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Cornell University
New York State School of Industrial and Labor Relations
Department of Social Statistics
Spring Term 2012
ILRST 5110: Statistical Methods for the Social Sciences II
ILRST 2110: Statistical Methods for the Social Sciences II
TEST #1
INSTRUCTIONS: Please answer all questions and write your answers neatly on the
test paper. For each question worth 1 point, a short answer is required that can
typically be read directly from the given computer printout or other information
provided; for these questions, simply report the required answer without further
explanation. For questions worth 2 or more points, additional calculation or
reasoning is necessary, and you are required to show how you derived your answer
by providing a relevant calculation or explanation; for these questions, please make
your answers as brief as possible subject to clarity, and be sure to include any
relevant computer printout (which you should append at the end of the test paper)
if you indeed used the computer in your solution. The test has a total of 105 points.
Remember that the solutions you submit should reflect your own reasoning and not
be copied from elsewhere.
A “hard-copy” of your solutions is due in class on Tuesday, April 3.
1. Write your name here:
2. Write your section day here:
3. Write your section time here:
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PART A. [33 points] A large multinational corporation has hired a team of HRM consultants to
evaluate the effectiveness of the human resource management practices at one of its divisions.
The consultants are famous for having developed a proprietary scale (ENG) that measures
employee engagement on a scale of one to fifty. To assess the level of engagement among the
employees at the division, a number of workers are sampled at random and administered an
extensive questionnaire. Based on the responses to the questionnaire, each of the employees is
assigned an ENG score by the consultants. The following printout shows one part of the statistical
analysis performed by the consultants on the data they obtained:
One-Sample T: ENG
Test of mu = 36.4 vs not = 36.4
Variable N Mean StDev SE Mean 95% CI T P
ENG 100 35.200 6.500 0.650 (33.910, 36.490) -1.85 0.068
In the questions that follow, let ? be the mean of the population of ENG scores for workers at
the division, and let ?2 be the population variance, so ? denotes the population standard
deviation. Similarly, let y denote the mean of the sample of ENG scores, and let 2
s denote the
sample variance, so s denotes the sample standard deviation.
1. [1 point] What is the sample size, i.e., how many employees were chosen at random and
administered the questionnaire?
2. [1 point] What is the value of the sample mean y ?
3. [1 point] Give an estimate of the population mean ? .
4. [1 point] What is the value of the sample standard deviation s ?
5. [2 points] What is the value of the sample variance 2
s ?
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6. [1 point] Give an estimate of the population standard deviation ? .
7. [1 point] Give an expression for the standard deviation of y in terms of the population
standard deviation ? .
8. [1 point] Give an estimate of the standard deviation of y .
9. [1 point] Give a 95% confidence interval for the population mean ? .
10. [3 points] Construct a 90% confidence interval for ? .
11. [1 point] State the null hypothesis being tested in the printout.
12. [1 point] State the alternative hypothesis being used for the test in the printout.
13. [1 point] Give the value of the test statistic for the hypothesis test performed in the printout.
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14. [1 point] Give the p-value for the hypothesis test performed in the printout.
15. [1 point] What distribution is being used to compute the p-value for the hypothesis test
performed in the printout, i.e., what is the assumed distribution of the test statistic under the null
hypothesis?
16. [2 points] For the test performed in the printout, would the null hypothesis be rejected in
favor of the alternative hypothesis in a test at the 5% level?
17. [2 points] Give the p-value for testing the null hypothesis ? ? 0 H : 36.4 against the alternative
hypothesis : 36.4 ? ? Ha .
18. [2 points] Would the null hypothesis ? ? 0 H : 36.4 be rejected in favor of the alternative
hypothesis : 36.4 ? ? Ha in a test at the 5% level?
19. [2 points] Give the p-value for testing the null hypothesis ? ? 0 H : 36.4 against the alternative
hypothesis : 36.4 ? ? Ha .
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20. [2 points] Would the null hypothesis ? ? 0 H : 36.4 be rejected in favor of the alternative
hypothesis : 36.4 ? ? Ha in a test at the 5% level? (Justify your answer briefly.)
21. [2 points] During a previous study commissioned by the corporation, the consultants
determined by a complete census that the population mean of the ENG scores for the employees
in a different, much smaller division which uses radically different HRM practices is 33.7 . The
corporation is interested in knowing if the population mean of the ENG scores for the division
currently under study is greater than that for the division that was investigated previously.
Consequently, the consultants decide to test the null hypothesis ? ? 0 H : 33.7 . What is the value
of the test statistic for testing this null hypothesis?
