## Find confidence interval from standard deviation,watch the secret saturdays episode 8 recap,the hundred secret senses author - New On 2016

23.03.2014
Our New BMJ website does not support IE6 please upgrade your browser to the latest version or use alternative browsers suggested below. Previously we have considered how to test the null hypothesis that there is no difference between the mean of a sample and the population mean, and no difference between the means of two samples. With small samples, where more chance variation must be allowed for, these ratios are not entirely accurate because the uncertainty in estimating the standard error has been ignored. The application of the t distribution to the following four types of problem will now be considered. The mean and standard deviation of a sample are calculated and a value is postulated for the mean of the population.

In each case the problem is essentially the same - namely, to establish multiples of standard errors to which probabilities can be attached. A rare congenital disease, Everley's syndrome, generally causes a reduction in concentration of blood sodium. To find the 95% confidence interval above and below the mean we now have to find a multiple of the standard error. Since the size of the sample influences the value of t, the size of the sample is taken into account in relating the value of t to probabilities in the table. Here we apply a modified procedure for finding the standard error of the difference between two means and testing the size of the difference by this standard error (see Chapter 5 for large samples). The addition of bran to the diet has been reported to benefit patients with diverticulosis. The table of the tdistribution Table B (appendix) which gives two sided P values is entered at degrees of freedom. If the standard deviations in the two groups are markedly different, for example if the ratio of the larger to the smaller is greater than two, then one of the assumptions of the ttest (that the two samples come from populations with the same standard deviation) is unlikely to hold. Although this may look very complicated, it can be evaluated very easily on a calculator without having to write down intermediate steps (see below). When the effects of two alternative treatments or experiments are compared, for example in cross over trials, randomised trials in which randomisation is between matched pairs, or matched case control studies (see Chapter 13 ), it is sometimes possible to make comparisons in pairs. Let us use as an example the studies of bran in the treatment of diverticulosis discussed earlier. In calculating t on the paired observations we work with the difference, d, between the members of each pair. This is quite wide, so we cannot really conclude that the two preparations are equivalent, and should look to a larger study. The second case of a paired comparison to consider is when two samples are chosen and each member of sample 1 is paired with one member of sample 2, as in a matched case control study.

It would seem logical that, because the t test assumes Normality, one should test for Normality first. Pairing provides information about an experiment, and the more information that can be provided in the analysis the more sensitive the test.

What is the difference between the mean levels in the two wards, and what is its significance? 7.4 A new treatment for varicose ulcer is compared with a standard treatment on ten matched pairs of patients, where treatment between pairs is decided using random numbers. What are the mean difference in the healing time, the value of t, the number of degrees of freedom, and the probability? Do you create products or organize events for UX professionals or manage a UX team that’s hiring? Our challenge, therefore, is to work out whether we can consider the average we’ve calculated from our sample as representative of our target audience.

That bit of magic is the Central Limit Theorem, which says: If you take a bunch of samples, then calculate the mean of each sample, most of the sample means cluster close to the true population mean. The chance that any individual mean is way off from the true population mean is quite small.

The probability of getting any particular value depends on only these two parameters—the mean and the standard deviation. To do the calculation, the first thing to decide is what we’re prepared to accept as likely. Finally, we plug in the mean, which is 54.0 seconds, and the number of participants, which is 8.

Remember that 95%, which says that about one time in 20 you’re likely to get it wrong?

You can see that, as we reduce the risk, we increase the confidence level and end up with a wider confidence interval—and in this example, also have an increasing level of depression about that launch date.

Have you come across Six Sigma, the quality improvement program that Motorola originated, which is now popular in many manufacturing companies? What to do if you want to get a higher confidence level, but also need to be sure you’ve met your target for the mean?

The confidence interval for the mean helps you to estimate the true population mean and lets you avoid the additional effort that gathering a lot of extra data would require. Once you have worked out what level of risk you are willing to accept, confidence intervals for the mean are easy to calculate.

Inexperienced researchers should not try to take unstructured data they’ve obtained through qualitative methods, and presto-changeo, turn it into quantitative data. This is a great introduction to some statistical approaches that usually confound most people.

