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# MAT222-2005Fall - MAT222 FINAL EXAM Fall 2005 STUDENT NAME...

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Unformatted text preview: MAT222 FINAL EXAM Fall 2005 STUDENT NAME: Instructions: Do all your work on this exam. Unsupponed answers may receive little or no credit. Be sure to give a complete solution to each problem. You may use a calculator. No books or notes are allowed. Do not share calculators. r.13robl em 1. Certain dosages of a new drug developed to reduce a smoker’s reliance on nicotine may reduce one’s pulse rate to dangerously low levels. To investigate the drug’s effect on pulse rate, different dosages ofthe drug were administrated to six randomly selected patients, and 30 minutes later the decrease in each patient’s pulse rate was recorded. Dosage x: 2.0 1.5 3.0 2.5 4.0 3.0 Decrease in pulse rate y: 15 9 18 16 23 20 For this data, 36 = 2.667, f : 16.83, 3x :.876, sy =4.79, r= .961. (a) Is there evidence ofa linear relationship between drug dosage and change in pulse rate? Test at the 0t 2 .10 level. (b) Find a 95% confidence interval for the slope ,61. (c) Find a 99% prediction interval for the decrease in pulse rate corresponding to a dosage of3.5 cc ofthe drug. 2. Particle size is an important property of latex paint and is varied to produce different ﬁnishes corresponding to different mean particle sizes. After adjusting the machine, particle size samples must be taken to determine the mean particle size being produced in order to grade the paint. The particles are normally distributed with standard deviation 100 angstroms. (a) Find the sample size necessary for a 95% CI for the mean particle size with margin of error less than 100 angstroms. You measure particles from a SRS of size 50 and determine the sample mean to be i = 2650. (b) Find a 95% conﬁdence interval for the mean particle size y. (c) Test the null hypothesis H0 2p : 2500 versus Ha :p i 2500 at the a z .05 level. 3. Does cocaine use by pregnant women cause their babies to have low birth weight? To study this question, birth weights of babies of women who tested positive for cocaine/crack during a drug-screening test were compared with the birth weights for women who either tested negative or were not tested, a group we call “other”. Here are the summary statistics. The birth weights are measured in grams. Group n 35 5 Positive test 134 2733 599 Other 5974 3118 672 (a) Formulate appropriate hypotheses to answer the question. (b) Assuming the two populations have different variances, use an appropriate test to test the null hypotheses that the mean cocaine use for the two groups are the same versus the alternative hypotheses that they are not the same. (0) Assuming the two populations have the same variance, use an appropriate test to test the null hypotheses that the mean cocaine use for the two groups are the same versus the alternative hypotheses that they are not the same. ((1) Give a 95% conﬁdence interval for the mean difference in birth weights. 4. A test question is considered good ifit differentiates between prepared and unprepared students. The first question on a test was answered correctly by 62 of80 prepared students and by 26 of 50 unprepared students. Perform a signiﬁcance test for the null hypothesis H0 .‘ p1 = p2 versus H{7 .‘ p1 > p2 where p1 is the population proportion of prepared students and [72 is the population proportion ofunprepared students. What do you conclude? 5. A researcher investigated the Spring break destinations of 50 college Freshmen and 60 college Seniors to see ifthey changed over time. The data are below. Freshmen Seniors Beach 1 1 17 Home 19 21 Other 20 22 (3) Use column percentages to describe the relationship between class and Spring break destination. Do you notice any trends? (b) Carry out a signiﬁcance test on the relationship between class and destination. 