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Unformatted text preview: STAT 252
Solutions to Practice Exam 1 1) For each of the following questions, provide a formula or name a method which might be used to
help answer the question. If none of the methods we’ve discussed this term so far will help
answer the question, you should indicate an answer of “not covered.” a) Does the average life expectancy differ by region in the US? Independent samples t—test (or z-test since the sample sizes are not given) about the difference in
population means (or one-way ANO VA) b) How much higher is the average blood pressure of men than of women.
Independent samples confidence interval about the difference in population means c) Is major related to how likely a student is to have attended a Cal Poly football game?
Not Covered Yet d) How much is the average rent paid by Cal Poly student who live off—campus?
Conﬁdence interval for a population mean e) Is the average starting salary of students who graduate with an undergraduate degree in Finance higher than the average starting salary of students who graduate with an undergraduate degree in Mathematics? Independent samples t—test (or z-test since the sample sizes are not given) about the dr'ﬁerence in
population means i) Based on pre-test and post-test (Graduate Management Admission Test) GMAT scores of people who took the Princeton Review GMAT course, does the course have a positive effect
on GMAT scores? Paired samples t-test about the difference in population means 2) Starting annual salary for individuals entering the public accounting and ﬁnancial planning
professions were presented in Fortune, June 26, 1995. The summary statistics for the starting
salaries for a sample of 12 public accountants and 14 ﬁnancial planners are shown in the table
below. Data are in thousands of dollars. Profession N Mean Std. Deviation Public Accountant 30.517 3.347
I]! 27-000 a) State the null and alternative (in words and symbols) for testing if there is a difference
between the population mean starting annual salaries for the two professions. H 0 : pi = ,uz The average salaries of all public accountants and ﬁnancial planners are the same H a : ,ul at M The average salaries of all public accountants and ﬁnancial planners are different b) Calculate the test statistic and ﬁnd an approximate p-value for the test. Assume that the
population variances are equal. 2 _ (12—1)3.3472 +(14—1)2.6412
1’ 12+14—2 t = 30.517—27 =2‘99 obs
8.9125 i + i
12 14 Note that the df = 12 + 14 - 2 = 24, so since p — value = 2P(t >| 2.99 |) and P(t > 2.99) is
between .001 and .005: S = 8.9125 .002 < p — value < .01 c) Using a = .1 , what is your decision about the average starting salaries of public accountants
versus ﬁnancial planners? Since p—value is less than .1, reject H 0 and conclude at the .1 level that the average starting
salaries of public accountants is diﬂ'erent from that of ﬁnancial planners. 3) A large group of learning—disabled college freshmen who experience debilitating anxiety before
major tests were matched on an index of test anxiety. Members of these matched pairs were
randomly assigned to two diﬂ‘erent groups. The ﬁrst group was given two weeks of relaxation
exercises (Relax). The second group of students was given two weeks of study skills training
(Study). Using the data below determine if there is a signiﬁcant difference between their ﬁnal exam scores.
N Mean SE Mean StDev
Relax 11 47.636 0.778 2.580
Study 11 50364 0.560 1.859
Difference 11 —2.73 1.14 3.80 a) Find the 95% conﬁdence interval for the mean difference between their ﬁnal exam scores. This is paired data so the CI for the mean difference is given by: £1" i rm sd = -2.73 i 2.2283 = (—5.28,—.177) E ii b) Interpret the CI and state a conclusion about average ﬁnal exam scores for those who are
given relaxation exercises and those who are given study skills training. We can be 95% conﬁdent that the unknown mean difference between the ﬁnal exam scores of
those who were given relaxation exercises and those who were given study skills training is
between -5. 28 and -. 177. We can conclude that there is a signiﬁcant diﬁ‘erence in the mean ﬁnal
exam scores between the two groups. c) What assumption about the diﬁ'erences in the scores do we need to make in order for the CI to
be valid? Assume that the population of differences between the ﬁnal scores of those who were given
relaxation exercises and those who were given study skills training is approximately normally
distributed. d) Now suppose we had incorrectly analyzed this data with an independent samples procedure.
Compute the conﬁdence interval for the difference in average ﬁnal exam scores. Note that
under the assumption of equal population variances, the pooled estimator of the variance would be S: = 5.056. i + = 47.636 — 50.364 i 2.086 5.056(3— + "1 "2 11 11 = (—4.728,—.728) e) How does your answer in ((1) compare to your answer in (a)?
