Question

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 x₃represent 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.)

Yes, the spread is about the same; No, the tests have different levels 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.No, the spread is different; Yes, the tests are about the same level 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; 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; No, the other 2 exams do not have a lot of influence because of their low 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.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.

(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 look at all coefficients together, each one can be thought of 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 a student were to study "extra hard" for exam 3 and increase his or her score on that exam by 5 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 fail to reject all null hypotheses, there is insufficient 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 reject all null hypotheses, there is sufficient 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 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 contributes to the regression equation.    If a coefficient is found to be different from 0, then it contributes to the regression equation.If a coefficient is found to be not different from 0, then it does not contribute 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

Answer 1