaim globe and in the Columbus-Toscanelli concept. Copious marginal notes, in the handwriting of Bartholomew and his older brother Christopher, are found in the Admirala€™s copy of the printed book of Marco Polo published at Louvain between 1485 and 1490. The coasts charted by Diaz have been fitted into the circular outline of the world of the Fra Mauro hemisphere, representing such a marked trend to the southeast that the Cape of Good Hope seems to be due south of the Persian Gulf, whereas it is due south of the Adriatic. Yet the Martellus map shows South Africa extending across the frame of the map to 45A° south. He has described this voyage and plotted it league by league on a marine chart in order to place it under the eyes of the Most Serene King himself. This dates the legend as 1489, probably in January of that year, just before Bartholomew went to Seville.

Arthur Davies, in his discussion of this map refers to it as the Tiger-leg peninsula; in others it is referred to as Catigara. The best preserved copy is in the British Library and there is also a poorer copy in the University of Leiden.

Considering that in 1489 Martellus knew about the inner courses of many South American rivers, we have no reason to doubt that, forty years earlier, Andreas Walsperger (#245) knew of the Patagonians. The meeting with the Tewelche in Saint Julian was the full confirmation, for Magellan and Pigafetta, of what they already knew from their maps. The Laon globe (#259), made in France about 1493 (probably by Bartholomew Columbus when he served Anne, Regent of France) has the Tiger-leg to 40A° south.

She examined it and saw its connection with the Martellus map in the British Library and with the conceptions of Columbus. Fortunately, Martellus copied most of the nomenclature and legends on to the smaller map, now in the British Library, so the a€?Yalea€™ Martellus can recover them without difficulty. Next come three regional maps, striking because of the enormous nomenclature on the coasts. The three manuscript maps were larger in scale and showed more detail than the codex world map.

This large map, 180 cm by 120 cm, formed a standard Portuguese world map, continually added to by new discoveries, including those of CA?o and Diaz. Some historians claim that their presence indicates the map cannot be dated before 1498, when Vasco da Gama returned with news of his voyage to India. Modern Africa is so long, however, that it breaks through the frame at the lower edge of the map. Unlike Ptolemya€™s Asia, Martellusa€™ version has an eastern coast with the island-bedecked ocean beyond and additional names taken from Marco Polo. From this we can see that in the Insularia Martellus used modern information when he had it, incorporating it into a classical format. Martellusa€™s depiction of rivers and mountains in the interior of southern Africa, along with place names there, appear to be based on African sources. He suggests that the reader might prefer to stay safely at home, and learn about the world through his book. Like its model, the projection is the barrel-shaped (homeoteric, or modified spherical) a€?second projectiona€™ of Ptolemy. Columbus thus had documentary support for his beliefs about oceanic distances from his readings of earlier cosmographers, Cardinal Da€™Ailly (#238) and Paolo Toscanelli (#252). Its shape, outline and position are identical with the representation of Cipangu on the Behaim globe. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.

If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean. Kimble demonstrated that, as far as 13A° South, the nomenclature of West African coasts is 80% identical in the two maps but bears no relation to the nomenclature of any other map of the period. The minor differences in location between these three must be seen against the enormous exaggeration of the extent of Eurasia which they exhibit in comparison with all previous estimates.

The only other claim that the Cape was at 45 degrees south is in the hand of Bartholomew Columbus. It suited Columbus admirably and it is likely that Bartholomew made the change at his direction. It is shown to more than 25A° south in the Cantino map of 1502 (Book IV, #306), Canerio map of 1504 (Book IV, #307), the WaldseemA?ller maps of 1507, 1513 (Book IV, #310) and many others. The map is surrounded by a dozen puffing classical wind heads, and for the first time in a non-Ptolemaic map, there is a latitude and longitude scale on the side.

At her request, Roberto Almagia and Raleigh Skelton examined it and pronounced it authentic.

The first is of Western Europe and Morocco, cut off in the east through the center of France; the second starts from this line and extends east as far as Naples and Tunis.

The Martelli were a prominent merchant family in Florence, and Enrico Martello sounds Italian enough.

If the two Ptolemy atlases can be dated as the first and last of his works, it is interesting to see that he reverts to the pure Ptolemaic form for the world in the later edition, with the lengthened Mediterranean and the closed Indian Ocean. Ita€™s likely that this information came from an African delegation that visited the Council of Florence in 1441 and interacted with European geographers.

This provided him with the ammunition to further promote his plan to sail west to reach the Indies.

