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8.3 T- TEST FOR A MEAN

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T- TEST The t test is a statistical test for the mean of a population and is used when the population is normally or approximately normally distributed and is unknown. The formula for the t test is The degrees of freedom are d.f. = n – 1.

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FINDING CRITICAL VALUES Find the critical t value for α = 0.05 with d.f. = 16 for a right- tailed t test.

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EXAMPLES 1. Find the critical t value for α = 0.01 with d.f. = 22 for a left-tailed test. 2. Find the critical value for α = 0.10 with d.f. = 18 for a two-tailed t test.

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T-TEST STEPS Use the same steps as for the z-test. 1.State null and alternative hypothesis. 2.Find the critical values from Table F. 3.Compute the test value. 4.Reject/Not Reject 5.Sentence.

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EXAMPLE 1 A medical investigation claims that the average number of infections per week at a hospital in southwestern Pennsylvania is 16.3. A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is there enough evidence to reject the investigator’s claim at α = 0.05? Step 1: State the hypotheses and identify the claim. H 0 : μ = 16.3 and H 1 : μ 16.3 Step 2: Find the critical value. The critical values are 2.262 and –2.262 for α = 0.05 and d.f. = 9.

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Step 3: Find the test value. Step 4: Reject/Not Step 5: Sentence There is enough evidence to reject the claim that the average number of infections is 16.3.

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An educator claims that the average salary of substitute teachers in school districts in Allegheny County, Pennsylvania, is less than $60 per day. A random sample of eight school districts is selected, and the daily salaries (in dollars) are shown. Is there enough evidence to support the educator’s claim at α = 0.10? 60 56 60 55 70 55 60 55

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