M12 Q21

Professor Gill has taught General Psychology for many years.During the semester, she gives three multiple-choice exams, eachworth 100 points. At the end of the course, Dr. Gill gives acomprehensive final worth 200 points. Let x₁,x₂, and x₃ represent astudent's scores on exams 1, 2, and 3, respectively. Letx₄ represent the student's score on the finalexam. Last semester Dr. Gill had 25 students in her class. Thestudent 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 lastsemester to the present semester, the preceding data should beuseful for constructing a regression model that describes thissemester as well.

(a) Generate summary statistics, including the mean and standarddeviation of each variable. Compute the coefficient of variationfor 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 thesame spread of scores? Most professors do not wish to give an examthat is extremely easy or extremely hard. Would you say that all ofthe exams were about the same level of difficulty? (Consider bothmeans and spread of test scores.)

No, the spread is different; Yes, the tests are about the samelevel of difficulty.

Yes, the spread is about the same; Yes, the tests are about thesame level of difficulty.     

No, the spread is different; No, the tests have different levelsof difficulty.

Yes, the spread is about the same; No, the tests have differentlevels of difficulty.

(b) For each pair of variables, generate the correlationcoefficient r. Compute the corresponding coefficient ofdetermination 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 mostinfluence on the final exam 4? Although one exam had more influenceon the final exam, did the other two exams still have a lot ofinfluence 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 theirlow 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 theirhigh 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 theirhigh 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 theirhigh correlations with exam 4.

(c) Perform a regression analysis with x₄ asthe response variable. Use x₁,x₂, and x₃ as explanatoryvariables. Look at the coefficient of multiple determination. Whatpercentage of the variation in x₄ can beexplained by the corresponding variations inx₁, x₂, andx₃ 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, theintercept can be thought of as a"slope."     

If we look at all coefficients together, the sum of them can bethought of as the overall "slope" of the regression line.

If we look at all coefficients together, each one can be thoughtof as a "slope."

If a student were to study "extra hard" for exam 3 and increase hisor her score on that exam by 13 points, what corresponding changewould you expect on the final exam? (Assume that exams 1 and 2remain "fixed" in their scores.) (Use 1 decimal place.)


(e) Test each coefficient in the regression equation to determineif it is zero or not zero. Use level of significance 5%. (Use 2decimal places for t and 3 decimal places for theP-value.)

	t	P-value
?1
?2
?3

Conclusion

We reject all null hypotheses, there is insufficient evidencethat ?₁, ?₂ and?₃ differ from 0.

We reject all null hypotheses, there is sufficient evidence that?₁, ?₂ and?₃ differ from0.     

We fail to reject all null hypotheses, there is sufficientevidence that ?₁, ?₂ and?₃ differ from 0.

We fail to reject all null hypotheses, there is insufficientevidence that ?₁, ?₂ and?₃ differ from 0.

Why would the outcome of each hypothesis test help us decidewhether or not a given variable should be used in the regressionequation?

If a coefficient is found to be not different from 0, then itcontributes to the regression equation.

If a coefficient is found to be different from 0, then it doesnot contribute to the regressionequation.     

If a coefficient is found to be not different from 0, then itdoes not contribute to the regression equation.

If a coefficient is found to be different from 0, then itcontributes to the regression equation.

(f) Find a 90% confidence interval for each coefficient. (Use 2decimal places.)

	lower limit	upper limit
?1
?2
?3

(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 thefinal exam and find a 90% confidence interval for your prediction(if your software supports prediction intervals). (Round allanswers to nearest integer.)

prediction
lower limit
upper limit