The Martellus map in the British Library is less than 20 inches (50 cm) from west to east, and is on a scale one-quarter that of the Yale map. South of that limit, however, the Martellus map gives the outlines and nomenclature of the voyage of Bartholomew Diaz in 1487-88, while the Behaim globe has invented nomenclature corresponding to nothing in Portuguese cartography. It is difficult to avoid the conclusion that he copied a map which was originally designed to support the ideas of Columbus. Due south of the Malay peninsula, at 28A° South, there appears an enormous peninsula which widens and turns north to join China again with the largely circular concept of the Fra Mauro map. On the face of it, seeing it on the Martellus map, it asserts that Diaz had proceeded north along the cast coast of South Africa to beyond Natal. Columbus hoped Spain would support a voyage westward to Cipangu, 85A° away, and to Cathay 130A° to the west. Of course, this is mostly symbolic, as neither Henricus Martellus nor anyone else had a complete set of accurate coordinates. The sheets of the a€?Yalea€™ Martellus, together with certain regional maps, were acquired by the patron. The one piece of information he gives us about himself, that he had traveled extensively, suggests a career in business. Again, with the Tigerleg or the Dragona€™s Tail, there has been an controversial attempt to identify it as an early map of South America, turning Ptolemy's a€?Sinus Magnusa€? into the Pacific Ocean. Clearly, Ptolemy maps were considered as illustrations for a revered and ancient text and did not have to include the latest news. Three other surviving maps contain some of this same information, but the Martellus map covers more territory than any of them, making it the most complete surviving representation of Africansa€™ geographic knowledge of their continent in the 15th century. The splendid manuscript of Martellus preserved in the National Library at Florence contains thirteen tabulA¦ modernA¦, but is probably later than the earliest printed editions of Ptolemya€™s Geography.A However, Martellus revised the Ptolemaic world map based on Marco Poloa€™s information on Asia, and he incorporated the recent Portuguese voyages to Africa. Due south of the Malay peninsula, at 28A° South, there appears an enormous peninsula which widens and turns north to join China again with the largely circular concept of the Fra Mauro map.A This peninsula does not exist in fact, and seems to be a repeat of the Donus Nicolaus map of the world in the Ulm Ptolemy, cut away by the circular outline of Fra Mauro. On the face of it, seeing it on the Martellus map, it asserts that Diaz had proceeded north along the cast coast of South Africa to beyond Natal.A His furthest point, in fact, was the Rio de Infante [Great Fish River] on the south coast, at 34A° south. Although the Columbus brothers knew that Marco Polo had returned from China by this sea route, they inserted this great obstruction of Tiger-leg by 1485. The correspondence between the Behaim globe and the 1489 Martellus map consequently ended in 1485, when Diego CA?o had returned from his first voyage after reaching 13A° south (Cape Santa Maria in the Congo, 1482-1484).

Yet that map could not have been completed until early in 1489 for it had complete details of the discoveries of Bartholomew Diaz in his voyage of 1487-88, when he circumnavigated the Cape of Good Hope (capo da€™ esperanza) and reached the Indian Ocean.

At that latitude one degree was thought to be 50 miles (80 km), according to the Toscanelli letter, so that Japan was only 4,250 miles (7,200 km) west. No wonder that, having regard to his maps, he concluded that this river flowed from Paradise. Martellus assembled the paper sheets, stuck them on a canvas backing, made his characteristic rectangular picture frame for it and colored the seas in dark blue. His surviving work includes two editions of Ptolemya€™s Geography, a large wall map now at Yale University, and five editions of the Insularium Illustratum of Cristoforo Buondelmonti. The correspondence between the Behaim globe and the 1489 Martellus map consequently ended in 1485, when Diego CA?o had returned from his first voyage after reaching 13A° south (Cape Santa MariaA in the Congo, 1482-1484).

It would show King John that even if the Portuguese reached India they could not reach the Spice Islands (which were on the equator east of Tiger-leg) without having to make long voyages into the southern stormy seas.

To have included Cipangu would have required a scale reduced one-sixth, too small to permit of legible nomenclature.

This was the time when Martin Behaim, as a member of the King Johna€™s mathematical junta, was able to study the map and proposals of Columbus.

He returned from this voyage in December 1488 and, within a year, full details, including rich nomenclature, had appeared on the map of Martellus made in Italy; this despite the utmost secrecy on the part of King John of Portugal. The fact that the map had been put away in a Swiss bank for perhaps 60 years explains why it went unobserved.

Martellus had been required by his patron to include in the codex a world map and regional maps which had just come into his possession in 1489. The historian Roberto Almagia has pronounced that Henricus Martellus was an excellent draftsman, who drew upon the latest information and improved the maps he adapted for his collections. This was the time when Martin Behaim, as a member of the King Johna€™s mathematical junta, was able to study the map and proposals of Columbus.A On the 1489 Martellus map there is an inscription next to the Congo that mentions the commemorative stone (PadrA?o) that CA?o erected at Cape Negro during his second voyage (1485-87) when he reached as far as Cape Cross. For King John and for the Catholic Sovereigns it showed that Spain could easily reach Cathay and then the Spice Islands, secure from all interference from Arabs or Portuguese in the Indian Ocean.A Columbus may have deceived himself regarding the existence of the Tiger-leg but, since it suited his plans so admirably, one suspects that he asserted its existence just as he later made the Cape of Good Hope to be at 45A° south. The second major difference is that, while the coasts from Normandy to Sierra Leone, on the Yale map, are based on the world map of Donus Nicolaus of 1482, the corresponding coasts on the British Library Martellus uses a Portuguese map for them. The furthest point reached by Diaz, the Rio de Infante [the great Fish River], is duly recorded as ilha de fonti. His great talent for such an exacting task was his skill as a draughtsman-cartographer experienced in altering the scale of maps of islands that he was copying. Someone with access to it and to the reports of Bartholomew Diaz, drew the prototype of the Martellus map. According to the historian Arthur Davies, this is conclusive evidence that the prototype originally terminated at 35A°, with the legend correctly placed near it. Even today, the best source for information on the voyage of Diaz is the Martellus map of 1489. The extra ten degrees in shifting the Cape to 45A° south meant more than twenty degrees extra distance in a voyage to India.

Two peculiar features in the region of South Africa suggest that Bartholomew Columbus was that someone. When Bartholomew altered the prototype map to 45A° south, he was unable to remove the legend. The policy of secrecy of King John was shattered in one great leakage by someone in a unique position to know all the details.