This means our small usability test samples of 5-8 are not enough for this sort of quantitative analysis.

It is a very good and simple explanation for people like me who are not very knowledgeable about six sigma.

Based on a 90% confidence interval, which of the following would not be a plausible value of the population mean? Can someone tell me whether one can say a confidence interval is used in quantitative studies? We obtained the difference between the means by subtraction, and then divided this difference by the standard error of the difference. Some modification of the procedure of dividing the difference by its standard error is needed, and the technique to use is the t test. What is the significance of the difference between the means of the two sets of observations? This is thought to provide a useful diagnostic sign as well as a clue to the efficacy of treatment.

Assuming that blood sodium concentration is Normally distributed what is the 95% confidence interval within which the mean of the total population of such cases may be expected to lie? For large samples we used the standard deviation of each sample, computed separately, to calculate the standard error of the difference between the means. Several different bran preparations are available, and a clinician wants to test the efficacy of two of them on patients, since favourable claims have been made for each. By random allocation the clinician selects two groups of patients aged 40-64 with diverticulosis of comparable severity. An approximate test, due to Sattherwaite, and described by Armitage and Berry, (1)which allows for unequal standard deviations, is as follows. It can produce a degree of freedom which is not an integer, and so not available in the tables. Whether treatment A or treatment B is given first or second to each member of the sample should be determined by the use of the table of random numbers Table F (Appendix).

The clinician wonders whether transit time would be shorter if bran is given in the same dosage in three meals during the day (treatment A) or in one meal (treatment B). As the aim is to test the difference, if any, between two types of treatment, the choice of members for each pair is designed to make them as alike as possible.

It is not valid to compare each treatment with each other treatment using t tests because the overall type I error rate will be bigger than the conventional level set for each individual test. What is the 95% confidence interval within which the mean of the population of such cases whose specimens come to the same laboratory may be expected to lie? I wanted to know that recently and turned to one of my favorite books: Measuring the User Experience, by Tom Tullis and Bill Albert.

In fact, the Central Limit Theorem also says that means are normally distributed, as in the bell-curve normal distribution shown in Figure 3.

Any mean from a random sample is likely to be quite close to the true population mean, and a normal distribution models the chance that it might be different from the true population mean. That’s because with more data, you have more chance of the sample being a pretty good match to the whole population and, therefore, of its mean being similar to the true population value. For example, let’s say we wanted the average time it took to complete the New York Marathon across 45,000 runners.

Those users ought to be happy—the time it took them to complete the task is much shorter than the advertised time. But it won’t be meaningful, because there is no Central Limit Theorem or any equivalent for modes. You can compare the confidence interval you calculated with the target you were aiming for. I’m a qual person to the core, but I recognized that quant data can be both useful to the researcher and very powerful as a way of convincing stakeholders. I would like to ask whether the 95% confidence interval incorporates any information on bias and misclassification? I am doing a degree in psychology and I have to do a report based on an experiment and run off results using SPSS.

If the difference is 196 times its standard error, or more, it is likely to occur by chance with a frequency of only 1 in 20, or less.

Its foundations were laid by WS Gosset, writing under the pseudonym "Student" so that it is sometimes known as Student's t test. Little is known about the subject, but the director of a dermatological department in a London teaching hospital is known to be interested in the disease and has seen more cases than anyone else.

Whether it should be regarded clinically as abnormally high is something that needs to be considered separately by the physician in charge of that case. Among the consequences of administering bran that requires testing is the transit time through the alimentary canal. Sample 1 contains 15 patients who are given treatment A, and sample 2 contains 12 patients who are given treatment B. The null hypothesis that there is no difference between the means is therefore somewhat unlikely.

In this way any effect of one treatment on the other, even indirectly through the patient's attitude to treatment, for instance, can be minimised.

A random sample of patients with disease of comparable severity and aged 20-44 is chosen and the two treatments administered on two successive occasions, the order of the treatments also being determined from the table of random numbers.

The null hypothesis is that there is no difference between the mean transit times on these two forms of treatment.

