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M12 Q21 Professor Gill has taught General Psychology for many years. During the semester, she gives...

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

Solutions

Expert Solution

Answer:----                                                             Date:---30/07/2019


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