Moreover, and perhaps this was the decisive factor, it would take Portuguese ships to nearly 50A° south to round Africa, into what Diaz had already found to be the roughest seas encountered anywhere in the world. Moreover, and perhaps this was the decisive factor, it would take Portuguese ships to nearly 50A° south to round Africa, into what Diaz had already found to be the roughest seas encountered anywhere in the world.A Bartholomew, in 1512, gave evidence in the Pleitos (the great lawsuit of the Columbus family versus the Crown of Spain) and declared that he had gone about with his brother in Spain helping to gain support for his enterprise. Each island-picture is surrounded by a frame, drawn and painted in ivory to give the impression of a carved wood frame, suggesting a picture hanging on a wall. Such picture-islands occupy nearly all the folios, making an Atlas of Islands, and are then followed by a superb picture-map of Italy, taken from the 1482 world map of Donus Nicolaus. The islands in this codex were copies from the Ulm Ptolemy, from the Isolario of Bartolomeo de li Sonetti printed in Venice in 1485, and from other sources. This account of Gallo was copied, almost verbatim by Serenega in 1499 and by Giustiniani in 1516. Bartholomew left Genoa as a youth in 1479 and officially made only one further visit to Italy, in 1506. Gallo could have acquired knowledge of the role of Bartholomew only from his own lips, in 1489.

When the population standard deviation ? is known, we use a normal distribution to calculate the error bound.

To construct a confidence interval estimate for an unknown population mean, we need data from a random sample.

Write a sentence that interprets the estimate in the context of the situation in the problem. We will first examine each step in more detail, and then illustrate the process with some examples.

When we know the population standard deviation ?, we use a standard normal distribution to calculate the error bound EBM and construct the confidence interval. The interpretation should clearly state the confidence level (CL), explain what population parameter is being estimated (here, a population mean), and state the confidence interval (both endpoints). Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. Explanation of 90% Confidence Level Ninety percent of all confidence intervals constructed in this way contain the true mean statistics exam score.

Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user’s body when using the handset.

Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Solution ATo find the confidence interval, start by finding the point estimate: the sample mean. Notice the difference in the confidence intervals calculated in [link] and the following Try It exercise. Explanation of 95% Confidence Level Ninety-five percent of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score. Increasing the confidence level increases the error bound, making the confidence interval wider. Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

Suppose we change the original problem in [link] to see what happens to the error bound if the sample size is changed.

Increasing the sample size causes the error bound to decrease, making the confidence interval narrower. Decreasing the sample size causes the error bound to increase, making the confidence interval wider. When we calculate a confidence interval, we find the sample mean, calculate the error bound, and use them to calculate the confidence interval. If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. Use n = 217: Always round the answer UP to the next higher integer to ensure that the sample size is large enough.

Therefore, 217 Foothill College students should be surveyed in order to be 95% confident that we are within two years of the true population mean age of Foothill College students. The population standard deviation for the height of high school basketball players is three inches.

In this module, we learned how to calculate the confidence interval for a single population mean where the population standard deviation is known.

The calculation of EBM depends on the size of the sample and the level of confidence desired. Given a confidence interval, you can work backwards to find the error bound (EBM) or the sample mean. Sometimes researchers know in advance that they want to estimate a population mean within a specific margin of error for a given level of confidence. The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size. Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds.

What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? As the sample size increases, there will be less variability in the mean, so the interval size decreases. If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make? If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence?

The level of confidence would decrease because decreasing n makes the confidence interval wider, so at the same error bound, the confidence level decreases. Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Construct a 90% confidence interval for the population mean weight of the heads of lettuce. Construct a 95% confidence interval for the population mean weight of the heads of lettuce. In complete sentences, explain why the confidence interval in [link] is larger than in [link]. What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same? What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?

As a result of your answer to [link], state the exact distribution to use when calculating the confidence interval.

Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the next seven exercises.

Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample mean. Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? The error bound for the mean would decrease because as the CL decreases, you need less area under the normal curve (which translates into a smaller interval) to capture the true population mean.

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches.

What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Announcements for 84 upcoming engineering conferences were randomly picked from a stack of IEEE Spectrum magazines. Construct a 95% confidence interval for the population mean length of engineering conferences. Suppose that an accounting firm does a study to determine the time needed to complete one person’s tax forms. Construct a 90% confidence interval for the population mean time to complete the tax forms. If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?

If the firm did another survey, kept the error bound the same, and only surveyed 49 people, what would happen to the level of confidence? Suppose that the firm decided that it needed to be at least 96% confident of the population mean length of time to within one hour.

In complete sentences, explain why the confidence interval in part f is larger than the confidence interval in part e. A camp director is interested in the mean number of letters each child sends during his or her camp session.

Construct a 90% confidence interval for the population mean number of letters campers send home. What is meant by the term “90% confident” when constructing a confidence interval for a mean? If we took repeated samples, approximately 90% of the samples would produce the same confidence interval. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.

If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.

If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples. The Federal Election Commission collects information about campaign contributions and disbursements for candidates and political committees each election cycle.

Notice the small difference between the two solutions–these differences are simply due to rounding error in the hand calculations.

We estimate with 95% confidence that the mean amount of contributions received from all individuals by House candidates is between $287,109 and $850,637. The American Community Survey (ACS), part of the United States Census Bureau, conducts a yearly census similar to the one taken every ten years, but with a smaller percentage of participants.

The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

When you use the appropriate formula for CIs built from that distribution, you will find that you need a slightly different critical value, which will result in a slightly wider CI.

Not the answer you're looking for?Browse other questions tagged self-study confidence-interval or ask your own question.

To maximize the chance of correctly guessing the result of a coin flip, should I always choose the most probable outcome?

DESCRIPTION: Henricus Martellus was the one mapmaker who linked the late medieval cartography, just emerging from social, religious, academic and technological constraints, to mapping that reflected the Renaissance and the new discoveries.

No one at that time had any knowledge of the true position and outlines of East Asia, yet the representation of East Asia is identical on both the Martellus maps and the Behaim globe. Who had most to gain from such a reckless exaggeration of the extent of Eurasia and who was the first to do so?

