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16.03.2015
Confidence Interval Calculator is an online Probability and Statistics tool for data analysis programmed to calculate the statistical accuracy of a survey-based estimate. DefinitionIn statistics it is important to measure how certain we can be in the results of an analysis or experiment. When quoting confidence It is common to refer to the 95% confidence interval around an experiment assessment or test result, even though other confidence intervals can be reported (e.g.
The collection of tools employs the study of methods and procedures used for gathering, organizing, and analyzing data to understand theory of Probability and Statistics. In Part 4 of this series on Monte Carlo Simulation, we plotted the results of our Sales Forecast Example as a histogram to visualize the uncertainty in profit.
Download the Sales Forecast ExampleFigure 1: Summary statistics for the sales forecast example. The sample mean and median statistics describe the central tendancy or "location" of the distribution. If you sort the results from lowest to highest, the median is the "middle" value or the 50th Percentile, meaning that 50% of the results from the simulation are less than the median. Extreme values can have a large impact on the mean, but the median only depends upon the middle point(s). The range is also a helpful statistic, and it is simply the maximum value minus the minimum value. Extreme values have a large effect on the range, so another measure of spread is something called the Interquartile Range. If you repeated the Monte Carlo simulation and recorded the sample mean each time, the distribution of the sample mean would end up following a Normal distribution (based upon the Central Limit Theorem). The standard error can be used to calculate confidence intervals for the true population mean.
Popescu et al., 2003, "Monte Carlo Simulation using Excel for Predicting Reliability of a Geothermal Plant", International Geothermal Conference, Reykjavik, Sept. Excel is the widely used statistical package, which serves as a tool to understand statistical concepts and computation to check your hand-worked calculation in solving your homework problems. This site provides illustrative experience in the use of Excel for data summary, presentation, and for other basic statistical analysis. Microsoft Excel 2000 (version 9) provides a set of data analysis tools called the Analysis ToolPak which you can use to save steps when you develop complex statistical analyses.
If the Data Analysis command is selectable on the Tools menu, then the Analysis ToolPak is installed on your system.
Microsoft Excel is a powerful spreadsheet package available for Microsoft Windows and the Apple Macintosh. To save your document with its current filename, location and file format either click on File - Save. When you start Excel, a blank worksheet is displayed which consists of a multiple grid of cells with numbered rows down the page and alphabetically-titled columns across the page.
It is important to be able to move around the worksheet effectively because you can only enter or change data at the position of the cursor. Note that the size of a scroll box indicates the proportional amount of the used area of the sheet that is visible in the window. To select several cells which are not adjacent, press "control" and click on the cells you want to select. One worksheet can have up to 256 columns and 65,536 rows, so it'll be a while before you run out of space.
Note that as you type information into the cell, the information you enter also displays in the formula bar.
If you are creating a long worksheet and you will be repeating the same label information in many different cells, you can use the AutoComplete function.
Numbers are assumed to be positive; to enter a negative number, use a minus sign "-" or enclose the number in parentheses "()". An entry interpreted as a value (number, date, or time) is aligned to the right side of the cell, to reformat a value. Select a cell in the region, and press Ctrl+Shift+* (in Excel 2003, press this or Ctrl+A) to select the Current Region. Repeat step 4, select a different color than you selected for Condition 1, and then click OK.
Note: Be sure to distinguish between absolute reference and relative reference when entering the formulas. Problem: Rounding all the numbers in column A to zero decimal places, except for those that have "5" in the first decimal place. Copying the entire sheet means copying the cells, the page setup parameters, and the defined range Names. In the Sort by option of the Sort dialog box, select the row number by which the columns will be sorted and click OK.
The Data Analysis ToolPak has a Descriptive Statistics tool that provides you with an easy way to calculate summary statistics for a set of sample data.
