Testing Hypotheses about a Population Mean: Z and t Tests - Prof. Mihails Levins, Study notes of Data Analysis & Statistical Methods

Download Testing Hypotheses about a Population Mean: Z and t Tests - Prof. Mihails Levins and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Lecture 15: Tests about a Population Mean Devore: Section 8.2 Aug, 2006 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 A Normal Population with known σ • This case is not common in practice. We will use it to illustrate basic principles of test procedure design • Let X1, . . . , Xn be a sample size n from the normal population. The null value of the mean is usually denoted μ0 and we consider testing either of the three possible alternatives μ > μ0, μ < μ0 and μ = μ0 • The test statistic that we will use is Z = X̄ − μ0 σ/ √ n It measures the distance of X̄ from μ0 in standard deviation units. Aug, 2006 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Summary • Let H0 : μ = μ0; define the test statistic Z = X̄−μ0σ/√n . 1. Ha : μ > μ0 has the rejection region z ≥ zα and is called an upper-tailed test 2. Ha : μ < μ0 has the rejection region z ≤ −zα and is called an lower-tailed test 3. Ha : μ = μ0 has the rejection region z ≥ zα/2 or z ≤ −zα/2 and is called a two-tailed test Aug, 2006 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Recommended Steps for Testing Hypotheses about a Parameter 1. Identify the parameter of interest and describe it in the context of the problem situation. 2. Determine the null value and state the null hypothesis. 3. State the alternative hypothesis. 4. Give the formula for the computed value of the test statistic. 5. State the rejection region for the selected significance level 6. Compute any necessary sample quantities, substitute into the formula for the test statistic value, and compute that value. • The formulation of hypotheses (steps 2 and 3) should be done before examining the data. Aug, 2006 Page 6 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Example • Consider Ex. 8.6 in Devore. • Parameter of interest is μ = true average activation temperature. • H0 : μ = 130; Ha : μ = 130. • Test statistic is Z = x̄ − μ0 σ/ √ n = x̄ − 130 1.5/ √ n Aug, 2006 Page 7 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 • Similar derivations can help us to derive Type II error probabilities for a lower-tailed test and a two-tailed test. Results can be summarized as follows: 1. Ha : μ > μ0 has the probability of Type II Error Φ ( zα + μ0−μ′ σ/ √ n ) 2. Ha : μ < μ0 has the probability of Type II Error 1 − Φ ( −zα + μ0−μ ′ σ/ √ n ) 3. Ha : μ = μ0 has the probability of Type II Error Φ ( zα/2 + μ0−μ′ σ/ √ n ) − Φ ( −zα/2 + μ0−μ ′ σ/ √ n ) Aug, 2006 Page 10 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Sample Size Determination • Sometimes, we want to bound the value of Type II error for a specific value μ ′ . • Consider Ex. 8.6 again, fix α and specify β for such an alternative value. For μ ′ = 132 we may want to require β(132) = 0.1 in addition to α = .01. • The sample size required for that purpose is such that Φ ( zα + μ0 − μ′ σ/ √ n ) = β Aug, 2006 Page 11 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 • Solving for n, we obtain n = [ σ(zα + zβ) μ0 − μ′ ]2 and the same answer is true for a lower-tailed test • For a two-tailed test, it is only possible to give an approximate solution. It is n ≈ [ σ(zα/2 + zβ) μ0 − μ′ ]2 Aug, 2006 Page 12 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Summary of the Three Possible t-Tests • If Ha : μ > μ0, the rejection region of the level α test is t ≥ tα,n−1 • If Ha : μ < μ0, the rejection region of the level α test is t ≤ −tα,n−1 • If Ha : μ = μ0, the rejection region of the level α test is t ≥ tα/2,n−1 or t ≤ −tα/2,n−1 Aug, 2006 Page 15 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Example • The Edison Electric Institute publishes figures on the annual number of kilowatt hours expended by various home appliances. • It is claimed that a vacuum cleaner expends an average of 46 kilowatt hours per year. • Suppose a planned study includes a random sample of 12 homes and it indicates that VC’s expend an average of 42 kilowatt hours per year with s = 11.9 kilowatt hours. • Assuming the population normality, design a 0.05 level test to see whether VC’s spend less than 46 kilowatt hours annually Aug, 2006 Page 16 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 • H0 : μ = 46 kilowatt hours and Ha : μ < 46 kilowatt hours • Assuming α = 0.05, we have a critical region t < −1.796 where t = x̄ − μ0 s/ √ n with 11 df • The value of the statistic is t = 42 − 46 11.9/ √ 12 = −1.16 • Since t is not in the rejection region, we fail to reject H0. Aug, 2006 Page 17 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 Calculating β • First, we select μ′ and the estimated value for unknown σ. Then, we find an estimated value of d = |μ0 − μ′ |/σ. Finally, the value of β is the height of the n − 1 df curve above the value of d • If n − 1 is not the value for which the corresponding curve appears visual interpolation is necessary Aug, 2006 Page 20 Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2006 • One can also calculate the sample size n needed to keep the Type II Error probability below β for specified α. 1. First, we compute d 2. Then, the point (d, β) is located on the relevant set of graphs 3. The curve below and closest to the point gives n − 1 and thus n 4. Interpolation, of course, is often necessary Aug, 2006 Page 21