handout_15a - Apr 2110 HYPOTHESIS TESTING: ILLUSTRATIVE...

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HYPOTHESIS TESTING: ILLUSTRATIVE EXAMPLES #15a INVOLVING STANDARD NORMAL DISTRIBUTION We’ll consider examples of the testing of a hypothesis about a population mean under the condition that sampling is from a normally (or approximately normally) distributed population with known variance . This is a rare case but it may serve us to illustrate the procedure of two -sided hypothesis test (read at first the handout #15 ). Example 1a (#7.2.1 on pg. 219 of the textbook). Researchers are interested in the mean age of a certain population that is assumed to be approximately normally distributed with a known variance of σ 2 = 20 . They are asking the following question: Can we conclude that the mean age of this population is different from 30 years? They selected a simple random sample of 10 individuals from the population of interest and computed the mean age of 27 years for that sample. Let α = 0.05 . Solution . Researchers can conclude that the mean age of this population is different from 30 years if they can reject the null hypothesis that the mean age of the population is equal to 30 years . Let us use the following ten-step procedure of hypothesis testing to illustrate the decision-making process. 1. Data . The available data are the ages of 10 individuals randomly selected from the population of interest from which a sample mean of 27 years has been computed. 2. Assumptions . It is assumed that the population in question is approximately normally distributed with a known variance of σ 2 = 20 . 3. Hypotheses . The null hypothesis H 0 to be tested is that the mean age of the population is equal to 30 years (thus, the hypothesized population mean is µ 0 = 30 years). The alternative hypothesis H A is that the mean age of the population is not equal to 30 years. Symbolically these hypotheses may be written as H 0 : µ = 30 and H A : µ ≠ 30 , or in more general terms, H 0 : µ = µ 0 and H A : µ ≠ µ 0 . (As we agreed earlier in handout #15 , what we hope or expect to be able to conclude if the null hypothesis H 0 will be rejected as a result of testing usually should be placed in the alternative hypothesis H A . ) 4. Test statistic . Since we are testing the hypothesis about a population mean assuming that the population in question is approximately normally distributed with a known variance, σ , the test statistic of interest for the corresponding sampling distribution of the sample means is _ _ z = ( x – µ 0 )/(σ/√n) (15a.1) 5. Distribution of test statistic . Based on our knowledge of properties of normal and sampling distributions (see handouts #8, 9 ), we know that the test statistic z (15a.1) is normally distributed with a mean of 0 and a variance of 1 , if H 0 is true and µ = µ 0 (see the standard normal curve
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handout_15a - Apr 2110 HYPOTHESIS TESTING: ILLUSTRATIVE...

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