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Unformatted text preview: Chapter 9 Hypothesis testing 9.1 Introduction Confidence intervals are one of the two most common types of statistical inference. Use them when our goal is to estimate a population parameter. The second common type of inference has a different goal: to assess the evidence provided by the data in favor of some claim about the population. Example 9.1 The rapid increases in college tuition over the past few years have been a great concern for college students and their parents. According to Digest of Education Statistics 1997 , the average annual cost of in-state tuition and fees for 4-year public colleges in the United States was $ 3,321 in 1997. A recent sample of 40 four-year public colleges yielded a mean in-state tuition of $ 3393 . Can we conclude that the mean in-state tuition for 4-year public colleges increases this year? Outline of a test: I. Set hypotheses Null hypothesis H : μ = 3321 Alternative hypothesis H 1 : μ > 3321 A null hypothesis H is a claim (or statement) about a population parameter that is being tested. H can be interpreted as “there is no difference” An alternative hypothesis H a is a statement that we intend to prove true • A test of significance is intended to assess the evidence provided by data against H in favor of an H 1 9-1 • Both hypotheses are in terms of parameters , never about statistics. Example (a): A person has been indicted for committing a crime and is being tried in a court. The jury will make one of two possible decisions: Null hypothesis: H : The defendant is not guilty Alternative hypothesis: H 1 : The defendant is guilty Example (b): State H and H 1 in each case. • Your car averages 32 miles per gallon on the highway. You now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with new oil, you want to determine if your gas mileage actually has increased • A man claims to have extrasensory perception . You don’t believe it. • The English mathematician John Kerrich tossed a coin 10,000 times and got 5067 heads. You suspect that the coin is not balanced. Suppose that H : θ = θ . If H 1 : θ > θ or H 1 : θ < θ , then it is referred to as a one-tailed test (one-sided test); If H 1 : θ 6 = θ , then it is referred to as a two-tailed test (two-sided test). II. Choose a test statistic III. Assess if the observed value of the test statistic is surprising when H is true. How unlikely the observed value of the test statistic would be if H were really true? Assume that σ = 150. Under H , ¯ x ≈ N (3321 , 150 √ 40 ) Hence, P (¯ x ≥ 3393) = P ( ¯ x- 3321 150 / √ 40 ≥ 3393- 3321 150 / √ 40 ) ≈ P ( Z ≥ 3 . 04) = . 0012 Or we partition the possible values of the test statistic into two subsets: an acceptance region for H , and a rejection region for H . The rejection region is also called critical region ....
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