The more alike they are, the more apparent will be any differences due to treatment, because they will not be confused with differences in the results caused by disparities between members of the pair. With a small sample a non-significant result does not mean that the data come from a Normal distribution. A rule of thumb is that if the ratio of the larger to smaller standard deviation is greater than two, then the unequal variance test should be used.

By repeating measures within subjects, each subject acts as its own control, and the between subjects variability is removed.

One doctor is responsible for treatment and a second doctor assesses healing without knowing which treatment each patient had. So I trawled through various statistics books to gain a better understanding of confidence intervals, and this column is the result. Once we’ve seen three or four test participants in a row fail for the same reason, we just want to get on with fixing the problem. Some values of the true population mean would make it very likely that I’d get this sample mean, while other values would make it very unlikely. I didn’t bother calculating the confidence interval for our sample to get a Six-Sigma level of risk, because it would constitute the whole range of our data. If we took a random sample of just 1000 runners, we would get a narrow confidence interval.

But to ensure that a high volume of users can achieve those task times once the system gets rolled out, we’ll have to make sure that the system can cope with that high peak of very fast task times. However, to be able to identify the confidence interval, there are some prerequisites regarding the study design, variables measured, and sample selection that don’t map to many UX projects. The procedure does not differ greatly from the one used for large samples, but is preferable when the number of observations is less than 60, and certainly when they amount to 30 or less.

With small samples these multiples are larger, and the smaller the sample the larger they become. The transit times of food through the gut are measured by a standard technique with marked pellets and the results are recorded, in order of increasing time, in Table 7.1 . Since it is possible for the difference in mean transit times for A-B to be positive or negative, we will employ a two sided test. Many statistical packages now carry out this test as the default, and to get the equal variances I statistic one has to specifically ask for it. Occasionally it is possible to give both treatments simultaneously, as in the treatment of a skin disease by applying a remedy to the skin on opposite sides of the body. On the other hand, with a large sample, a significant result does not mean that we could not use the t test, because the t test is robust to moderate departures from Normality - that is, the P value obtained can be validly interpreted. With a computer one can easily do both the equal and unequal variance t test and see if the answers differ. In general this means that if there is a true difference between the pairs the paired test is more likely to pick it up: it is more powerful. The likely values represent the confidence interval, which is the range of values for the true population mean that could plausibly give me my observed value. That would be 18.6 seconds greater than our task-time target, which is disappointing all around. But if we took the times of the first 1000 runners across the finish line, we’d get something very far indeed from the true population mean.

So our colleagues who are managing system performance are likely to be far more interested in the most frequent value, which is the mode, than in the mean.

Conversely, as the sample becomes larger t becomes smaller and approaches the values given in table A, reaching them for infinitely large samples. The unequal variance t test tends to be less powerful than the usual t test if the variances are in fact the same, since it uses fewer assumptions. For instance, in a test for a drug reducing blood pressure the colour of the patients' eyes would probably be irrelevant, but their resting diastolic blood pressure could well provide a basis for selecting the pairs.

There is something illogical about using one significance test conditional on the results of another significance test.

Would the data from eight test participants be enough to represent the experience of all users? But we wouldn’t be nearly as worried if the true population mean happened to fall at the low end of the range.

In practice the degrees of freedom amount in these circumstances to one less than the number of observations in the sample. However, it should not be used indiscriminantly because, if the standard deviations are different, how can we interpret a nonsignificant difference in means, for example? Another (perhaps related) basis is the prognosis for the disease in patients: in general, patients with a similar prognosis are best paired. Many people would indeed notice if a task that they anticipated taking less than an hour actually took over two hours. Often a better strategy is to try a data transformation, such as taking logarithms as described in Chapter 2. Whatever criteria are chosen, it is essential that the pairs are constructed before the treatment is given, for the pairing must be uninfluenced by knowledge of the effects of treatment.

One can "eyeball" the data and if the distributions are not extremely skewed, and particularly if (for the two sample t test) the numbers of observations are similar in the two groups, then the t test will be valid. This is because only 17 observations plus the total number of observations are needed to specify the sample, the 18th being determined by subtraction.

