## Statistics MATH 324 Assignment 4

Johanna G. NeÅ¡lehovÃ¡Statistics MATH 324McGill University, Fall Term 2015Assignment 4: Due November 19 at 11:59 PMQ 22 Solid copper produced by sintering (heating without melting) a powder under speciﬁedenvironmental conditions is then measured for porosity (the volume fraction due tovoids) in a laboratory. A sample of n = 4 independent porosity measurements havemean xn = 0.22 and variance s2 = 0.001. A second laboratory repeats the sameÂ¯nprocess on solid copper formed from an identical powder and gets m = 5 independentporosity measurements with ym = 0.17 and s2 = 0.002.Â¯m(a) List all the assumptions you have to make in order to construct an exact twosided conﬁdence interval for the true diﬀerence between the population meansfor these two laboratories.(b) Construct the exact two sided 95% conﬁdence interval for the true diﬀerencebetween the population means for these two laboratories.(c) Is there evidence that the population means diﬀer for the two laboratories?Q 23 In a study of the relationship between birth order and college success, an investigatorfound that 126 in a sample of 180 college graduates were ﬁrstborn or only children; ina sample of 100 nongraduates of comparable age and socioeconomic background, thenumber of ﬁrstborn or only children was 54.(a) Construct an approximate two-sided 90% conﬁdence interval for the diﬀerence inthe proportions of ﬁrstborn or only children for the two populations from whichthese samples were drawn.(b) Based on the interval in part (a), is there evidence that the proportion of ﬁrstbornor only children is higher in the population of college graduates?(c) If the researcher wants to estimate the diﬀerence in proportions to within 0.05with 90% conﬁdence, how many graduates and nongraduates must be interviewed?Q 24 Historically, biology has been taught through lectures, and assessment of learning wasaccomplished by testing vocabulary and memorized facts. A teacher-developed newcurriculum, Biology: A Community Content (BACC), is standards based, activityoriented, and inquiry centered. Students taught using the historical and new methods were tested in the traditional sense on biology concepts that featured biologicalknowledge and process skills. The results of a test on biology concepts were publishedin the The American Biology Teacher and are given in the following table.Pretest: all BACC classesPretest: all traditionalPosttest: all BACC classesPosttest: all traditionalSampleMeanSize13.3837214.0636818.5036516.50298StandardDeviation5.595.458.036.96Johanna G. NeÅ¡lehovÃ¡Statistics MATH 324McGill University, Fall Term 2015Assignment 4: Due November 19 at 11:59 PM(a) Give a 90% conﬁdence interval for the mean posttest score for all BACC students.(b) Find a 95% conﬁdence interval for the diﬀerence in the mean posttest scores forBACC and traditionally taught students.(c) Does the conﬁdence interval in part (b) provide evidence that there is a diﬀerencein the mean posttest scores for BACC and traditionally taught students? Explain.(d) Another similar study is to be undertaken to compare the mean posttest scoresfor BACC and traditionally taught high school biology students. The objective isto produce a 99% conﬁdence interval for the true diﬀerence in the mean posttestscores. If we need to sample an equal number of BACC and traditionally taughtstudents and want the width of the conﬁdence interval to be 1.0, how manyobservations should be included in each group?(e) Repeat the calculations from part (d) if we are interested in comparing the meanpretest scores.(f) Suppose that the researcher wants to construct 99% conﬁdence intervals to compare both pretest and posttest scores for BACC and traditionally taught biologystudents. If her objective is that both intervals have widths no larger than 1 unit,what sample sizes should be used?Q 25 Suppose that the proportion p of defective items in a large population of items isunknown, and that it is desirable to test the following hypotheses:H0 : p = 0.2H1 : p = 0.2.Suppose also that a random sample of n = 20 items is drawn from the population. LetY denote the number of defective items in the sample, and consider a test procedurethat rejects the null hypothesis if either Y ≥ 7 or Y ≤ 1.