2.(12 total points)Textbook problem 6.28on page 234In 1993 a typical American family spent 22% of its after-tax income on clothing, entertain-ment, and other activities, while the remaining 78% was spent on essentials like housing, food,transportation, health care and insurance/pensions.It is suggested that because of the in-crease of cost of essentials since 1993, the mean percentage income spent on the first threeitems has decreased. To test this proposition a random sample of 50 households is taken andthe percentage spent on these three items is determined. Assume that this percentage cariesacross families according to a normal distribution with unknown meanμand knownσ= 5%.(Note that this percentage is not a binomial proportion.)(a)(2 points) Set up the hypotheses onμ.(b)(5 points) Suppose the average percentage for 50 families is 20.5%. What is the rejectionregion for theα=.01 level test? Justify which test statistic you used for this calculation.Would you conclude that this is a significant decrease compared to the 1993 level?(c)(5 points) Repeat the test in (b) by calculating theP-value. Solution: Stat 23400: Winter 2013 Homework #8 2 of 8
3.(12 total points)Textbook problem 7.8 on page 263The mean pH value of a certain chemical is to be controlled at 5. Deviation from this targetvalue in either direction is to be detected with high probability. For this purpose it is proposedto measure a certain number of samples from each batch and decide that the mean pH isdifferent from 5 if the sample mean differs significantly from 5 at the 10% level of significance.(a)(2 points) State the hypotheses tested by the above decision rule.Interpret the testparameterμ.(b)(5 points) What sample size is needed if the probability ofnotdetecting a change of onestandard deviation is to be no more than 1%?(c)(5 points) Suppose that 16 samples are measured. The sample meanxis 4.915 and thesample standard deviationsis 0.2. Ignoring that we have a small sample and assumingthats≈σ, calculate theP-value. Do you conclude that the mean pH has changed fromthe target value of 5? Useα= 0.10.
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