The Arctic coasts on the Martellus map and those of north and northwest Europe resemble those of the Fra Mauro planisphere of 1459 (#249) rather than those of the Ulm Ptolemy of 1482. In the volume of Imago Mundi, found amongst the possessions of Christopher Columbus after his death, there are numerous notes or postils written in the margins or below the printed matter. Note that in the year a€?88 in the month of December arrived in Lisbon Bartholomew Diaz, Captain of three caravels which the Most Serene King of Portugal had sent to try out the land in Guinea. Another peculiar feature of the Martellus map is the enormous peninsula commencing due south of Aureus Chersonesus [the Malay peninsula] at 28A° south, thereafter widening to reach China. It has become clear to some cartographic scholars that South America was represented as a huge peninsula of southeastern Asia on many world maps of the 16th century, from the Zorzi sketches of 1506 to the Livio Sanuto map of 1574.

It is not so well known that this very same peninsula existed already under the name of India Meridionalis on earlier maps, drawn before the arrival in the western hemisphere of Christopher Columbus. However, the Martellus maps show a very good representation of the South American hydrographic system, including all the great rivers in the sub-continent. A deeper study of the same maps has made possible the identification of several capes on the Atlantic coast, the swamps of the Rio Negro in Brazil, and Lake Titicaca.

There was thus no known pre-Columbian historical exploration of South America by the European nations.

The earlier maps extant include the so-called mappaemundi drawn by medieval churchmen in Western Christendom. In order to detect this peninsula on pre-Martellus maps, we needed an identifying criterion. The following is a theory expressed by Paul Gallez regarding the depiction of South America prior to Columbusa€™ voyages.

In the southernmost part of South America (?), there are the words, next to a strait: Hic sunt gigantes pugnantes cum draconious [Here live some giants who fight against the dragons]. Traditionally, the so-called Legend of the Patagonian Giants is attributed to Antonio Pigafetta, the chronicler of Magellana€™s voyage.

The east-west extent from the Canaries to the coast of China is about 235A°, which agrees with the Toscanelli-Columbus concept and with the 1489 Martellus map now in the British Library. The sheets of paper on which the Yale map is drawn are of different sizes, which excluded the possibility that they were printed map sheets, for they would then have had to be the same size to fit within the map portfolio. Roberto Almagia (1940) stated that he had identified no less than three manuscript maps of the world signed by Martellus, all virtually identical with that in the codex in the British Library. When Columbus left Lisbon in 1485 for Spain, Bartholomew, with his highly trained skills as a cartographer in the Genoese style, stayed on in the map workshop of King John II. These two world maps by Martellus represent, along with Martin Behaima€™s famous globe of 1492, the last view of the old pre-Columbian world as perceived by Western Europe before the great expansion of the world picture during the subsequent twenty years. The Yale example shows more of the Ocean Sea in the Far East than does the British Library manuscript. Very little is known about Henricus Martellus Germanus, a mapmaker working in Florence from 1480 to 1496.

The Indian Ocean is open to the south, but Martellus retains its eastern coast, which Ptolemy had called Sinae, home of the fish-eating Ethiopians. Martellus retains other Ptolemaic features, such as the appearance of the northern coast of the Indian Ocean with a flattened India and a huge Taprobana. Although Martellusa€™s world maps in the Ptolemy manuscripts show an extension of the inhabited world as 180A°, those in the Insularium make it 220A° or more, an attractive feature to those who dreamed of reaching Asia by sailing west. Most of the text still isna€™t legible, but some of the parts that are appear to be drawn from the travels of Marco Polo through east Asia. The work of Henricus Martellus Germanus epitomizes the best of European cartography at the end of the 15th century. Wolff, Hans, a€?The Conception of the World on the Eve of the Discovery of America - Introduction,a€? in Wolff, Hans (ed.), America. AlmagiA , Roberto, a€?I mappamondi di Enrico Martello e alcuni concetti geografici di Cristoforo Colombo,a€? La Bibliofolia, Firenze, XLII (1940), pp.

AlmagiA , Roberto, a€?On the cartographic work of Francesco Roselli,a€? Imago Mundi, VIII (1951), pp. Caraci, Ilaria L., a€?La€™opera cartografica di Enrico Martello e la a€?prescopertoa€™ della€™America,a€? Rivista Geografica Italiana 83 (1976), Florence, pp.

Caraci, Ilaria L., a€?Il planisfero di Enrico Martello della Yale University Library e i fratelli Colombo,a€? Rivista Geografica Italiana 85 (1978), Florence, pp.

Crino, Sebastiano, a€?I Planisferi di Francesco Roselli,a€? LaBibliofilia, Firenze, XLI (1939), pp. Francesco Rosselli was one of the earliest known map stockists and map sellers; in addition he was an important map-maker whose cartographic output spans the decades of the great discoveries.

The mapa€™s most prominent feature is, like the Martellus model, the new outline of Africa, now quite separate from Asia, and reflecting the rounding of the Cape by Bartolomeu Diaz in 1487. First, by 1486 the mathe-matical junta had solved the problem of establishing latitude by measuring the height of the mid-day sun. The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter. If we include the central 90%, we leave out a total of ? = 10% in both tails, or 5% in each tail, of the normal distribution. We need to find the value of z that puts an area equal to the confidence level (in decimal form) in the middle of the standard normal distribution Z ~ N(0, 1). This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the standard normal distribution. For example, if we constructed 100 of these confidence intervals, we would expect 90 of them to contain the true population mean exam score.

A random sample of 28 pizza delivery restaurants is taken and has a sample mean delivery time of 36 minutes. Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. These intervals are different for several reasons: they were calculated from different samples, the samples were different sizes, and the intervals were calculated for different levels of confidence. The population standard deviation is six minutes and the sample mean deliver time is 36 minutes.

However, sometimes when we read statistical studies, the study may state the confidence interval only.

