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Unformatted text preview: Chapter 8 Section 2 Z Test for a Mean Mr. Kenneth Horwitz December 1, 2010 z Test for a Mean The z test is a statistical test for the mean of a population. It can be used when n ‡ 30 , or when the population is normally distributed and s is known. The formula for the z test is where = sample mean μ = hypothesized population mean s = population standard deviation = X z n μ σ X Professors’ Salaries A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 per year. The standard deviation of the population is $5230. Step 1: State the hypotheses and identify the claim. H 0: μ = $42,000 and H 1: μ > $42,000 (claim) Step 2: Find the critical value. Since α = 0.05 and the test is a righttailed test, the critical value is z = 1.65. Professors’ Salaries A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 per year. The standard deviation of the population is $5230. Step 3: Compute the test value. = X z n μ σ 43260 42000 5230 30 = 1.32 = Professors’ Salaries Step 4: Make the decision. Since the test value, 1.32, is less than the critical value, 1.65, and is not in the critical region, the decision is to not reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that assistant professors earn more on average than $42,000 per year. Important Comments Even though in the last example the sample mean of $43,260 is higher than the hypothesized population mean of $42,000, it is not significantly higher. Hence, the difference may be due to chance. When the null hypothesis is not rejected, there is still a probability of a type II error, i.e., of not rejecting the null hypothesis when it is false. When the null hypothesis is not rejected, it cannot be accepted as true. There is merely not enough evidence to say that it is false. Cost of Men’s Shoes A researcher claims that the average cost of men’s athletic shoes is less than $80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following costs (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researcher’s claim at...
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 Fall '10
 Nielson
 Statistical hypothesis testing, researcher, Average cost

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