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Chapter 7 Hypothesis Testing with One Sample § 7.3 Hypothesis Testing for the Mean (Small Samples) Critical Values in a t-Distribution Finding Critical Values in a t-Distribution 1. Identify the level of significance . 2. Identify the degrees of freedom d.f. = n – 1. 3. Find the critical value(s) using Table 5 in Appendix B in the row with n – 1 degrees of freedom. If the hypothesis test is a. left-tailed, use “One Tail, ” column with a negative sign, b. right-tailed, use “One Tail, ” column with a positive sign, c. two-tailed, use “Two Tails, ” column with a negative and a positive sign. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Finding Critical Values for t Example: Find the critical value t0 for a right-tailed test given = 0.01 and n = 24. The degrees of freedom are d.f. = n – 1 = 24 – 1 = 23. To find the critical value, use Table 5 with d.f. = 23 and 0.01 in the “One Tail, “ column. Because the test is a right-tail test, the critical value is positive. t0 = 2.500 Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Finding Critical Values for t Example: Find the critical values t0 and t0 for a two-tailed test given = 0.10 and n = 12. The degrees of freedom are d.f. = n – 1 = 12 – 1 = 11. To find the critical value, use Table 5 with d.f. = 11 and 0.10 in the “Two Tail, “ column. Because the test is a two-tail test, one critical value is negative and one is positive. t0 = 1.796 and t0 = 1.796 Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 t-Test for a Mean μ (n < 30, Unknown) The t-test for the mean is a statistical test for a population mean. The t-test can be used when the population is normal or nearly normal, is unknown, and n < 30. The test statistic is the sample mean test statistic is t. and the standardized x μ t s n The degrees of freedom are d.f. = n – 1 . Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 t-Test for a Mean μ (n < 30, Unknown) Using the t-Test for a Mean μ (Small Sample) In Words In Symbols 1. State the claim mathematically and verbally. Identify the null and alternative hypotheses. State H0 and Ha. 2. Specify the level of significance. Identify . 3. Identify the degrees of freedom and sketch the sampling distribution. d.f. = n – 1. 4. Determine any critical values. Use Table 5 in Appendix B. 5. Determine any rejection region(s). Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 t-Test for a Mean μ (n < 30, Unknown) Using the t-Test for a Mean μ (Small Sample) In Words In Symbols 6. Find the standardized test statistic. 7. Make a decision to reject or fail to reject the null hypothesis. 8. Interpret the decision in the context of the original claim. t x μ s n If t is in the rejection region, reject H0. Otherwise, fail to reject H0. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Testing μ Using Critical Values Example: A local telephone company claims that the average length of a phone call is 8 minutes. In a random sample of 18 phone calls, the sample mean was 7.8 minutes and the standard deviation was 0.5 minutes. Is there enough evidence to support this claim at = 0.05? H0: = 8 (Claim) H a: 8 The level of significance is = 0.05. The test is a two-tailed test. Degrees of freedom are d.f. = 18 – 1 = 17. The critical values are t0 = 2.110 and t0 = 2.110 Larson & Farber, Elementary Statistics: Picturing the World, 3e Continued. 9 Testing μ Using Critical Values Example continued: A local telephone company claims that the average length of a phone call is 8 minutes. In a random sample of 18 phone calls, the sample mean was 7.8 minutes and the standard deviation was 0.5 minutes. Is there enough evidence to support this claim at = 0.05? Ha: 8 H0: = 8 (Claim) The standardized test statistic is The test statistic falls in the nonrejection region, so H0 is not rejected. t x μ 7.8 8 s n 0.5 18 1.70. z0 = 2.110 0 z0 = 2.110 z At the 5% level of significance, there is not enough evidence to reject the claim that the average length of a phone call is 8 minutes. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Testing μ Using P-values Example: A manufacturer claims that its rechargeable batteries have an average life greater than 1,000 charges. A random sample of 10 batteries has a mean life of 1002 charges and a standard deviation of 14. Is there enough evidence to support this claim at = 0.01? H0: 1000 Ha: > 1000 (Claim) The level of significance is = 0.01. The degrees of freedom are d.f. = n – 1 = 10 – 1 = 9. The standardized test statistic is t x μ 1002 1000 s n 14 10 0.45 Larson & Farber, Elementary Statistics: Picturing the World, 3e Continued. 11 Testing μ Using P-values Example continued: A manufacturer claims that its rechargeable batteries have an average life greater than 1,000 charges. A random sample of 10 batteries has a mean life of 1002 charges and a standard deviation of 14. Is there enough evidence to support this claim at = 0.01? Ha: > 1000 (Claim) H0: 1000 t 0.45 0 0.45 z Using the d.f. = 9 row from Table 5, you can determine that P is greater than = 0.25 and is therefore also greater than the 0.01 significance level. H0 would fail to be rejected. At the 1% level of significance, there is not enough evidence to support the claim that the rechargeable battery has an average life of at least 1000 charges. Larson & Farber, Elementary Statistics: Picturing the World, 3e 12