We may know that the sample mean is 68, or perhaps our source only gave the confidence interval and did not tell us the value of the sample mean. A researcher planning a study who wants a specified confidence level and error bound can use this formula to calculate the size of the sample needed for the study.

If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed? If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed? When estimating a population mean, the margin of error is called the error bound for a population mean (EBM).

The confidence level is the percent of all possible samples that can be expected to include the true population parameter. To find the error bound, find the difference of the upper bound of the interval and the mean. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. If the only change made in the analysis is a change in confidence level, then all we are doing is changing how much area is being calculated for the normal distribution.

Recall, when all factors remain unchanged, an increase in sample size decreases variability. The firm needs to determine what the confidence level should be, then apply the error bound formula to determine the necessary sample size. Since we increase the confidence level, we need to increase either our error bound or the sample size.

During the 2012 campaign season, there were 1,619 candidates for the House of Representatives across the United States who received contributions from individuals. The most recent survey estimates with 90% confidence that the mean household income in the U.S. You want to estimate the mean height of students at your college or university to within one inch with 93% confidence. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just study for that next big test).

However, even though you report normality, you could still adjust for your smaller sample size by using a different distribution. Unfortunately, using pivotal quantities as a way to construct intervals is usually not taught at this level.

Little is known of this important German cartographer, probably from Nuremberg, who worked in Italy from 1480 to 1496 and produced a number of important manuscript maps. No meridians, parallels or scales of longitude are given, but estimates based on measurements of the map indicate about 230A° from Lisbon to the coast of China, or 240A° from the Canaries. The actual latitude of the Cape of Good Hope is 34A° 22a€™ south based on land measurements by Diaz, who landed three times on the south coast.

He reported to the same Most Serene King that he had sailed beyond Yan 600 leagues, namely 450 to the south and 250 to the north, up to a promontory which he called Cabo de Boa Esperanza [Cape of Good Hope] which we believe to be in Abyssinia. It indicates that he was high in the confidence of the King as an expert cartographer, otherwise he would not have been present at such a secret occasion. It was done probably in Seville after he joined his brother, entering the postil in Imago Mundi and altering his prototype map at the same time. It seems that the prototype originally showed the Cape at 35A° south, well clear of the frame at about 41A° south, as one would expect of a competent cartographer. It is a relic of the continuous coastline that linked Southeast Asia to South Africa in Ptolemaic world maps that displayed a land-locked Indian Ocean, and it needs a name to identify it in argument.

This is the India that Columbus was looking for, because it was marked in the right place on his maps.

In the post-Columbian series, the isthmus of Panama is represented with its true width, because it had been heard of by Columbus and other explorers from the aborigines; in the pre-Columbian series, the union of the peninsula with Asia is much broader, because nobody had exact information about it.

The pre-Magellanic maps have South America extending only to some degrees South; on post-Magellanic maps the land extends to 53 degrees South.

On these pre-Columbian maps, the drainage net is much better drawn than on any other representation made before 1850. So Gallez believes that the deep and sound European knowledge of South America before its exploration by Columbus and his Spanish and Portuguese challengers has been firmly established.

But the detail of its hydrographic features mapped by Martellus in 1489 is a fact, even if this fact remains historically unexplained.

In this way we have identified the Dragona€™s Tail on three maps drawn between 1440 and 1470. As cultured persons, both Pigafetta and Magellan would have seen such maps as Walspergera€™s or others of the same family, and they would surely have taken aboard some copies of them. Long in the possession of a family from Lucca, it went to Austria in the 19th century and was bought for Yale in 1961.

Vietor, then Map Curator of Yale University, reported a gift by an anonymous donor a€?in the form of a magnificent painted world map signed by Henricus Martellus, approximately six feet by four feet (180 x 120 cm). In a private communication of June 1972, Vietor stated that X-ray examination had revealed no evidence of printing on the paper sheets and that it was hand-drawn, lettered and colored.

It was not foretold on Folio 1 and is clearly an unexpected addition to the codex of picture-islands. He was engaged in building up a large map of the world based on Donus Nicolaus and on Portuguese charts.

On his globe, South Africa, between 23A° and 28A° south, extends a horn of land for hundreds of miles due eastward into the Indian Ocean. Among the thousands of islands Marco Polo reported off the coast of Asia, an enormous Sumatra and Java are found in the south, while to the northeast is the huge island of Cipangu [Japan]. The map is remarkable for its exciting new information, although being imperfect because of its acceptance of classical and medieval antecedents.

On the world map in his Insularium at the British Library, he shows the results of Bartholomew Diasa€™ rounding of the Cape of Good Hope in 1488, listing the names of the various ports and landmarks along the way. This was the edge of Ptolemya€™s map - he did not show the full extent of Asia, noting that a€?unknown landsa€? lay beyond. In the north, Greenland is a long, skinny peninsula attached to Europe and north of Scandinavia, a concept derived from the Claudius Clavus map of 1427.

One of the Insularium manuscripts at the Laurentian Library in Florence seems to be a working copy, as it has many cross-outs and corrections on the maps and in the texts. His intelligent effort to reconcile modern discoveries with traditional knowledge, especially Ptolemy, provides a workable model for the world map in a time of rapid change. He worked in Florence prior to 1480, then was away from Italy for about two years before returning to his home town where he was active until his death some time after 1513. The cartography of such maps is very poor: for instance, on the maps of Hieronymo Girava 1556, Johann Honter 1561, Giacomo Gastaldi 1562 and Francesco Basso 1571, the Rio Amazonas has its source in Patagonia and flows from south to north.

However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).

