In: Math
M12 Q21
Professor Gill has taught General Psychology for many years. During the semester, she gives three multiple-choice exams, each worth 100 points. At the end of the course, Dr. Gill gives a comprehensive final worth 200 points. Let x1, x2, and x3 represent a student's scores on exams 1, 2, and 3, respectively. Let x4 represent the student's score on the final exam. Last semester Dr. Gill had 25 students in her class. The student exam scores are shown below.
x1 | x2 | x3 | x4 |
73 | 80 | 75 | 152 |
93 | 88 | 93 | 185 |
89 | 91 | 90 | 180 |
96 | 98 | 100 | 196 |
73 | 66 | 70 | 142 |
53 | 46 | 55 | 101 |
69 | 74 | 77 | 149 |
47 | 56 | 60 | 115 |
87 | 79 | 90 | 175 |
79 | 70 | 88 | 164 |
69 | 70 | 73 | 141 |
70 | 65 | 74 | 141 |
93 | 95 | 91 | 184 |
79 | 80 | 73 | 152 |
70 | 73 | 78 | 148 |
93 | 89 | 96 | 192 |
78 | 75 | 68 | 147 |
81 | 90 | 93 | 183 |
88 | 92 | 86 | 177 |
78 | 83 | 77 | 159 |
82 | 86 | 90 | 177 |
86 | 82 | 89 | 175 |
78 | 83 | 85 | 175 |
76 | 83 | 71 | 149 |
96 | 93 | 95 | 192 |
Since Professor Gill has not changed the course much from last semester to the present semester, the preceding data should be useful for constructing a regression model that describes this semester as well.
(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)
x | s | CV | |
x1 | % | ||
x2 | % | ||
x3 | % | ||
x4 | % |
Relative to its mean, would you say that each exam had about the same spread of scores? Most professors do not wish to give an exam that is extremely easy or extremely hard. Would you say that all of the exams were about the same level of difficulty? (Consider both means and spread of test scores.)
No, the spread is different; Yes, the tests are about the same level of difficulty.
Yes, the spread is about the same; Yes, the tests are about the same level of difficulty.
No, the spread is different; No, the tests have different levels of difficulty.
Yes, the spread is about the same; No, the tests have different levels of difficulty.
(b) For each pair of variables, generate the correlation
coefficient r. Compute the corresponding coefficient of
determination r2. (Use 3 decimal places.)
r | r2 | |
x1, x2 | ||
x1, x3 | ||
x1, x4 | ||
x2, x3 | ||
x2, x4 | ||
x3, x4 |
Of the three exams 1, 2, and 3, which do you think had the most influence on the final exam 4? Although one exam had more influence on the final exam, did the other two exams still have a lot of influence on the final? Explain each answer.
Exam 3 because it has the highest correlation with Exam 4; No, the other 2 exams do not have a lot of influence because of their low correlations with exam 4.
Exam 2 because it has the lowest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.
Exam 3 because it has the highest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.
Exam 1 because it has the highest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.
(c) Perform a regression analysis with x4 as
the response variable. Use x1,
x2, and x3 as explanatory
variables. Look at the coefficient of multiple determination. What
percentage of the variation in x4 can be
explained by the corresponding variations in
x1, x2, and
x3 taken together? (Use 1 decimal place.)
%
(d) Write out the regression equation. (Use 2 decimal places.)
x4 = | + x1 | + x2 | + x3 |
Explain how each coefficient can be thought of as a slope.
If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."
If we hold all explanatory variables as fixed constants, the intercept can be thought of as a "slope."
If we look at all coefficients together, the sum of them can be thought of as the overall "slope" of the regression line.
If we look at all coefficients together, each one can be thought of as a "slope."
If a student were to study "extra hard" for exam 3 and increase his
or her score on that exam by 13 points, what corresponding change
would you expect on the final exam? (Assume that exams 1 and 2
remain "fixed" in their scores.) (Use 1 decimal place.)
(e) Test each coefficient in the regression equation to determine
if it is zero or not zero. Use level of significance 5%. (Use 2
decimal places for t and 3 decimal places for the
P-value.)
t | P-value | |
β1 | ||
β2 | ||
β3 |
Conclusion
We reject all null hypotheses, there is insufficient evidence that β1, β2 and β3 differ from 0.
We reject all null hypotheses, there is sufficient evidence that β1, β2 and β3 differ from 0.
We fail to reject all null hypotheses, there is sufficient evidence that β1, β2 and β3 differ from 0.
We fail to reject all null hypotheses, there is insufficient evidence that β1, β2 and β3 differ from 0.
Why would the outcome of each hypothesis test help us decide
whether or not a given variable should be used in the regression
equation?
If a coefficient is found to be not different from 0, then it contributes to the regression equation.
If a coefficient is found to be different from 0, then it does not contribute to the regression equation.
If a coefficient is found to be not different from 0, then it does not contribute to the regression equation.
If a coefficient is found to be different from 0, then it contributes to the regression equation.
(f) Find a 90% confidence interval for each coefficient. (Use 2
decimal places.)
lower limit | upper limit | |
β1 | ||
β2 | ||
β3 |
(g) This semester Susan has scores of 68, 72, and 75 on exams 1, 2,
and 3, respectively. Make a prediction for Susan's score on the
final exam and find a 90% confidence interval for your prediction
(if your software supports prediction intervals). (Round all
answers to nearest integer.)
prediction | |
lower limit | |
upper limit |