Transformations that render distributions closer to Normality often also make the standard deviations similar. The main problem is often that outliers will inflate the standard deviations and render the test less sensitive. Also, it is not generally appreciated that if the data originate from a randomised controlled trial, then the process of randomisation will ensure the validity of the I test, irrespective of the original distribution of the data.

In each case the problem is essentially the same - namely, to establish multiples of standard errors to which probabilities can be attached. A rare congenital disease, Everley's syndrome, generally causes a reduction in concentration of blood sodium. To find the 95% confidence interval above and below the mean we now have to find a multiple of the standard error. Since the size of the sample influences the value of t, the size of the sample is taken into account in relating the value of t to probabilities in the table. Here we apply a modified procedure for finding the standard error of the difference between two means and testing the size of the difference by this standard error (see Chapter 5 for large samples). The addition of bran to the diet has been reported to benefit patients with diverticulosis. The table of the tdistribution Table B (appendix) which gives two sided P values is entered at degrees of freedom. If the standard deviations in the two groups are markedly different, for example if the ratio of the larger to the smaller is greater than two, then one of the assumptions of the ttest (that the two samples come from populations with the same standard deviation) is unlikely to hold. Although this may look very complicated, it can be evaluated very easily on a calculator without having to write down intermediate steps (see below). When the effects of two alternative treatments or experiments are compared, for example in cross over trials, randomised trials in which randomisation is between matched pairs, or matched case control studies (see Chapter 13 ), it is sometimes possible to make comparisons in pairs. Let us use as an example the studies of bran in the treatment of diverticulosis discussed earlier. In calculating t on the paired observations we work with the difference, d, between the members of each pair. This is quite wide, so we cannot really conclude that the two preparations are equivalent, and should look to a larger study. The second case of a paired comparison to consider is when two samples are chosen and each member of sample 1 is paired with one member of sample 2, as in a matched case control study.

It would seem logical that, because the t test assumes Normality, one should test for Normality first. Pairing provides information about an experiment, and the more information that can be provided in the analysis the more sensitive the test.

What is the difference between the mean levels in the two wards, and what is its significance? 7.4 A new treatment for varicose ulcer is compared with a standard treatment on ten matched pairs of patients, where treatment between pairs is decided using random numbers. What are the mean difference in the healing time, the value of t, the number of degrees of freedom, and the probability? Do you create products or organize events for UX professionals or manage a UX team that’s hiring? Our challenge, therefore, is to work out whether we can consider the average we’ve calculated from our sample as representative of our target audience.

That bit of magic is the Central Limit Theorem, which says: If you take a bunch of samples, then calculate the mean of each sample, most of the sample means cluster close to the true population mean. The chance that any individual mean is way off from the true population mean is quite small.

The probability of getting any particular value depends on only these two parameters—the mean and the standard deviation. To do the calculation, the first thing to decide is what we’re prepared to accept as likely. Finally, we plug in the mean, which is 54.0 seconds, and the number of participants, which is 8.

Remember that 95%, which says that about one time in 20 you’re likely to get it wrong?

You can see that, as we reduce the risk, we increase the confidence level and end up with a wider confidence interval—and in this example, also have an increasing level of depression about that launch date.

Have you come across Six Sigma, the quality improvement program that Motorola originated, which is now popular in many manufacturing companies? What to do if you want to get a higher confidence level, but also need to be sure you’ve met your target for the mean?

The confidence interval for the mean helps you to estimate the true population mean and lets you avoid the additional effort that gathering a lot of extra data would require. Once you have worked out what level of risk you are willing to accept, confidence intervals for the mean are easy to calculate.

Inexperienced researchers should not try to take unstructured data they’ve obtained through qualitative methods, and presto-changeo, turn it into quantitative data. This is a great introduction to some statistical approaches that usually confound most people.

This means our small usability test samples of 5-8 are not enough for this sort of quantitative analysis.

It is a very good and simple explanation for people like me who are not very knowledgeable about six sigma.