What happens to the error bound and the confidence interval if we increase the sample size and use n = 100 instead of n = 36? If we know the confidence interval, we can work backwards to find both the error bound and the sample mean. If you do not know the sample mean, you can find the error bound by calculating half the difference of the upper and lower bounds. With this installment from Internet pedagogical superstar Salman Khan's series of free math tutorials, you'll learn how to calculate a confidence interval in basic statistics.

Ptolemya€™s geographical writings, largely unknown to Western Europeans during the earlier Christian Middle Ages in Europe, became the basis for the Renaissance in geography. His entire hopes of gaining support from King John in 1485 for such an enterprise as sailing westward to Cathay rested on his argument that it lay only 130A° to 140A° to the west. A manuscript copy of Polo and his travels was given by the Doge of Venice to Prince Pedro in 1427 and was thereafter in the Kinga€™s Treasury in Lisbon. Measurements of altitudinal height of the sun by astrolabe or quadrant were accurate on land but could be 2A° or more out on the rolling deck of a ship.

23 is in the handwriting of Bartholomew, and was identified as his by Bishop Bartolomeo de Las Casas, who knew him well. He says that in this place he found by the astrolabe that he was 45A° below the equator and that this place is 3,100 leagues distant from Lisbon [19,850 km].

It was to influence the Catholic Sovereigns who were in the dark owing to the intense secrecy by Portugal regarding discovery. There is no record extant of anyone reporting that they had seen any land or island south of the equator, nor did anybody pretend to have explored the inner part of a trans-Atlantic continent and to have mapped its rivers.

We may thus believe that this knowledge already existed before Martellus, and we should look at older maps in search of the sources that he could have had at his disposal.

They thus knew that, following their maps, they would have to sail to the south along the coast of the Dragona€™s Tail until they reached the Land of Giants, and that at the end of that land, they would find a passage to the West, to the Sinus Magnus, and thus a way to the Moluccas. It shows a great deal more ocean to the east of Asia with several large islands in it, including Japan in the far northeast.

Signora Carla Marzoli of Milan, in a private communication, stated that this large map a€?had left Italy into the possession of family centuries ago and had been lodged in a Swiss bank for safety, for a long time.a€™ In 1959, through trade channels, she learned that this map was for sale. It uses the homeopathic (heart-shaped) projection, as far as can be judged, for it lacks meridians and parallels and scales. They all omitted Cipangu but he found a separate map of Cipangu in a codex which had the same outlines as those in the Behaim globe of 1492. It was, like all important maps at that time, drawn on sheets of parchment which could be joined together almost invisibly, and mounted on linen. In the Indian Ocean, the islands of Madagascar and Zanzibar, rather poorly drawn, add an intriguing aspect. It was the most accurate delineation available to Martin Behaim when he constructed his globe exhibiting the pre-Columbian world. On the Martellus map there is a long peninsula to the east of the Golden Chersonese (Indochina), featuring the mysterious port of Cattigara.

The Mediterranean and Black Seas and the Atlantic coasts are taken from sea charts, while the east coast of Africa, as yet unexplored, also follows Ptolemya€™s design. Here the habitable world is extended to 265A°, with northern Asia coming right up against the right-hand border. According to historian Chet Van Duzer who is participating in a new (2014) digital image scanning of the Yale Martellus, a€?In northern Asia, Martellus talks about this race of wild people who dona€™t have any wine or grain but live off the flesh of deer and ride deer-like horsesa€?.

The role of experience was clearly important to Martellus, for in his copy of the Laurentian Insularium he includes a verse about his own travels, noting that he has been traveling around for many years, and adding that it is worthwhile but difficult to set a white sail upon the stormy sea.

It is a cleanly-engraved copperplate, with finely-stippled sea and six characteristic loose-haired wind-heads grouped round the border of the map.

A legend at the bottom referring to the date 1498 is clearly an error for 1489, and has been copied as such from the 1489 Martellus manuscript world map with obviously similar geographic content to that of Rosselli. These maps depicted graphically the theory that Cipangu [Japan] was but 3,500 miles (5,635 kilometers) westward, and only 1,500 miles (2,415 kilometers) further lay the shores of Cathay [China]. 23 is in the handwriting of Bartholomew, and was identified as his by Bishop Bartolomeo de Las Casas, who knew him well.A Arthur Davies has made a long study of the writing of the Columbus brothers and states, without a shadow of doubt that it is in the hand of Bartholomew.

Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.

To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than 1.6 watts per kilogram. To find the sample mean given a confidence interval, find the difference of the upper bound and the error bound. Construct a 95% confidence interval estimate for the population mean number of eggs laid per chicken per month. The Martellus delineation included some Ptolemaic dogma in its continental contours and projection but significantly modified and improved upon the ancient model with regards to its contents. The coast of Cathay is approximately 130A° West of Lisbon on the Martellus map, on the Behaim globe and in the Columbus-Toscanelli concept. Copious marginal notes, in the handwriting of Bartholomew and his older brother Christopher, are found in the Admirala€™s copy of the printed book of Marco Polo published at Louvain between 1485 and 1490. The coasts charted by Diaz have been fitted into the circular outline of the world of the Fra Mauro hemisphere, representing such a marked trend to the southeast that the Cape of Good Hope seems to be due south of the Persian Gulf, whereas it is due south of the Adriatic. Yet the Martellus map shows South Africa extending across the frame of the map to 45A° south. He has described this voyage and plotted it league by league on a marine chart in order to place it under the eyes of the Most Serene King himself. This dates the legend as 1489, probably in January of that year, just before Bartholomew went to Seville.

Arthur Davies, in his discussion of this map refers to it as the Tiger-leg peninsula; in others it is referred to as Catigara. The best preserved copy is in the British Library and there is also a poorer copy in the University of Leiden.

