How does sample size affect the width of the confidence interval for the population mean?
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One issue with using tests of significance is that black and white cut-off points such as 5 percent or 1 percent may be difficult to justify. Show
Significance tests on their own do not provide much light about the nature or magnitude of any effect to which they apply. One way of shedding more light on those issues is to use confidence intervals. Confidence intervals can be used in univariate, bivariate and multivariate analyses and meta-analytic studies. What Determines the Width of the Confidence Interval?A narrow confidence interval enables more precise population estimates. The width of the confidence interval is a function of two elements:
The greater the confidence level, the wider the confidence interval. If we assume the confidence level is fixed, the only way to obtain more precise population estimates is to minimize sampling error. Sampling error is measured by the standard error statistic. The size of the standard error is due to two elements:
Usually there is little that we can do about changing variation in the population. One thing we can do is to increase the sample size. As a general guide, to halve the standard error the sample size must be quadrupled. Very precise population estimates with little margin for error require large sample sizes and/or resampling techniques like bootstrapping. However, in cases where such precision is not required there is a point where the gain in precision is not worth the cost of increasing the size of the sample. How to Interpret Confidence Intervals for MeansThe figures in Table 1 below were obtained for the average income of males and females in a fictitious survey for unemployment. How much better do males do than females in the income stakes? The sample estimate, based on 1698 respondents, is that males, on average, earn $5299 more than females ($44,640 – $39,341). That, of course, is the difference in the sample. What is the difference between males and females likely to be in the population? The table indicates this difference in the sample ($5299) and provides the standard error of this difference ($1422). Applying the 95 percent rule, the table also displays the confidence interval: we can be 95 percent confident that the real male-female income difference in the population is between $2509 and $8088.  Confidence intervals are focused on precision of estimates — confidently use them for that purpose! Effect Size Statistics Statistical software doesn't always give us the effect sizes we need. Learn some of the common effect size statistics and the ways to calculate them yourself. Please note that, due to the large number of comments submitted, any questions on problems related to a personal study/project will not be answered. We suggest joining Statistically Speaking, where you have access to a private forum and more resources 24/7. In the preceding discussion we have been using s, the population standard deviation, to compute the standard error. However, we don't really know the population standard deviation, since we are working from samples. To get around this, we have been using the sample standard deviation (s) as an estimate. This is not a problem if the sample size is 30 or greater because of the central limit theorem. However, if the sample is small (<30) , we have to adjust and use a t-value instead of a Z score in order to account for the smaller sample size and using the sample SD. Therefore, if n<30, use the appropriate t score instead of a z score, and note that the t-value will depend on the degrees of freedom (df) as a reflection of sample size. When using the t-distribution to compute a confidence interval, df = n-1. Calculation of a 95% confidence interval when n<30 will then use the appropriate t-value in place of Z in the formula: The T-distributionOne way to think about the t-distribution is that it is actually a large family of distributions that are similar in shape to the normal standard distribution, but adjusted to account for smaller sample sizes. A t-distribution for a small sample size would look like a squashed down version of the standard normal distribution, but as the sample size increase the t-distribution will get closer and closer to approximating the standard normal distribution. The table below shows a portion of the table for the t-distribution. Notice that sample size is represented by the "degrees of freedom" in the first column. For determining the confidence interval df=n-1. Notice also that this table is set up a lot differently than the table of Z scores. Here, only five levels of probability are shown in the column titles, whereas in the table of Z scores, the probabilities were in the interior of the table. Consequently, the levels of probability are much more limited here, because t-values depend on the degrees of freedom, which are listed in the rows.
Notice that the value of t is larger for smaller sample sizes (i.e., lower df). When we use "t" instead of "Z" in the equation for the confidence interval, it will result in a larger margin of error and a wider confidence interval reflecting the smaller sample size. With an infinitely large sample size the t-distribution and the standard normal distribution will be the same, and for samples greater than 30 they will be similar, but the t-distribution will be somewhat more conservative. Consequently, one can always use a t-distribution instead of the standard normal distribution. However, when you want to compute a 95% confidence interval for an estimate from a large sample, it is easier to just use Z=1.96. Because the t-distribution is, if anything, more conservative, R relies heavily on the t-distribution. Problem #1 Using the table above, what is the critical t score for a 95% confidence interval if the sample size (n) is 11? Answer Problem #2 A sample of n=10 patients free of diabetes have their body mass index (BMI) measured. The mean is 27.26 with a standard deviation of 2.10. Generate a 90% confidence interval for the mean BMI among patients free of diabetes. Link to Answer in a Word file Confidence Intervals for a Mean Using RInstead of using the table, you can use R to generate t-values. For example, to generate t values for calculating a 95% confidence interval, use the function qt(1-tail area,df). For example, if the sample size is 15, then df=14, we can calculate the t-score for the lower and upper tails of the 95% confidence interval in R: > qt(0.025,14) Then, to compute the 95% confidence interval we could plug t=2.144787 into the equation: Confidence Intervals from Raw Data Using RIt is also easy to compute the point estimate and 95% confidence interval from a raw data set using the " t.test" function in R. For example, in the data set from the Weymouth Health Survey I could compute the mean and 95% confidence interval for BMI as follows. First, I would load the data set and give it a short nickname. Then I would attach the data set, and then use the following command: > t.test(bmi) The output would look like this: One Sample t-test data: bmi R defaults to computing a 95% confidence interval, but you can specify the confidence interval as follows: > t.test(bmi,conf.level=.90) This would compute a 90% confidence interval. Lozoff and colleagues compared developmental outcomes in children who had been anemic in infancy to those in children who had not been anemic. Some of the data are shown in the table below.
Source: Lozoff et al.: Long-term Developmental Outcome of Infants with Iron Deficiency, NEJM, 1991 Compute the 95% confidence interval for verbal IQ using the t-distribution Link to the Answer in a Word file return to top | previous page | next page How sample size affects the width of the confidence interval?A larger sample size or lower variability will result in a tighter confidence interval with a smaller margin of error. A smaller sample size or a higher variability will result in a wider confidence interval with a larger margin of error. The level of confidence also affects the interval width.
How does sample size affect the width of the confidence interval for the population mean quizlet?How does sample size affect the width of the confidence interval for the population mean? Larger sample sizes result in narrow intervals.
What happens to the width of the confidence interval for the population mean as sample size decreases?This means that when the sample size is decreased the width of the confidence interval will increase.
Does population size affect width of confidence interval?Populations (and samples) with more variability generate wider confidence intervals. Sample Size: Smaller sample sizes generate wider intervals.
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