Question.3172 - 1. Confidence interval for μ when σ is known Suppose n = 9 people are selected at random from a large population. Assume the heights of the people in this population are normal, with mean μ = 68.71 inches and σ = 3 inches. Simulate the results of this selection 20 times and in each case find a 90% confidence interval for μ. The following commands may be used: MTB > random 9 c1-c20; SUBC> normal 68.71 3. MTB > zinterval 0.90 3 c1-c20 a. [2] How many of your intervals contain μ? b. [2] What is the probability that 100 (not 20) such intervals would contain μ? c. Do all the intervals have the same width? [1] ........ Why [1] (what is the theoretical width)?———————————— d. [1] Suppose you constructed 80% intervals instead of 90%. Would they be narrower or wider?—————– e. [1] How many of your intervals contained the value 71?———————— f. Suppose you took samples of size n = 4 instead of n = 9. Would you expect more or fewer intervals to contain 71? [1] ...... What about 68.71? [1] ........ What about the width of the intervals for n = 4: Would they be narrower or wider than for n = 9? [1] ........12. Confidence interval for μ when σ is NOT known Repeat the simulation of Question 1 but now assume σ is unknown and use the tintervals command to get the 20 90% intervals: MTB > random 9 c1-c20; SUBC > normal 68.71 3. MTB > tinterval 0.90 c1-c20a. [2] How many of your intervals contain μ?———————————- b. Would you expect all 20 of the intervals to contain μ? [0.5]........ Why? [1.5]————— c. Do all the intervals have the same width? [1]......... Why (what is the theoretical width)? [1]—————— d. [1] Suppose you took 95% intervals instead of 90%. Would they be narrower or wider?— ————————– e. [2] How many of your intervals contain the value 71?—————————- f. Suppose you took samples of size n = 64 instead of n = 9. Would you expect more or fewer intervals to contain 71? [1] ...... What about 68.71? [1]——————— What about the width of the intervals for n = 64: Would they be narrower or wider than for n = 9? [1] ......... 3. Hypothesis testing for μ when σ is known Imagine choosing n = 16 women at random from a large population and measuring their heights. Assume that the heights of the women in this population are normal with μ = 63.8 inches and σ = 3 inches. Suppose you then test the null hypothesis H0 : μ = 63.8 versus the alternative that Ha : μ 6= 63.8, using α = 0.10. Assume σ is known. Simulate the results of doing this test 30 times as follows: MTB > random 16 c1-c30; SUBC > normal 63.8 3. MTB > ztest 63.8 3 c1-c30 a. [2] In how many tests did you reject H0. That is, how many times did you make an “incorrect decision”? ........2b. [2] Are the p-values all the same for the 30 tests? ...... c. Suppose you used α = 0.001 instead of α = 0.10. Does this change any of your decisions to reject or not?[1]........... In general, should the number of rejections increase or decrease if α = 0.001 is used instead of α = 0.10?[2]............. d. Now assume that the population really has a mean of μ = 63, instead of 63.8, and carry out the above 30 simulations, (thus, use the above minitab commands with ’normal 63.8 3’ changed to ’normal 63 3’. Once again, using α = 0.10 and assuming σ known, in how many tests did you reject H0?[2] ............. A rejection of H0 in part (a) is a “correct decision”. True or False? [0.5] ......... A rejection of H0 in part (d) is a “correct decision”. True or False?[0.5] .......... 4. Hypothesis testing for μ when σ is NOT known Repeat Question 3, using ttest instead of ztest, and answer parts (a), (b), and (c) again. (Thus ‘ztest 63.8 3 c1-c30’ changes to ‘ttest 63.8 c1-c30’) a. [2] In how many tests did you reject H0. That is, how many times did you make an “incorrect decision”? .......... b. [1] Are the p-values all the same for the 30 tests? ....... c. Suppose you used α = 0.00008 instead of α = 0.10. Does this change any of your decisions to reject or not? [2] ........... In general, should the number of rejections increase or decrease if α = 0.00008 is used instead of α = 0.10? [2] ............ Part II Comprehension questions 5. A fast food franchiser is considering building a restaurant at a certain location. According to a financial analysis, a site is acceptable only if the number of pedestrians passing the location averages more than 100 per hour. A random sample of 50 hours produced ?x = 110 and s = 12 pedestrians per hour. (a) [6] Do these data provide sufficient evidence to establish that the site is acceptable? Use α = 0.05. (b) [2] What are the consequences of Type I and Type II errors? Which error is more expensive to make? (c) [4] Considering your answer in part (b), should you select α to be large or small? Explain.3(d) [3] What assumptions about the number of pedestrians passing the location in an hour are necessary for your hypothesis test to be valid? 6. An experiment was conducted to test the effect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50. The first group, the control group, received no treatment for the infection. The second group received the drug. After a 30-day period, the proportions of survivors, ˆp1 and ˆp2, in the two groups were found to be 0.36 and 0.60, respectively. (a) [6] Is there sufficient evidence to indicate that the drug is effective in treating the viral infection? Test at 5% significance level. (Make sure to state your null and alternative hypotheses.) (b) [6] Use a 95% confidence interval to estimate the actual difference in the cure rates, i.e. p1 −p2, for the treatment versus the control groups. [3] Based on this confidence interval can you conclude that the drug is effective? Why?7. In an investigation of pregnancy-induced hypertension, one group of women with this disor- der was treated with low-dose aspirin, and a second group was given a placebo. A sampleconsisting of 23 women who received aspirin has mean arterial blood pressure 111 mm Hg and standard deviation 8 mm Hg; a sample of 24 women who were given the placebo has mean blood pressure 109 mm Hg and standard deviation 8 mm Hg. (a) [6] At the 0.01 level of significance, test the null hypothesis that the two populations of women have the same mean arterial blood pressure. Justify any procedure you use. (b) [5] Construct a 99% confidence interval for the true difference in population means. [1] Does this interval contain the value 0? [3] Based on this confidence interval, what is you conclusion regarding the effect of the two treatments on the blood pressure of pregnant women? 8. [7] A company is interested in offering its employees one of two employee benefit packages. A random sample of the company’s employees is collected, and each person in the sample is asked to rate each of the two packages on an overall preference scale of 0 to 100. Results were4Employee Program A Program B 1 45 56 2 67 70 3 63 60 4 59 45 5 77 85 6 69 79 7 45 50 8 39 46 9 52 50 10 58 60 11 70 82Do you believe that the employees of this company prefer, on the average, one package over the other? Explain. 9. In an attempt to compare the starting salaries for university graduates who majored in education and the social sciences, random samples of 100 recent university graduates were selected from each major and the following sample information was obtained:Major Mean St. Dev. Education $50,554 $2225 Social Science $48,348 $2375Conduct an appropriate hypothesis test at the 5% level of significance to determine if there is a difference in the average starting salaries for all university graduates who majored in education and the social sciences. Conduct an appropriate hypothesis test at the 5% level of significance to determine if there is a difference in the average starting salaries for all university graduates who majored in education and the social sciences. Conduct this test using (i) [3] the p-value method , (ii) [3] the critical value method , and (iii) [3] the confidence interval method.
