8-3 Testing a Claim About a Mean 405 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. Testing Claims About a Population Mean with s Not Known Objective Use a formal hypothesis test to test a claim about a population mean m. Notation n = sample size x = sample mean s = sample standard deviation mx = population mean (this value is taken from the claim and is used in the statement of the null hypothesis H0) Requirements KEY ELEMENTS 1. The sample is a simple random sample. 2. Either or both of these conditions are satisfied: The population is normally distributed or n 7 30. (If the second requirement is not satisfied, test the claim using a resampling method of bootstrapping or randomization described in Section 8-5, or use a nonparametric method such as the sign test (Section 13-2) or Wilcoxon signed-ranks test (Section 13-3).) Test Statistic for Testing a Claim About a Mean t = x - mx s2 n (Round t to three decimal places, as in Table A-3.) P-values: Use technology or use the Student t distribution (Table A-3) with degrees of freedom given by df = n - 1. (Figure 8-3 on page 380 summarizes the procedure for finding P-values.) Critical values: Use the Student t distribution (Table A-3) with degrees of freedom given by df = n - 1. (When Table A-3 doesn’t include the number of degrees of freedom, you could be conservative by using the next lower number of degrees of freedom found in the table, you could use the closest number of degrees of freedom in the table, or you could interpolate.) er we of Go Figure There are now 2.7 zettabytes 110212 of data in our digital universe.
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