Enter A1:A4 in the input range box, A1 is a value in column A and row 1, in this case this value is 2. Consider the problem of finding the probability of getting less than a certain value under any normal probability distribution. A: What is the probability that a randomly selected student score will be less than 600 points?
C: What is the probability that a randomly selected student score will be between 400 and 600? Hint: Using Excel you can find the probability of getting a value approximately less than or equal to a given value. As you see the value 0.84134474 appears in A1, indicating the probability that a randomly selected student's score is below 600 points.
You can use NORMINV from the function box to calculate a value for the random variable - if the probability to the left side of this variable is given. At the end of this screen you will see the formula result which is approximately 628 points.
Now we would like to show how Excel is used to develop a certain confidence interval of a population mean based on a sample information. After entering the data, we followed the descriptive statistic procedure to calculate the unknown quantities.
On the other hand, we can say that of all the intervals formed this way 95% contains the mean of the population.
If the sample n is less than 30 or we must use the small sample procedure to develop a confidence interval for the mean of a population. Now you would like to see how Excel is used to develop a certain confidence interval of a population mean based on this small sample information. Again, the only things you have to do are: add the margin of error to the mean of the sample,, find the upper limit of the interval and to subtract the margin of error from the mean to find the lower limit of the interval.
To demonstrate how Excel finds these quantities we will use the data set, which contains the hourly incomes of 10 work-study students here, at the University of Baltimore. After entering the data we follow the descriptive statistic procedure to calculate the unknown quantities (exactly the way we found quantities for large sample). Now, like the calculation of the confidence interval for the large sample, calculate the confidence interval of the population based on this small sample information. In this section you wish to know how Excel can be used to conduct a hypothesis test about a population mean. The null hypothesis indicates that the average hourly income of a work-study student is equal to $7 per hour; however, the alternative hypothesis indicates that the average hourly income is not equal to $7 per hour.
I will repeat the steps taken in descriptive statistics and at the very end will show how to find the value of the test statistics in this case, z, using a cell formula. Using steps taken the large sample size case, Excel can be used to conduct a hypothesis for small-sample case. The null hypothesis indicates that average hourly income of a work-study student is equal to $7 per hour .The alternative hypothesis indicates that average hourly income is not equal to $7 per hour. In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means assuming that populations have equal variances. In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means.
We can also draw this conclusion by comparing the p-value for a two-tail test and the alpha value. In this section the objective is to see whether or not means of three or more populations based on random samples taken from populations are equal or not. In this case we were interested to see whether there a significant difference among hourly wages of student assistants in three different service departments here at the University of Baltimore. In this section, the study involves six students who were offered different hourly wages in three different department services here at the University of Baltimore. The CHI-SQUARE distribution can be used in a hypothesis test involving a population variance. In this example the objective is to see whether or not based on a randomly selected sample information the standards set for a population is met.


As you see the observed sample numbers for groups two and three are lower than their expected values unlike group one which has a higher expected value.
Step 1.Select a cell in the work sheet, the location which you like the p value of the CHI-SQUARE to appear.
Step 3.Refer to function category box and choose statistical, from function name box select CHITEST and click on OK. The CHI-SQUARE distribution is also used to test and see whether two variables are independent or not. In this section we would like to examine whether or not the variances of two populations are equal. Now let see whether or not the variances of hourly income of student-assistant and work-study students based on samples taken from populations previously are equal. In this section the objective is to see whether there is a correlation between two variables and to find a model that predicts one variable in terms of the other variable.
Now lets use Excel to find the linear correlation coefficient and the regression line equation. As you see the correlation between the number of cases of soda demanded and the temperature is a very strong positive correlation.