Considering that in 1489 Martellus knew about the inner courses of many South American rivers, we have no reason to doubt that, forty years earlier, Andreas Walsperger (#245) knew of the Patagonians. The meeting with the Tewelche in Saint Julian was the full confirmation, for Magellan and Pigafetta, of what they already knew from their maps. The Laon globe (#259), made in France about 1493 (probably by Bartholomew Columbus when he served Anne, Regent of France) has the Tiger-leg to 40A° south.

She examined it and saw its connection with the Martellus map in the British Library and with the conceptions of Columbus. Fortunately, Martellus copied most of the nomenclature and legends on to the smaller map, now in the British Library, so the a€?Yalea€™ Martellus can recover them without difficulty. Next come three regional maps, striking because of the enormous nomenclature on the coasts. The three manuscript maps were larger in scale and showed more detail than the codex world map.

This large map, 180 cm by 120 cm, formed a standard Portuguese world map, continually added to by new discoveries, including those of CA?o and Diaz. Some historians claim that their presence indicates the map cannot be dated before 1498, when Vasco da Gama returned with news of his voyage to India. Modern Africa is so long, however, that it breaks through the frame at the lower edge of the map. Unlike Ptolemya€™s Asia, Martellusa€™ version has an eastern coast with the island-bedecked ocean beyond and additional names taken from Marco Polo. From this we can see that in the Insularia Martellus used modern information when he had it, incorporating it into a classical format. Martellusa€™s depiction of rivers and mountains in the interior of southern Africa, along with place names there, appear to be based on African sources. He suggests that the reader might prefer to stay safely at home, and learn about the world through his book. Like its model, the projection is the barrel-shaped (homeoteric, or modified spherical) a€?second projectiona€™ of Ptolemy. Columbus thus had documentary support for his beliefs about oceanic distances from his readings of earlier cosmographers, Cardinal Da€™Ailly (#238) and Paolo Toscanelli (#252). Its shape, outline and position are identical with the representation of Cipangu on the Behaim globe. To be more confident that the confidence interval actually does contain the true value of the population mean for all statistics exam scores, the confidence interval necessarily needs to be wider.

If the error bound is unknown, then average the upper and lower bounds of the confidence interval to find the sample mean. Kimble demonstrated that, as far as 13A° South, the nomenclature of West African coasts is 80% identical in the two maps but bears no relation to the nomenclature of any other map of the period. The minor differences in location between these three must be seen against the enormous exaggeration of the extent of Eurasia which they exhibit in comparison with all previous estimates.

The only other claim that the Cape was at 45 degrees south is in the hand of Bartholomew Columbus. It suited Columbus admirably and it is likely that Bartholomew made the change at his direction. It is shown to more than 25A° south in the Cantino map of 1502 (Book IV, #306), Canerio map of 1504 (Book IV, #307), the WaldseemA?ller maps of 1507, 1513 (Book IV, #310) and many others. The map is surrounded by a dozen puffing classical wind heads, and for the first time in a non-Ptolemaic map, there is a latitude and longitude scale on the side.

At her request, Roberto Almagia and Raleigh Skelton examined it and pronounced it authentic.

The first is of Western Europe and Morocco, cut off in the east through the center of France; the second starts from this line and extends east as far as Naples and Tunis.

The Martelli were a prominent merchant family in Florence, and Enrico Martello sounds Italian enough.

If the two Ptolemy atlases can be dated as the first and last of his works, it is interesting to see that he reverts to the pure Ptolemaic form for the world in the later edition, with the lengthened Mediterranean and the closed Indian Ocean. Ita€™s likely that this information came from an African delegation that visited the Council of Florence in 1441 and interacted with European geographers.

This provided him with the ammunition to further promote his plan to sail west to reach the Indies.

The Martellus map in the British Library is less than 20 inches (50 cm) from west to east, and is on a scale one-quarter that of the Yale map. South of that limit, however, the Martellus map gives the outlines and nomenclature of the voyage of Bartholomew Diaz in 1487-88, while the Behaim globe has invented nomenclature corresponding to nothing in Portuguese cartography. It is difficult to avoid the conclusion that he copied a map which was originally designed to support the ideas of Columbus. Due south of the Malay peninsula, at 28A° South, there appears an enormous peninsula which widens and turns north to join China again with the largely circular concept of the Fra Mauro map. On the face of it, seeing it on the Martellus map, it asserts that Diaz had proceeded north along the cast coast of South Africa to beyond Natal. Columbus hoped Spain would support a voyage westward to Cipangu, 85A° away, and to Cathay 130A° to the west. Of course, this is mostly symbolic, as neither Henricus Martellus nor anyone else had a complete set of accurate coordinates. The sheets of the a€?Yalea€™ Martellus, together with certain regional maps, were acquired by the patron. The one piece of information he gives us about himself, that he had traveled extensively, suggests a career in business. Again, with the Tigerleg or the Dragona€™s Tail, there has been an controversial attempt to identify it as an early map of South America, turning Ptolemy's a€?Sinus Magnusa€? into the Pacific Ocean. Clearly, Ptolemy maps were considered as illustrations for a revered and ancient text and did not have to include the latest news. Three other surviving maps contain some of this same information, but the Martellus map covers more territory than any of them, making it the most complete surviving representation of Africansa€™ geographic knowledge of their continent in the 15th century. The splendid manuscript of Martellus preserved in the National Library at Florence contains thirteen tabulA¦ modernA¦, but is probably later than the earliest printed editions of Ptolemya€™s Geography.A However, Martellus revised the Ptolemaic world map based on Marco Poloa€™s information on Asia, and he incorporated the recent Portuguese voyages to Africa. Due south of the Malay peninsula, at 28A° South, there appears an enormous peninsula which widens and turns north to join China again with the largely circular concept of the Fra Mauro map.A This peninsula does not exist in fact, and seems to be a repeat of the Donus Nicolaus map of the world in the Ulm Ptolemy, cut away by the circular outline of Fra Mauro. On the face of it, seeing it on the Martellus map, it asserts that Diaz had proceeded north along the cast coast of South Africa to beyond Natal.A His furthest point, in fact, was the Rio de Infante [Great Fish River] on the south coast, at 34A° south. Although the Columbus brothers knew that Marco Polo had returned from China by this sea route, they inserted this great obstruction of Tiger-leg by 1485. The correspondence between the Behaim globe and the 1489 Martellus map consequently ended in 1485, when Diego CA?o had returned from his first voyage after reaching 13A° south (Cape Santa Maria in the Congo, 1482-1484).