Based on a 90% confidence interval, which of the following would not be a plausible value of the population mean? Can someone tell me whether one can say a confidence interval is used in quantitative studies? We obtained the difference between the means by subtraction, and then divided this difference by the standard error of the difference. Some modification of the procedure of dividing the difference by its standard error is needed, and the technique to use is the t test. What is the significance of the difference between the means of the two sets of observations? This is thought to provide a useful diagnostic sign as well as a clue to the efficacy of treatment.

Assuming that blood sodium concentration is Normally distributed what is the 95% confidence interval within which the mean of the total population of such cases may be expected to lie? For large samples we used the standard deviation of each sample, computed separately, to calculate the standard error of the difference between the means. Several different bran preparations are available, and a clinician wants to test the efficacy of two of them on patients, since favourable claims have been made for each. By random allocation the clinician selects two groups of patients aged 40-64 with diverticulosis of comparable severity. An approximate test, due to Sattherwaite, and described by Armitage and Berry, (1)which allows for unequal standard deviations, is as follows. It can produce a degree of freedom which is not an integer, and so not available in the tables. Whether treatment A or treatment B is given first or second to each member of the sample should be determined by the use of the table of random numbers Table F (Appendix).

The clinician wonders whether transit time would be shorter if bran is given in the same dosage in three meals during the day (treatment A) or in one meal (treatment B). As the aim is to test the difference, if any, between two types of treatment, the choice of members for each pair is designed to make them as alike as possible.

It is not valid to compare each treatment with each other treatment using t tests because the overall type I error rate will be bigger than the conventional level set for each individual test. What is the 95% confidence interval within which the mean of the population of such cases whose specimens come to the same laboratory may be expected to lie? I wanted to know that recently and turned to one of my favorite books: Measuring the User Experience, by Tom Tullis and Bill Albert.

In fact, the Central Limit Theorem also says that means are normally distributed, as in the bell-curve normal distribution shown in Figure 3.

Any mean from a random sample is likely to be quite close to the true population mean, and a normal distribution models the chance that it might be different from the true population mean. That’s because with more data, you have more chance of the sample being a pretty good match to the whole population and, therefore, of its mean being similar to the true population value. For example, let’s say we wanted the average time it took to complete the New York Marathon across 45,000 runners.

Those users ought to be happy—the time it took them to complete the task is much shorter than the advertised time. But it won’t be meaningful, because there is no Central Limit Theorem or any equivalent for modes. You can compare the confidence interval you calculated with the target you were aiming for. I’m a qual person to the core, but I recognized that quant data can be both useful to the researcher and very powerful as a way of convincing stakeholders. I would like to ask whether the 95% confidence interval incorporates any information on bias and misclassification? I am doing a degree in psychology and I have to do a report based on an experiment and run off results using SPSS.

If the difference is 196 times its standard error, or more, it is likely to occur by chance with a frequency of only 1 in 20, or less.

Its foundations were laid by WS Gosset, writing under the pseudonym "Student" so that it is sometimes known as Student's t test. Little is known about the subject, but the director of a dermatological department in a London teaching hospital is known to be interested in the disease and has seen more cases than anyone else.

Whether it should be regarded clinically as abnormally high is something that needs to be considered separately by the physician in charge of that case. Among the consequences of administering bran that requires testing is the transit time through the alimentary canal. Sample 1 contains 15 patients who are given treatment A, and sample 2 contains 12 patients who are given treatment B. The null hypothesis that there is no difference between the means is therefore somewhat unlikely.

In this way any effect of one treatment on the other, even indirectly through the patient's attitude to treatment, for instance, can be minimised.

A random sample of patients with disease of comparable severity and aged 20-44 is chosen and the two treatments administered on two successive occasions, the order of the treatments also being determined from the table of random numbers.

The null hypothesis is that there is no difference between the mean transit times on these two forms of treatment.

The more alike they are, the more apparent will be any differences due to treatment, because they will not be confused with differences in the results caused by disparities between members of the pair. With a small sample a non-significant result does not mean that the data come from a Normal distribution. A rule of thumb is that if the ratio of the larger to smaller standard deviation is greater than two, then the unequal variance test should be used.