Moving Average Models: Use the Add Trendline option to analyze a moving average forecasting model in Excel. Click on the chart to select it, and click on any point on the line to select the data series. Exponential Smoothing Models: The simplest way to analyze a timer series using an Exponential Smoothing model in Excel is to use the data analysis tool. Select "Descriptive Statistics" by pointing at it and clicking twice, or by highlighting it and clicking on the "Okay" button. Variance: The variance of a set of data is the average of the cumulative measure of the squares of the difference of all the data values from the mean. To construct the probability function for bank robberies, first define the random variable x, bank robbery take. Normal Example: If the time required to complete an examination by those with a certain learning disability is believed to be distributed normally, with mean of 65 minutes and a standard deviation of 15 minutes, then when can the exam be terminated so that 99 percent of those with the disability can finish? Solution: Because the average and standard deviation are known, what needs to be established is the amount of time, above the mean time, such that 99 percent of the distribution is lower. Alternatively, instead of standardizing with the Z distribution using Excel we can simply work directly with the normal distribution with a mean of 65 and standard deviation of 15 and enter "=NORMINV(0.99,65,15)".
Another Example: In the early 1980s, the Toro Company of Minneapolis, Minnesota, advertised that it would refund the purchase price of a snow blower if the following winter's snowfall was less than 21 percent of the local average. Solution: Within limits, snowfall is a continuous random variable that can be expected to vary symmetrically around its mean, with values closer to the mean occurring most often. If a random sample of size n = 1 (day) is selected, what is the probability this x value will exceed 28? If a random sample of size n = 4 (days) is selected, what is theprobability that xbar ³ 28? From this we can see that five countries with the same speed limit have very different positions on the safety list.
Solution: Enter the ten paired y and x data into cells A2 to A11 and B2 to B11, with the "death" rate label in A1 and "speed" limits label in B1, the following steps produce the regression output. Because the Y and X ranges include the "Death" and "Speed" labels in A1 and B1, select the "Labels" box with a click. Click the "Output Range" button and type reference cell, which in this demonstration is A13. To get the predicted values of Y (Death rates) and residuals select the "Residuals" box with a click.
Under the "ANOVA" section are the estimated regression coefficients and related statistics that will be discussed in detail in the next chapter. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Not the answer you're looking for?Browse other questions tagged confidence-interval or ask your own question. How to add standard error or confidence interval to a plot of a predicted vs observed values?
In a formal paper, should I censor "brainf**k", the name of a programming language?
Is there a more effective way to build the vocabulary of a fictional language than a word frequency list?
Can Bootstrap Resampling be used to Calculate a Confidence Interval for the Variance of a Data Set? I know that if you re-sample from a data set many times and calculate the mean each time, these means will follow a normal distribution (by the CLT).
I know that if that original data set is normal, then the variances would follow a chi-squared distribution.
It is not always the case that if you bootstrap a mean the bootstrap means will follow a normal distribution, even for distributions for which the CLT applies.
The problem is that in this case the sample size (100) is too small for the CLT to apply with this sort of distribution shape; it doesn't matter how many times we resample it. However, if the original sample size is much larger, the resampling distribution of sample means for something like this will be more normal-looking (though always discrete). As we see, the distribution function when resampling the large sample does look much more normal. However, the CLT does also apply to the variance*; it's just that you can't argue that the CLT applies to bootstrap resampling simply by taking many resamples.
Not the answer you're looking for?Browse other questions tagged distributions confidence-interval bootstrap resampling or ask your own question. Restricting minimum subgroup size in a bootstrap resampling study - why is this approach wrong?
New employer wants me to pay for ISO 9001 accreditation before employment begins - is this a scam? What is the smallest number of people in a group, so that it is guaranteed that three of them will have their birthday in the same month? One way of computing the measure of confidence in statistical results is to review the confidence interval reported by researchers. The set of ideas which is intended to offer the way for making scientific implication from such resulting summarized data. Next, to provide a concise summary of the results, it is customary to report the mean, median, standard deviation, standard error, and a few other summary statistics to describe the results with numbers.