(a) Are the hypotheses simple or composite? Provide a brief explanation.(b) Write down the critical region of the test.(c) Compute the size of the test and the probability of type I error.(d) Calculate the value of the power function π(p) at the pointsp ∈ {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}and sketch the power function, preferably using R. How does the power functionrelate to the probabilities of type I and type II error?(e) Redo part (d) when the sample size increases to n = 50 and to n = 100. Do youthink the test is any good?(f) Suggest a way to modify the above test so that it remains meaningful at samplesizes other than 20.Johanna G. NeÅ¡lehovÃ¡Statistics MATH 324McGill University, Fall Term 2015Assignment 4: Due November 19 at 11:59 PMQ 26 Suppose that X1 , . . . , Xn form a random sample from a Poisson distribution for whichthe value of the parameter λ is unknown. Let λ0 and λ1 be speciﬁed values such that0 < λ0 < λ1 and suppose that it is desired to test the following simple hypotheses:H0 : λ = λ0H1 : λ = λ1 .To solve the problems below, you can use, without proof, the result that if Z1 and Z2are independent Poisson variables with parameters Âµ1 and Âµ2 , respectively, Z1 + Z2 isPoisson with parameter Âµ1 + Âµ2 .Â¯(a) Show that the optimal test at level α rejects H0 when Xn > c. Specify the value c.Â¯(b) Show that the test δ that minimizes α(δ) + β(δ) rejects H0 when Xn > c∗ . Findthe value c∗ .(c) Suppose that n = 20, λ0 = 1/4, and λ1 = 1/2. Find the value c in part (a) whenα = 0.068 and compute the probabilities of type I and type II error.(d) Suppose that, as in part (c), n = 20, λ0 = 1/4, and λ1 = 1/2. Compute thevalue c∗ in part (b) and determine the minimum value of α(δ) + β(δ) that can beattained. What are the probabilities of type I and type II error?(e) Suppose that n = 20, λ0 = 1/4, and λ1 = 1/2. Construct a randomized test forwhich the probability of type I error is exactly α = 0.05 and the probability oftype II error is the smallest possible among all tests at level α.Q 27 Nutritional information provided by Kentucky Fried Chicken (KFC) claims that eachsmall bag of potato wedges contains 4.8 ounces of food and 280 calories. A sample often orders from KFC restaurants in New York and New Jersey averaged 358 calories.Assume that the sample is approximately normally distributed.(a) If the sample deviation was s = 54 calories, is there suﬃcient evidence to indicatethat the average number of calories in small bags of KFC potato wedges is greaterthan advertised? Clearly formulate the null and the alternative hypotheses, selectan appropriate test and test at the 1% level of signiﬁcance.(b) Compute the p-value of the test conducted in part (a).(c) Construct a 99% upper conﬁdence interval for for the true mean number of calories in small bags of KFC potato wedges.(d) On the basis of the interval computed in part (c), what would you concludeabout the claim that the mean number of calories exceeds 280? How does yourconclusion here compare with your conclusion in part (a) where you conducteda formal test of hypothesis?Johanna G. NeÅ¡lehovÃ¡Statistics MATH 324McGill University, Fall Term 2015Assignment 4: Due November 19 at 11:59 PMQ 28 (optional) Recall the data on losses to content above 1 mio. Danish Kroner arising from ﬁreclaims made to the Copenhagen Re insurance company between 1980 and 1990 usedin Q 21. Recall also that the exponential distribution arises as a special case of thegamma distribution when the shape parameter α = 1.(a) Fit the gamma distribution to the data and construct a 95% conﬁdence intervalfor α. Does it contain the value α = 1? What can you conclude?(b) Test the hypothesis that α = 1 using the Wald test with 0.05 level of signiﬁcance.Compare your conclusion to the conclusion made in part (a).(c) Test the hypothesis that the exponential model provides an adequate simpliﬁcation of the gamma model using a likelihood ratio test with 0.05 level of signiﬁcance. Compare your conclusion to the one obtained in part (b).

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