22. [3 points] The consultants decide to test the null hypothesis ? ? 0 H : 33.7 against the
alternative : 33.7 ? ? Ha . Would the null hypothesis be rejected in favor of the alternative
hypothesis in a test at the 1% level?
PART B. [32 points] In addition to measuring the engagement of each of the sampled
employees, the consultants also measured the amount of supervisor support that each of the
sampled employees perceive they receive. The amount of supervisor support is measured by a
SUP score on a scale of one to fifty. Using the ENG and SUP scores, the consultants performed the
following simple linear regression analysis:
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Regression Analysis: ENG versus SUP
The regression equation is
ENG = 14.2 + 0.600 SUP
Predictor Coef SE Coef T P
Constant 14.200 2.646 5.37 0.000
SUP 0.60000 0.07420 8.09 0.000
S = 5.05966 R-Sq = 40.0% R-Sq(adj) = 39.4%
Analysis of Variance
Source DF SS MS F P
Regression 1 65.39 0.000
Residual Error 98 2508.8 25.6
Total 99 4182.8
Predicted Values for New Observations
New
Obs Fit SE Fit 95% CI 95% PI
1 39.400 0.725 (37.961, 40.839) (29.256, 49.543)
Values of Predictors for New Observations
New
Obs SUP
1 42.0
In the questions that follow, let the population regression line be denoted by ? ? ? ? 0 1 Ey x x (|) ,
where the response variable y is ENG and the predictor variable x is SUP. Recall that the simple
linear regression model assumes constant variance, i.e., ??2 Var y x (|) .
1. [1 point] Give the fitted least-squares regression line.
2. [1 point] Give the value of b0 , the estimate of ? 0 , the population intercept.
3. [1 point] Give the value of b1 , the estimate of ?1 , the population slope.
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4. [1 point] Estimate the change in the mean ENG score per one-point increase in SUP score.
5. [2 points] Estimate the mean of the population of ENG scores for employees who have an SUP
score of 35 .
6. [1 point] Estimate the standard deviation of the population of ENG scores for employees who
have an SUP score of 35 .
7. [1 point] Estimate the standard deviation of b0 .
8. [1 point] Give the value of the test statistic for testing the null hypothesis ? ? 0 1 H : 0 .
9. [1 point] Which distribution is used to calculate the p-value for testing ? ? 0 1 H : 0 ?
10. [2 points] Would you reject the null hypothesis ? ? 0 1 H : 0 in favor of the alternative
hypothesis ? ?1 : 0 Ha in a test at the 1% level?
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11. [2 points] Is there evidence that the SUP scores are useful for explaining the variability in the
ENG scores?
12. [1 point] What is the amount of variability in the ENG scores that is left unexplained after the
SUP scores are taken into account through a linear relationship?
13. [2 points] What is the amount of variability in the ENG scores that is explained by taking the
SUP scores into account through a linear relationship?
14. [1 point] What is the proportion of variability in the ENG scores that is explained by taking
the SUP scores into account through a linear relationship?
15. [1 point] What is the coefficient of determination for the regression?
16. [2 points] Give the value of the correlation coefficient between the ENG and SUP scores.
17. [3 points] Observation 9 in the data set has SUP 39.3 ? and ENG 35.6 ? . What are the fitted
value and the residual for this observation?
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18. [3 points] Construct a 95% confidence interval for ?1 .
19. [1 point] Give a 95% confidence interval for the mean ENG score of employees who have an
SUP score of 42 .
20. [2 points] What is the value of the test statistic for testing ? ? 0 H Ey x : ( | 42) 37 , i.e., for
testing the null hypothesis that the mean ENG score is 37 for employees having an SUP score of
42 ?
21. [1 point] Suppose that a randomly chosen employee has an SUP score of 42 . Give a point
prediction of the employee’s ENG score.
22. [1 point] Give a 95% prediction interval for the ENG score of a randomly chosen employee
whose SUP score is 42 .
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PART C. [33 points] A team of Human Resource consultants, who are investigating the salaries
of recent college graduates working in the area of software development, fit the following
regression model to their data:
? ? 01 2 3 4 5 6 7 E sal yr G E G E G E yr G yr E yr G E yr | ,, * * * ** ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ,
where
annual salary in tens of thousands of dollars per year;
gender (male 0; female 1);
educational background (computer science 0; engineering 1);
completed years of work experience,
sal
G
E
yr
?
? ??
? ??