Yet that map could not have been completed until early in 1489 for it had complete details of the discoveries of Bartholomew Diaz in his voyage of 1487-88, when he circumnavigated the Cape of Good Hope (capo da€™ esperanza) and reached the Indian Ocean.

At that latitude one degree was thought to be 50 miles (80 km), according to the Toscanelli letter, so that Japan was only 4,250 miles (7,200 km) west. No wonder that, having regard to his maps, he concluded that this river flowed from Paradise. Martellus assembled the paper sheets, stuck them on a canvas backing, made his characteristic rectangular picture frame for it and colored the seas in dark blue. His surviving work includes two editions of Ptolemya€™s Geography, a large wall map now at Yale University, and five editions of the Insularium Illustratum of Cristoforo Buondelmonti. The correspondence between the Behaim globe and the 1489 Martellus map consequently ended in 1485, when Diego CA?o had returned from his first voyage after reaching 13A° south (Cape Santa MariaA in the Congo, 1482-1484).

It would show King John that even if the Portuguese reached India they could not reach the Spice Islands (which were on the equator east of Tiger-leg) without having to make long voyages into the southern stormy seas.

To have included Cipangu would have required a scale reduced one-sixth, too small to permit of legible nomenclature.

This was the time when Martin Behaim, as a member of the King Johna€™s mathematical junta, was able to study the map and proposals of Columbus.

He returned from this voyage in December 1488 and, within a year, full details, including rich nomenclature, had appeared on the map of Martellus made in Italy; this despite the utmost secrecy on the part of King John of Portugal. The fact that the map had been put away in a Swiss bank for perhaps 60 years explains why it went unobserved.

Martellus had been required by his patron to include in the codex a world map and regional maps which had just come into his possession in 1489. The historian Roberto Almagia has pronounced that Henricus Martellus was an excellent draftsman, who drew upon the latest information and improved the maps he adapted for his collections. This was the time when Martin Behaim, as a member of the King Johna€™s mathematical junta, was able to study the map and proposals of Columbus.A On the 1489 Martellus map there is an inscription next to the Congo that mentions the commemorative stone (PadrA?o) that CA?o erected at Cape Negro during his second voyage (1485-87) when he reached as far as Cape Cross. For King John and for the Catholic Sovereigns it showed that Spain could easily reach Cathay and then the Spice Islands, secure from all interference from Arabs or Portuguese in the Indian Ocean.A Columbus may have deceived himself regarding the existence of the Tiger-leg but, since it suited his plans so admirably, one suspects that he asserted its existence just as he later made the Cape of Good Hope to be at 45A° south. The second major difference is that, while the coasts from Normandy to Sierra Leone, on the Yale map, are based on the world map of Donus Nicolaus of 1482, the corresponding coasts on the British Library Martellus uses a Portuguese map for them. The furthest point reached by Diaz, the Rio de Infante [the great Fish River], is duly recorded as ilha de fonti. His great talent for such an exacting task was his skill as a draughtsman-cartographer experienced in altering the scale of maps of islands that he was copying. Someone with access to it and to the reports of Bartholomew Diaz, drew the prototype of the Martellus map. According to the historian Arthur Davies, this is conclusive evidence that the prototype originally terminated at 35A°, with the legend correctly placed near it. Even today, the best source for information on the voyage of Diaz is the Martellus map of 1489. The extra ten degrees in shifting the Cape to 45A° south meant more than twenty degrees extra distance in a voyage to India.

Two peculiar features in the region of South Africa suggest that Bartholomew Columbus was that someone. When Bartholomew altered the prototype map to 45A° south, he was unable to remove the legend. The policy of secrecy of King John was shattered in one great leakage by someone in a unique position to know all the details.

Moreover, and perhaps this was the decisive factor, it would take Portuguese ships to nearly 50A° south to round Africa, into what Diaz had already found to be the roughest seas encountered anywhere in the world. Moreover, and perhaps this was the decisive factor, it would take Portuguese ships to nearly 50A° south to round Africa, into what Diaz had already found to be the roughest seas encountered anywhere in the world.A Bartholomew, in 1512, gave evidence in the Pleitos (the great lawsuit of the Columbus family versus the Crown of Spain) and declared that he had gone about with his brother in Spain helping to gain support for his enterprise. Each island-picture is surrounded by a frame, drawn and painted in ivory to give the impression of a carved wood frame, suggesting a picture hanging on a wall. Such picture-islands occupy nearly all the folios, making an Atlas of Islands, and are then followed by a superb picture-map of Italy, taken from the 1482 world map of Donus Nicolaus. The islands in this codex were copies from the Ulm Ptolemy, from the Isolario of Bartolomeo de li Sonetti printed in Venice in 1485, and from other sources. This account of Gallo was copied, almost verbatim by Serenega in 1499 and by Giustiniani in 1516. Bartholomew left Genoa as a youth in 1479 and officially made only one further visit to Italy, in 1506. Gallo could have acquired knowledge of the role of Bartholomew only from his own lips, in 1489.

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