In relation to physics, if the probability distribution represented mass, then the mean would be the balancing point, or the center of mass. If there is an even number of data points, then the median is the average of the middle two points. This property makes the median useful for describing the center of skewed distributions such as the Lognormal distribution. The standard error is a good estimate of the standard deviation of this distribution (the distribution of multiple mean values), assuming that the sample is sufficiently large (n >= 30).
It can be opened either by selecting Start - Programs - Microsoft Excel or by clicking on the Excel Short Cut which is either on your desktop, or on any PC, or on the Office Tool bar.
You can move the cursor by using the arrow keys or by moving the mouse to the required cell and clicking. The position of a scroll box indicates the relative location of the visible area within the worksheet. You can also enter information into the formula bar, and the information will appear in the selected cell. This function will look at other entries in the same column and attempt to match a previous entry with your current entry. Right click on a cell, then select "Pick From List." This will give you a menu of all other entries in cells in that column. The Move or Copy dialog box enables one to copy the sheet either to a different location in the current workbook or to a different workbook. Select the Tools *pull-down menu, if you see data analysis, click on this option, otherwise, click on add-in.. In a problem, when the mean and the standard deviation of the population are given, you have to use common sense to find different probabilities based on the question since you know the area under a normal curve is 1. After clicking on the Function option, the Paste Function dialog appears from Function Category.
We will use the hourly incomes of different work-study students than those introduced earlier in the confidence interval section. Let's use the hourly income of 10 work-study students at UB to conduct the following hypothesis. The Excel procedure for testing the difference between the two population means will require information on the variances of the two populations. Assuming independents samples are taken from normally distributed populations with equal variances, Excel would do this analysis if you choose one way anova from the menus.


This means there is a significant difference between means of hourly incomes of student assistants in these departments.
However, in this section we would like to test and see how close a sample results are to the expected results. From the menus, select insert then click on the Function option, Paste Function dialog box appears. For example based on sample data you might want to see whether smoking and gender are independent events for a certain population. When Paste Function dialog box appears, click on Statistical in function category and CHITEST in the function name then click OK. To find the chi-square value, use CHINV function, when Chinv box appears enter 0.477395 for probability part, then 1 for the degrees of freedom. There are so many examples that we could mention but we will mention the popular ones in the world of business. Referring to this expression we can approximately predict the number of cases of soda needed on June 14. The sample variance is simply the arithmetic mean of the squares of the difference between each data value in the sample and the mean of the sample. Assuming 60 percent of those caught lose their entire take and 40 percent lose half, graph the probability mass function using EXCEL.
This probability tree diagram shows the "branches" that must be followed to show the calculations captured in the binomial mass function for n = 4 and = 0.2. In general to obtain the x value for which alpha percent of a normal random variable's values are lower, the following "NORMINV" command may be used, where the first entry is a, the second is m , and the third is s.
If the average snowfall is 45.25 inches, with a standard deviation of 12.2 inches, what is the likelihood that Toro will have to make refunds?
Using Excel, this normal probability is obtained with the "NORMDIST" command, where the first entry is x, the second is mean m , the third is standard deviation s, and the fourth is CUMULATIVE (1). The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 4. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 2 using Excel, 0.02275 =1-NORMDIST(28,24,2,1).
Thus, you can calculate a confidence interval on the mean of the data set without making any assumptions on the probability distribution of the data set. That is, if I were to re-sample from a data set many times and calculate the variance each time, would these variances follow a certain distribution (regardless of what the original probability distribution of the data set was)?
If the original sample size is sufficiently large, that may (under the right conditions) tend to make the resampling distribution of means (and higher moments, if they exist) relatively close to a normal distribution (relative to its distribution in smaller samples, at least).
Let $y_i = (x_i - \bar x)^2$; then $s_n^2 = \bar y$, so if the CLT applies to the $y$-variable, it can be applied to $s_n^2$.
We can be rationally confident that the range comprises the true value for the population as a whole where a 95% confidence interval is reported.
If the distribution is symmetric (like the Normal distribution), then the mean and median will be identical.
