SECTION 7.3 Hypothesis Testing for the Mean (s Unknown) 379 The t@Test for a Mean M To test a claim about a mean m when s is not known, you can use a t@sampling distribution. The standardized test statistic takes the form of t = 1Sample mean2 - 1Hypothesized mean2 Standard error . Because s is not known, the standardized test statistic is calculated using the sample standard deviation s, as shown in the next definition. The t@test for a mean M is a statistical test for a population mean. The test statistic is the sample mean x. The standardized test statistic is t = x - m s 1n Standardized test statistic for m (s unknown) when these conditions are met. 1. The sample is random. 2. At least one of the following is true: The population is normally distributed or n Ú 30. The degrees of freedom are d.f. = n - 1. t-Test for a Mean M Using the t@Test for a Mean M (S Unknown) In Words In Symbols 1. Verify that s is not known, the sample is random, and either the population is normally distributed or n Ú 30. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Identify the degrees of freedom. d.f. = n - 1 5. Determine the critical value(s). Use Table 5 in Appendix B. 6. Determine the rejection region(s). 7. Find the standardized test statistic t = x - m s 1n and sketch the sampling distribution. 8. Make a decision to reject or fail to If t is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES In Step 8 of the guidelines, the decision rule uses rejection regions. You can also test a claim using P-values, as shown on page 382. Also, when the number of degrees of freedom you need is not in Table 5, use the closest number in the table that is less than the value you need (or use technology). For instance, for d.f. = 57, use 50 degrees of freedom. Picturing the World Exposure to lead may cause red blood cell, kidney, or brain damage.The Environmental Protection Agency established rules that require water systems to monitor drinking water at customer taps. If lead concentrations exceed 0.015 milligram per liter in more than 10% of customer taps sampled, the system must undertake a number of actions, such as source water treatment, public education, and lead service line replacement. On the basis of a t-test, a water system makes a decision on whether the mean level of lead in the water exceeds the allowable amount of 0.015 milligram per liter. Assume the null hypothesis is m … 0.015. (Source: Environmental Protection Agency) Do not reject H0. Reject H0. H0 is true. H0 is false. Describe the possible type I and type II errors of this situation.
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