By repeating measures within subjects, each subject acts as its own control, and the between subjects variability is removed.

One doctor is responsible for treatment and a second doctor assesses healing without knowing which treatment each patient had. So I trawled through various statistics books to gain a better understanding of confidence intervals, and this column is the result. Once we’ve seen three or four test participants in a row fail for the same reason, we just want to get on with fixing the problem. Some values of the true population mean would make it very likely that I’d get this sample mean, while other values would make it very unlikely. I didn’t bother calculating the confidence interval for our sample to get a Six-Sigma level of risk, because it would constitute the whole range of our data. If we took a random sample of just 1000 runners, we would get a narrow confidence interval.

But to ensure that a high volume of users can achieve those task times once the system gets rolled out, we’ll have to make sure that the system can cope with that high peak of very fast task times. However, to be able to identify the confidence interval, there are some prerequisites regarding the study design, variables measured, and sample selection that don’t map to many UX projects. The procedure does not differ greatly from the one used for large samples, but is preferable when the number of observations is less than 60, and certainly when they amount to 30 or less.

With small samples these multiples are larger, and the smaller the sample the larger they become. The transit times of food through the gut are measured by a standard technique with marked pellets and the results are recorded, in order of increasing time, in Table 7.1 . Since it is possible for the difference in mean transit times for A-B to be positive or negative, we will employ a two sided test. Many statistical packages now carry out this test as the default, and to get the equal variances I statistic one has to specifically ask for it. Occasionally it is possible to give both treatments simultaneously, as in the treatment of a skin disease by applying a remedy to the skin on opposite sides of the body. On the other hand, with a large sample, a significant result does not mean that we could not use the t test, because the t test is robust to moderate departures from Normality - that is, the P value obtained can be validly interpreted. With a computer one can easily do both the equal and unequal variance t test and see if the answers differ. In general this means that if there is a true difference between the pairs the paired test is more likely to pick it up: it is more powerful. The likely values represent the confidence interval, which is the range of values for the true population mean that could plausibly give me my observed value. That would be 18.6 seconds greater than our task-time target, which is disappointing all around. But if we took the times of the first 1000 runners across the finish line, we’d get something very far indeed from the true population mean.

So our colleagues who are managing system performance are likely to be far more interested in the most frequent value, which is the mode, than in the mean.

Conversely, as the sample becomes larger t becomes smaller and approaches the values given in table A, reaching them for infinitely large samples. The unequal variance t test tends to be less powerful than the usual t test if the variances are in fact the same, since it uses fewer assumptions. For instance, in a test for a drug reducing blood pressure the colour of the patients' eyes would probably be irrelevant, but their resting diastolic blood pressure could well provide a basis for selecting the pairs.

There is something illogical about using one significance test conditional on the results of another significance test.

Would the data from eight test participants be enough to represent the experience of all users? But we wouldn’t be nearly as worried if the true population mean happened to fall at the low end of the range.

In practice the degrees of freedom amount in these circumstances to one less than the number of observations in the sample. However, it should not be used indiscriminantly because, if the standard deviations are different, how can we interpret a nonsignificant difference in means, for example? Another (perhaps related) basis is the prognosis for the disease in patients: in general, patients with a similar prognosis are best paired. Many people would indeed notice if a task that they anticipated taking less than an hour actually took over two hours. Often a better strategy is to try a data transformation, such as taking logarithms as described in Chapter 2. Whatever criteria are chosen, it is essential that the pairs are constructed before the treatment is given, for the pairing must be uninfluenced by knowledge of the effects of treatment.

One can "eyeball" the data and if the distributions are not extremely skewed, and particularly if (for the two sample t test) the numbers of observations are similar in the two groups, then the t test will be valid. This is because only 17 observations plus the total number of observations are needed to specify the sample, the 18th being determined by subtraction.

Transformations that render distributions closer to Normality often also make the standard deviations similar. The main problem is often that outliers will inflate the standard deviations and render the test less sensitive. Also, it is not generally appreciated that if the data originate from a randomised controlled trial, then the process of randomisation will ensure the validity of the I test, irrespective of the original distribution of the data.

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