If you have made any changes since the file was last saved, you will be asked if you wish to save them.
Once selected the cell becomes the active cell and is identified by a thick border; only one cell can be active at a time.
Click OK on the descriptive statistics dialog, click on Summary Statistic, click on the confidence interval level and type in 90% or in other problems whichever confidence interval you desire. I collected the hourly income data of 36 randomly selected work-study students and 36 student assistants.
Since the variances of the two populations are unknowns they should be replaced with sample variances.
We can also draw this conclusion by comparing the p-value for a two tail -test and the alpha value. Similar to the above case, the data in this case are taken from various offices here at the University of Baltimore.
We can also choose Anova: two way factor with or without replication option and see whether there is significant difference between means when different factors are involved.
For example it is assumed the guidelines for hiring people with different ethnic background for the US government is set at 70%(WHITE), 20%(African American) and 10%(others), respectively. Since this value, F=1.984615385, falls in acceptance area we fail to reject the null hypothesis.
Usually independent variable is presented by the letter x and the dependent variable is presented by the letter y. On the other hand, the formula for an estimate for the variance in the population is similar to the formula for the sample variance, except that the denominator in the fraction is (n-1) instead of n. If criminals make their decisions strictly on this expected value, then it pays to rob banks.
Next to these coefficient estimates is information on the variability in the distribution of the least-squares estimators from which these specific estimates were drawn: the column titled "Std. Now $s_{n-1}^2$ is just a scaled version of $s_n^2$; if the CLT applies to $s_n^2$ it will apply to $s_{n-1}^2$.
The characteristics of a normal distribution describe the measure of a confidence interval.
The hourly income range for work-study students was $6 - $8 while the hourly income range for student assistants was $6-$9. I collected hourly income data of 11 randomly selected work-study students and 11 randomly selected student assistants.
Another example in this situation could involve the age range of an individual and his or her smoking habit. The following figure shows the graph of an F distribution with 10 degrees of freedom for both the numerator and the denominator.
Hence, the sample results do support the conclusion that student assistants hourly income variance is equal to the work study students hourly income variance. A business man would like to see whether there is a relationship between the number of cases of sold and the temperature in a hot summer day based on information taken from the past. However, you should not worry about this difference if the sample size is large, say over 30. A decision rule based only on an expected value, however, ignores the risks or variability in the returns. Draw a tree diagram showing the different ways in which a test taker could get 0, 1, 2, 3 and 4 correct answers.
A range may be typed in or selected by highlighting the cells with the mouse after clicking on the box.
Error" contains the standard deviations (standard errors) of the intercept and slope distributions; the "t-ratio" and "p" columns give the calculated values of the t statistics and associated p-values.
On the other hand if we were asked to do this problem by hand, we would have had to calculate the x value using the normal distribution formula x = m + zd. The main objective in this hypothesis testing is to see whether there is a significant difference between the means of the two populations. Similar to case one data may appear in a table but unlike the case one this table may contains several columns in addition to rows. He also would like to estimate the number cases of soda which will be sold in a particular hot summer day in a ball game.
In addition, our expected value calculations do not include the cost of jail time, which could be viewed by criminals as substantial. If the dialog box blocks the data, it can be moved on the screen by clicking on the title bar and dragging. The NULL and the ALTERNATIVE hypothesis is that the means are equal and the means are not equal, respectively.
The shape of an F distribution is positively skewed and depends on the degrees of freedom for the numerator and the denominator. He clearly recorded temperatures and number of cases of soda sold on those particular days. To find the value of each cell in the new table we should multiply the sum of the cell column by the sum of the cell row and divide the results by the grand total.
In this situation the variable y (the dependent variable) is the number of cases of soda and the x (independent variable) is the temperature. To calculate the expected frequency for a category, you can multiply the proportion of that category by the sample size (in here 1000).
Now based on the following table test whether or not the smoking habit and gender of the population that the following sample taken from are independent.



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