6. The following MINITAB output is the result ofmultiple regression analysis ofthc Cheese data described in the Data Appendix ofour test book. The variable “Case” is used to number the observations from 1 to 30. “taste” is the response variable ofinterest. Three ofthe chemicals whose concentrations were measured were Actic Acid, Hydrogen Sulﬁde (or H28), and Lactic Acid. The regression equation is taste 2 v 28.9 + 0.33 acetic + 3.91 H28 + 19.7 lactic Predictor Coef StDev T P Constant 728.88 19.74 —l.46 0.155 acetic 0.328 4.460 0.07 0.942 H25 3.912 1.248 3.13 0.004 lactic 19.671 8.629 2.28 0.031 S = 10.13 R—Sq : 65.2% R—Sq(adj) = 61.2% Analysis of Variance Source DF SS MS F P Regression 3 4994.5 1664.8 16.22 0.000 Residual Error 26 2668 4 102.6 Total 29 7662.9 (a) Give the model for the mean taste score for cheeses having Actic Acid : 4.5, HZS = 5.5, and Lactic Acid : l.5. (b) Calculate the estimate ofthis taste mean using the results given in the Minitab output. (c) What is the value of s, the estimate of o. (d) State the null hypothesis and alternative hypothesis by the ANOVA F statistic for this problem. After stating the hypotheses in symbols, explain them in words. (e) What is the distribution ofthe F statistic under the null hypothesis? What conclusion do you draw from the F test? (I) What percent ofthe variation in Cheese is explained by these three chemical concentration variables? (g) Calculate a 95% confidence interval for the regression coefficient of Actic Acid. (2 points each) 7. At what age do babies learn to crawl? Does it take longer to leam in the winter, when babies are often bundled in clothes that restrict their movement? Data were collected from parents who brought their babies into the University of Denver Infant Study Center to participate in one of a number of experiments between 1988 and 1991. Parents reported the birth month and the age at which their child was ﬁrst able to creep or crawl a distance of four feet within one minute. The resulting data were grouped by month of birth. The data are for January, May, and September. (2 points each for #1 —#6 and 8 points for #7.) Average Birth Month Crawling Age SD n January 2984 7.08 32 May 2858 8.07 27 September 33.83 6.93 38 Crawling age is given in weeks. Assume the data are 3 independent SRSs, one from each ofthe 3 populations (babies born in a particular month) and that the populations of crawling ages have normal distributions. A partial ANOVA table is given below. Analysis of Variance for crawling age. Source df Sums of Squares Mean Square F—ratio Birth Month 505.26 __ Error __ 53.45 Total #1. For this example, we notice A) this is a randomized, designed experiment. B) the data show no strong evidence ofa violation ofthe assumption that the 3 populations have the same standard deviation. C) ANOVA cannot be used on these data because the sample sizes are different. D) the data show very strong evidence ofa violation ofthe assumption that the 3 populations have the same standard deviation. Ans: #2. The degrees of freedom for birth month (group) are A) 2. B) 3. C) 94. D) 97. Ans: #3. The degrees of freedom for error are A) 2. B) 3. C) 94. D) 97. Ans: #4. The null hypothesis for the ANOVA F test is that the population mean crawling ages are equal for all three birth months. The altemative hypothesis is A) that the population mean crawling age is larger for January than for the other two months. B) that the population mean crawling age is larger for May than for the other two months. C) that the population mean crawling age is larger for September than for the other two months. D) none ofthe above. Ans: #5. The value ofthe ANOVA F-statistic for testing equality ofthe population means of the 3 birth months is A) 3.15. B) 4.73. C) 6.30. D) 9.45. Ans: #6. The P»value for the ANOVA F test for testing equality ofthe population means oftlie 3 birth months is A) less than 0.001. B) between 0.001 and 0.010. C) between 0.010 and 0.025. D) greater than 0.025. Ans: #7. Suppose we are interested in the contrast that compares the average crawling age for babies born in May and September. (a) What is the sample contrast? (b) What is the standard error for this contrast? (c) Find the test statistic and approximate p-value. What do you conclude? (d) Find a 90% conﬁdence interval for this contrast. ...
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MAT222-2005Fall - MAT222 FINAL EXAM Fall 2005 STUDENT NAME...

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