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**Hypothesis Testing for the Mean (Small Samples)**

Section 7.3 Hypothesis Testing for the Mean (Small Samples) Larson/Farber 4th ed.

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**Section 7.3 Objectives Find critical values in a t-distribution**

Use the t-test to test a mean μ Larson/Farber 4th ed.

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**Finding Critical Values in a t-Distribution**

Identify the level of significance α. Identify the degrees of freedom d.f. = n – 1. 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 left-tailed, use “One Tail, α ” column with a negative sign, right-tailed, use “One Tail, α ” column with a positive sign, two-tailed, use “Two Tails, α ” column with a negative and a positive sign. Larson/Farber 4th ed.

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**Example: Finding Critical Values for t**

Find the critical value t0 for a left-tailed test given α = 0.05 and n = 15. Solution: The degrees of freedom are d.f. = n – 1 = 15 – 1 = 14. Look at α = 0.05 in the “One Tail, α” column. Because the test is left-tailed, the critical value is negative. t -1.761 0.05 Larson/Farber 4th ed.

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**Example: Finding Critical Values for t**

Find the critical values t0 and -t0 for a two-tailed test given α = 0.10 and n = 23. Solution: The degrees of freedom are d.f. = n – 1 = 23 – 1 = 22. Look at α = 0.10 in the “Two Tail, α” column. Because the test is two-tailed, one critical value is negative and one is positive. t -1.717 0.05 1.717 Larson/Farber 4th ed.

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**t-Test for a Mean μ (n < 30, σ Unknown)**

A statistical test for a population mean. The t-test can be used when the population is normal or nearly normal, σ is unknown, and n < 30. The test statistic is the sample mean The standardized test statistic is t. The degrees of freedom are d.f. = n – 1. Larson/Farber 4th ed.

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**Using the t-Test for a Mean μ (Small Sample)**

In Words In Symbols State the claim mathematically and verbally. Identify the null and alternative hypotheses. Specify the level of significance. Identify the degrees of freedom and sketch the sampling distribution. Determine any critical value(s). State H0 and Ha. Identify α. d.f. = n – 1. Use Table 5 in Appendix B. Larson/Farber 4th ed.

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**Using the t-Test for a Mean μ (Small Sample)**

In Words In Symbols Determine any rejection region(s). Find the standardized test statistic. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim. If t is in the rejection region, reject H0. Otherwise, fail to reject H0. Larson/Farber 4th ed.

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**Example: Testing μ with a Small Sample**

7.3 #26 A computer company believes the mean repair cost for a damaged computer is more than $95. To test this claim, you determine the repair costs for 7 randomly selected computers and find that the mean repair cost is $100 per computer with a standard deviation of $ At α = 0.01, do you have enough information to support the repair’s claim? Larson/Farber 4th ed.

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**Solution: Testing μ with a Small Sample**

α = df = Rejection Region: μ ≤ $95 μ > $95 (claim) Test Statistic: Decision: 0.01 7 – 1 = 6 Fail to reject H0 At the 0.01 level of significance, there is not enough evidence to support the computer repair’s claim that the mean repair cost for damaged computers is more than $95. z 3.143 0.01 0.311 Larson/Farber 4th ed.

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**Example: Testing μ with a Small Sample**

7.3, #28 An employment information service claims the mean annual pay for full-time female workers over age 25 and without a high school diploma is $19,100. The annual pay for a random sample of 12 full time female workers without a high school diploma is gathered. At α = 0.05, test the claim that the mean salary is $19,100 Larson/Farber 4th ed.

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**Solution: Testing μ with a Small Sample**

α = df = Rejection Region: μ = $19,100 μ ≠ $19,100 Test Statistic: Decision: 0.05 12 – 1 = 11 Fail to reject H0 At the 0.05 level of significance, there is not enough evidence to reject the claim that the mean salary is $19,100 for full-time female workers, over 25, without a high school diploma. t -2.101 0.025 2.101 -0.529 Larson/Farber 4th ed.

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**Section 7.3 Summary Found critical values in a t-distribution**

Used the t-test to test a mean μ HW: 7, 17, 19, EO, 31, 33 Larson/Farber 4th ed.

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