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Unformatted text preview: MAT222 FINAL EXAM Spring 2003 STUDENT NAME: Instructor’s name:
( ) Buzaianu
( ) Chen ( ) Lynn Instructions: Do all your work on this exam. Unsupported 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. l. The following MINITAB output is the result of multiple regression analysis of the
Cheese data described in the Data Appendix of our test book. The variable “Case" is
used to number the observations from 1 to 30. “taste” is the response variable of
interest. Three of the chemicals whose concentrations were measured were Actic Acid, Hydrogen Sulfide (or H28), and Lactic Acid. The regression equation is taste 2 — 28.9 + 0.33 acetic + 3.91 H23 + 19.7 lactic
Predictor Coef StDev T P
Constant ~28.88 19.74 —1.46 0.155
acetic 0.328 4.460 0.07 0.942
HZS 3.912 1.248 3.13 0.004
lactic 19.671 8.629 2.28 0.031
S = 10.13 RSq = 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 Source DF Seq SS acetic 1 2314.1 H28 1 2147.0 lactic 1 533.3 (a) Give the model for the mean taste score for cheeses having Actic Acid : 4.5, H2S = 5.5, and Lactic Acid = 1.5.
(b) Calculate the estimate of this taste mean using the results given in the Minitab output. (c) What is the value of s, the estimate of 0'.
(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 of the F statistic under the null hypothesis? What conclusion do you draw from the F test?
(f) What percent of the 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) 2, At what age do babies learn to crawl? Does it take longer to learn 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 #l—#6 and 8 points for #7.) Average
Birth Month Crawling Age SD n
January 29.84 7.08 32
May 28.58 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
of the 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 of a violation of the 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 of a violation of the 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: .anﬂﬂ #4. The null hypothesis for the ANOVA F test is that the population mean crawling ages
are equal for all three birth months. The alternative 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 of the above.
Ans: #5. The value of the ANOVA F—statistic for testing equality of the 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 of the population means of the 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? ((1) Find a 90% confidence interval for this contrast. 3. 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 of the 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, )7: 2.667, y: 16.83, 5x 2.876, sy 24.79, r= .961. (a) Is there evidence of a linear relationship between drug dosage and change in pulse
rate? Test at the OL = .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 of 3.5 cc of the drug. 4. 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.
(3) 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 )7 = 2650.
(b) Find a 95% confidence interval for the mean particle size [I . (c) Test the null hypothesis H0 :u = 2500 versus Ha :u ¢ 2500 at the a = .05 level. 5. 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 drugscreening 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 )7 5
Positive test 134 2733 599
Other 5974 31 18 672 (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. (c) 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. 6. A test question is considered good if it differentiates between prepared and
unprepared students. The first question on a test was answered correctly by 62 of 80
prepared students and by 26 of 50 unprepared students. Perform a significance test for the null hypothesis H0 . p] = p2 versus Ha : p] > p2 where p1 is the population proportion of prepared students and p2 is the population proportion of unprepared
students. What do you conclude? 7. A researcher investigated the Spring break destinations of 50 college Freshmen and
60 college Seniors to see if they changed over time. The data are below. Freshmen Seniors
Beach 1 l 17
Home 19 21
Other 20 22
(a) 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. ...
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