?
and the variables G E* , G yr * , E yr * , and G E yr * * are interaction variables. The following
printout shows the results of the regression analysis:
Regression Analysis: sal versus G, E, G*E, yr, G*yr, E*yr, G*E*yr
The regression equation is
sal = 60.9 + 4.51 G + 8.17 E - 1.92 G*E + 5.44 yr - 2.09 G*yr + 1.67 E*yr
- 0.28 G*E*yr
Predictor Coef SE Coef T P
Constant 60.909 1.948 31.27 0.000
G 4.515 2.977 1.52 0.130
E 8.174 2.964 2.76 0.006
G*E -1.923 4.772 -0.40 0.687
yr 5.4388 0.5270 10.32 0.000
G*yr -2.0897 0.8368 -2.50 0.013
E*yr 1.6710 0.8175 2.04 0.041
G*E*yr -0.282 1.353 -0.21 0.835
S = 14.1002 R-Sq = 43.5% R-Sq(adj) = 42.7%
Analysis of Variance
Source DF SS MS F P
Regression 7 78280 11183 56.25 0.000
Residual Error 512 101794 199
Total 519 180074
1. [2 points] How many subjects are included in the data set?
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2. [2 points] What is the fitted regression relationship between sal and yr for males (G ? 0 ) who
are computer science graduates (E ? 0 )?
3. [2 points] What is the estimate of the mean starting salary for male computer science
graduates, i.e., what is the estimate of E sal yr G E ( | 0, 0, 0) ? ? ? ?
4. [3 points] Give a 95% confidence interval for the mean starting salary for males who are
computer science graduates, i.e., give a 95% confidence interval for E sal yr G E ( | 0, 0, 0) ? ?? .
5. [2 points] What is the estimate of the rate of change in mean salary of male computer science
graduates per year of work experience, i.e., how is E sal yr G E ( | , 0, 0) ? ? estimated to change as
yr is increased by a single unit?
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6. [3 points] What is the fitted regression relationship between sal and yr for females (G ?1)
who are computer science graduates (E ? 0 )?
7. [2 points] Among computer science graduates, estimate the difference between the mean
starting salary for females and the mean starting salary for males.
8. [2 points] Among computer science graduates, is there a statistically significant difference
between the mean starting salary for females and the mean starting salary for males? Perform a
test at the 5% level.
9. [3 points] Among computer science graduates, estimate the difference between males and
females in the rate of change of mean salary per year of work experience.
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10. [2 points] Among computer science graduates, is there a statistically significant difference
between males and females in the rate that mean salary changes per year of work experience?
Perform a test at the 5% level.
11. [3 points] Estimate the mean salary of a male engineering graduate having four years of work
experience. (Justify your answer briefly.)
12. [3 points] Estimate the mean salary of a female engineering graduate having four years of
work experience. (Justify your answer briefly.)
13. [4 points] What is the value of the test statistic for testing “the significance of the regression”?
State the null hypothesis in this test. What is the alternative hypothesis? Would the null
hypothesis be rejected in a test at the 1% level?
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PART D. [7 points] The consultants are especially interested in testing whether the effect of
gender is the same for computer science graduates as it is for engineering graduates.
Consequently, they fit the “reduced” regression model
? ? 01 2 3 4 5 E sal yr G E G E yr G yr E yr | ,, * * ?? ? ? ? ? ?? ? ? ? ? ,
thereby obtaining the following printout:
Regression Analysis: sal versus G, E, yr, G*yr, E*yr
The regression equation is
sal = 61.3 + 3.68 G + 7.34 E + 5.47 yr - 2.21 G*yr + 1.59 E*yr
Predictor Coef SE Coef T P
Constant 61.289 1.776 34.52 0.000
G 3.684 2.323 1.59 0.113
E 7.344 2.320 3.17 0.002
yr 5.4656 0.4848 11.27 0.000
G*yr -2.2071 0.6569 -3.36 0.001
E*yr 1.5894 0.6505 2.44 0.015
S = 14.0893 R-Sq = 43.3% R-Sq(adj) = 42.8%
Analysis of Variance
Source DF SS MS F P
Regression 5 78041 15608 78.63 0.000
Residual Error 514 102033 199
Total 519 180074
1. [3 points] What is the value of the test statistic for testing the null hypothesis that the reduced
model holds, i.e., for testing the null hypothesis that the coefficients of G E* and G E yr * * are
simultaneously zero in the full model of PART A? (Justify your answer briefly.)
2. [4 points] Would the reduced model be rejected in favor of the full model of PART A?
Perform a test at the 5% level. (Justify your answer briefly.)