Smaller width of the CI when we inconectly assumed independent samples. 4) In a study of cereal leaf beetle damage on oats, researchers measured the number of beetle larvae
per stem in small plots of oats after randomly applying one of two treatments: no pesticide, or the
pesticide malathion at a rate of .25 of a pound per acre. The summary statistics are given below: a) Construct the 99% conﬁdence interval for the difference in the mean number of beetle larvae
per stem between those that have and not been subjected to malathion. Assume unequal
variances. Note that df= min(13-1, 14-1) = 12 2 2 2 2
_ _ s s 1.21 .52
xl—xzitm,2 —l+—2-=3.47—1.36-|_-3.055 +
n1 n2 1 3 14
Conclude that the mean number of beetle larvae on the stems not treated is different from the
mean number of larvae on the stems that were treated with the pesticide Malathion. = (1,322) b) What do we need to assume about the distributions of number of beetle larvae per stem in
order to construct the interval in part (a)? Assume that the populations of the number of beetle larvae on the treated and untreated stems
are approximately normally distributed. 5) Researchers were interested in the weight gain of rats who were subjected to a high protein diet.
The MINITAB output below displays the summary statistics for the weight gains (in grams) of 60
female rats who were subjected to the diet. Variable N Mean StDev
Weight Gain 60 120.00 21.39 a) Set up the null and alternative hypothesis (in words and symbols) for testing whether the
average increase is more than 100 grams. H O 2 ,u = 100 The average weight gain is equal to 100 grams
H a : ,u > 100 The average weight gain is more than 100 grams b) Calculate the test statistic and p—value for the test. Since we have a large sample the test statistic is given by: _ 120—100 = 7.24 z ___—_.___.
“’5 21.39/J66 The p-value is P(z > 7.24) z 0 obs c) Make a decision and conclusion based on a level of signiﬁcance at = .01 . Since the p—value is less than .01, we would reject the null hypothesis and conclude at the .01
level that the average weight gain for the rats on the diet is more than 100 grams. 6) A university wanted to investigate the differences in the number of semester courses in
mathematics its students in three categories of majors took in high school. The three categories of
majors under study are: Computer science (CS), Science or Engineering (Sci/Eng), and Other.
Random samples of 103, 31, and 122 students in each of these majors were selected and the
following summary statistics on the number of semester courses in mathematics the student took
in high school were obtained: IEHIMMIEME
103 31 122
8.74 8.65 8.25
1.28 1.31 1.17 The number of semester mathematics courses students took in high school. a) What is the response variable? b) What is the factor and what are its levels?
College major with 3 levels: computer science, science or engineering, and other
C) Is this an observational or designed experiment? Brieﬂy discuss. This is an observational experiment because the university is randomly selecting students by
major and observing the number of math courses that the students took in high school. It has not
assigned the students (experimental units) to the particular major (treatment). d) We would like to conduct a one-way ANOVA to investigate the differences in the average
number of math courses taken by students in the different majors. Deﬁne the parameters of
interest, and write out the null and alternative hypotheses for the ANOVA test. ,u1 = The average number of mathematics courses taken by all computer science majors
,u2 = The average number of mathematics courses taken by all science and engineering majors
,u3 = The average number of mathematics courses taken by all other majors H 0 1/11 = #2 = #3
H a : The average number of math courses taken is dijﬁrent for at least two majors e) The overall sample mean number of mathematics courses taken in high school is given by
J? = 8.5 . Calculate the F-statistic for the one-way ANOVA test. SSW =103(8.74 — 8.5)2 + 31(8.65 — 8.5)2 +122(8.25 — 8.5)2 = 14.255 SSEW = (103 — 1)(1.282) + (31 — 1)(1.312) + (122 —1)(1.172) = 384.237 MST = = 7.1275
MSE = 384.237 2 1.5187
256 — 3
The F-statistic is given by: F = 7'1275 = 4.69
1.5187 1) Based on your answer in part (e), make a decision and give a conclusion in the context of the
problem. Let the level of signiﬁcance be a = .05
The critical value is given by: F05 taken from the F-distribution with dfmm = 3 —1 = 2 and dfden = 256 — 3 = 253 (since 253 is not on the chart, use dfdm = 120). So F05 = 3.07. Since Fm = 4.69 > F05 = 3.07, reject the null hypothesis. Conclude at the .05 level that the average number of high scth mathematics courses taken
differs for at least two of the majors. ...
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This homework help was uploaded on 02/13/2008 for the course STAT 252 taught by Professor Staff during the Spring '05 term at Cal Poly.
- Spring '05