7-2 Estimating a Population Mean 331 Requirement of “Normality or n + 30” Normality Requirement with Large Samples 1n + 302: It is common to consider the normality requirement to be satisfied if the sample is large 1n 7 302, because we know from the central limit theorem that for such large samples, the distribution of the sample means will tend to approximate a normal distribution. (For some population distributions that are extremely far from normal, the sample size might need to be much larger than 30.) Normality Requirement with Small Samples 1n " 302 : With small samples, the method for finding a confidence interval estimate of m is robust against a departure from normality, which means that the normality requirement is loose. The distribution need not be perfectly bell-shaped (it never is), but it should satisfy these conditions: 1. The distribution of the sample data should be somewhat close to being symmetric. 2. The distribution of the sample data should have one mode. 3. The sample data should not include any outliers. Interpreting the Confidence Interval The confidence interval is associated with a confidence level, such as 0.95 (or 95%). When interpreting a confidence interval estimate of m, know that the confidence level gives us the success rate of the procedure used to construct the confidence interval. For example, the 95% confidence interval estimate of 8.5901 g 6 m 6 9.0213 g can be interpreted as follows: “We are 95% confident that the interval from 8.5901 g to 9.0213 g actually does contain the true value of M.” By “95% confident” we mean that if we were to select many different samples of the same size and construct the corresponding confidence intervals, in the long run, 95% of the confidence intervals should contain the value of m. Student t Distribution In this section we use a Student t distribution, which is commonly referred to as a “t distribution.” It was developed by William Gosset (1876–1937), who was a Guinness Brewery employee who needed a distribution that could be used with small samples. The brewery prohibited publication of research results, but Gosset got around this by publishing under the pseudonym “Student.” (Strictly in the interest of better serving his readers, the author visited the Guinness Brewery and felt obligated to sample some of the product.) Here are some key points about the Student t distribution: ■ Student t Distribution If a population has a normal distribution, then the distribution of t = x - m s2 n is a Student t distribution for all samples of size n. A Student t distribution is commonly referred to as a t distribution. ■ Degrees of Freedom Finding a critical value t a>2 requires a value for the degrees of freedom (or df). In general, the number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. (Example: If 10 test scores have the restriction that their mean is 80, then their sum must be 800, and we can freely assign values to the first 9 scores, but the 10th score would then be determined,
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