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a xx my xxxxxxxxx contains xxxxx but xxxxxxxx above xx also xxxxxxxxxx b x Yes xxx theoretical xxxxx is x z x d xxxxxxxx e xx my xxxxxxxxx contained xxx value xxx any xxxxxx in xx acceptable x More xxxx Wider x of xx intervals xxxxxxx micro xxx anything xxxxx is xxxx acceptable x No xxxxxxxx is x t x s xxxxxx from xxxxxx to xxxxxx d xxxxx e xxx any xxxxxx in xx acceptable x Fewer xxxxx Narrower x p- xxxxxx less xxxx but xxx number xx is xxxxxxxxxx b xx c xxx but xxxx may xxxxxx No x p-values xxx any xxxxxx in xx acceptable x but xxx le xx acceptable x Yes xxx some xxx report xx Decrease x H xxxxx vs x alpha xxxxx amp xx since x is xxxxx enough xxxx statistic xx z xx micro x - xxxx hypothesis xx rejected xxxxx amp xx z x Type x error xx rejecting x when xx is xxxx and xxx result xx this xxxxx is xx construct xxx site xxxx there xxx not xxxxxx pedestrians xxxxxxx the xxxxxxxx bankruptcy xxxx II xxxxx is xx accept x when xx is xxx true xx the xxxx does xxx construct xx location xxxx than xxxxxxxxxxx missing x good xxxxxxxx Type x error xx more xxxxxxxxx It xx possible xxxx II xxxxx be xxxx expensive xxxxxx if xx supports xxxx good xxxxxxxxxxx c xxxxxxxxx to xxxx b xxxxx should xx a xxxxx number xx as xx reduce xxx errors xxxxxxx the xxxxx of xxxxx the xxxxx of x will xx larger xxx it xxxx accept xxx null xxxxxxxxxx d xxxxxxxxxxx are xxxxxxxx randomly x H x minus x D x alpha x minus x amp xx D xxx test xx a xxxxxxxxxx test xx the xxxxx cance xxxxx alpha xxx rejection xxxxxx z xxxx For xxx sake xx being xxxxxxxx note xxxx n x n x n x n x Since xxx of xxxxx are xxxxxxx than xx equal xx we xxx assume xxxxxxxxx b xxxxx note xxxx since xxx hypothesis xx centered xxxxx zero xx will xxx a xxxxxx estimator xxx p xxxx computing x p xxxxx p x x x n x sigma x minus x asymp x p xxxxx p xxx test xxxxxxxxx is x p xxxxx p xxxxx D xxxxx p xx minus xxxxx minus xxxxx z xxxxx is xx the xxxxxxxxx region xx reject x and xxxxxx H xxxxx There xx su xxxxx evidence xx indicate xxxx the xxxx is x ective xx treating xxx viral xxxxxxxxx a xxxxx of xxxxxxxxxxxx alpha xxxxxxx of xxxxxxx d x n x - x From xxxxxxx since xxx alternative xxxxxxxxxx is xxxxxxx the xxxxxxxx values x plusmn xxxxxxxx Rule xxxxxx H xx t xxx lt x or xx t xxx gt xxxx null xxxxxxxxxx holds xxxxx Test xxxxxxxxx t x -x x p x p x n x s xxxxxxx the xxxxxx in xxx above xxxxxxxx s x t x t xxxxx Since x amp xx therefore xxxx hypothesis xx accepted xx there xx no xxxxxxxxxxx evidence xxxx the xxx populations xx women xxxx the xxxx mean xxxxxxxx blood xxxxxxxx b xxxxxx n xxxxx s x B xx Cnnxxdf xxx ndash xxxxx t xxx critical xxxxxxx from xxx table xxxxx two xxxxxx is xxxxxxxxxx interval x Yes xxx interval xxxxxxxx the xxxxx Based xx this xxxxxxxxxx interval xx are xxxx that xxx effect xx the xxx treatments xx the xxxxx pressure xx pregnant xxxxx is x So xxxxx is xx statistical xxxxxxxx that xxx two xxxxxxxxxxx of xxxxx have xxx same xxxx arterial xxxxx pressure xxxxxxxx each xxxx set xxxxx from x normal xxxxxxxxxxxx and xxxxxx alpha x micro x micro xx H xxxxx micro x micro xx Or x micro x vs x alpha xxxxx d xx It xx paired xxxx where x i x - x - x - x sum x i x - x and x d x Test xxxxxxxxx t xx s x - x - xxx n- xxxxxxx of xxxxxxx t xx H xx not xxxxxxxx The xxxxxxxxxx are x micro x micro xx H xxxxx micro x micro xx Since xxx sample xxxxx are xxxxxxx than xxx test xxxxxxxxx is x x xx -D x At xxxxx the xxxxxxxx values xxx z x is xxxxxxxx since xxx gt xxxxxxxxx the xxxxxxx p-value x Z xxx gt xxxxx Compute xxxxxxxxxx interval xx follows x -x xxxxxx z x plusmnMore Articles From Statistics
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