(a)
To find: The
(a)
Answer to Problem 62E
Solution: The mean value of the ACT score is 21.133 and the mean value of the predicted value of ACT score is 21.162. Both the mean values are identical.
Explanation of Solution
Calculation: The predicted values can be obtained by using the referred Exercise 10.61.
The regression equation is
The predicted values for ACT corresponding to the SAT score of 1000 is calculated as shown below:
Similarly, obtain the predicted values for ACT corresponding to the remaining SAT scores as shown below:
Obs |
SAT |
Predicted values for ACT |
1 |
1000 |
23.03 |
2 |
1010 |
23.24 |
3 |
920 |
21.32 |
4 |
840 |
19.61 |
5 |
830 |
19.39 |
6 |
1440 |
32.45 |
7 |
490 |
12.12 |
8 |
1050 |
24.1 |
9 |
870 |
20.25 |
10 |
970 |
22.39 |
11 |
920 |
21.32 |
12 |
810 |
18.96 |
13 |
1080 |
24.74 |
14 |
1000 |
23.03 |
15 |
1030 |
23.67 |
16 |
870 |
20.25 |
17 |
880 |
20.46 |
18 |
850 |
19.82 |
19 |
780 |
18.32 |
20 |
830 |
19.39 |
21 |
1190 |
27.1 |
22 |
800 |
18.75 |
23 |
830 |
19.39 |
24 |
890 |
20.68 |
25 |
880 |
20.46 |
26 |
980 |
22.6 |
27 |
1030 |
23.67 |
28 |
1220 |
27.74 |
29 |
1080 |
24.74 |
30 |
970 |
22.39 |
31 |
1090 |
24.96 |
32 |
860 |
20.03 |
33 |
740 |
17.47 |
34 |
500 |
12.33 |
35 |
780 |
18.32 |
36 |
1120 |
25.6 |
37 |
590 |
14.26 |
38 |
990 |
22.82 |
39 |
700 |
16.61 |
40 |
930 |
21.53 |
41 |
860 |
20.03 |
42 |
420 |
10.62 |
43 |
800 |
18.75 |
44 |
1140 |
26.03 |
45 |
920 |
21.32 |
46 |
800 |
18.75 |
47 |
1040 |
23.89 |
48 |
840 |
19.61 |
49 |
1060 |
24.31 |
50 |
870 |
20.25 |
51 |
1120 |
25.6 |
52 |
800 |
18.75 |
53 |
960 |
22.17 |
54 |
880 |
20.46 |
55 |
1020 |
23.46 |
56 |
790 |
18.54 |
57 |
620 |
14.9 |
58 |
1150 |
26.24 |
59 |
970 |
22.39 |
60 |
1060 |
24.31 |
The mean value for the predicted values of ACT can be calculated as shown below:
With the help of Minitab, the mean of the ACT scores can be calculated as shown below:
Step 1: Enter the data into the worksheet of Minitab.
Step 2: Go to
Step 3: Go to statistics and select mean. Click on “OK” twice.
Hence, the required mean obtained is 21.133.
Interpretation: Therefore, it can be concluded that the mean value of the ACT score and the mean value of the predicted value of ACT score are identical.
(b)
To find: The standard deviation of the predicted values and its comparison with the standard deviation of the actual ACT scores.
(b)
Answer to Problem 62E
Solution: The standard deviation of the ACT score is 4.714 and the standard deviation of the predicted ACT scores is 3.854.
Explanation of Solution
Calculation: With the help of Minitab, the standard deviation of the ACT scores can be calculated as shown below:
Step 1: Enter the data into the worksheet of Minitab.
Step 2: Go to
Step 3: Go to statistics and select standard deviation. Click on “OK” twice.
Hence, the required standard deviation obtained is 4.714.
Use the predicted values of the ACT scores, which is obtained in the part (a). With the help of Minitab, the standard deviation of the predicted ACT scores or fitted values can be calculated as shown below:
Step 1: Enter the data of the predicted values of the ACT scores into the worksheet of Minitab.
Step 2: Go to
Step 3: Go to statistics and select standard deviation. Click on “OK” twice.
Hence, the required standard deviation obtained is 3.854.
Interpretation: Therefore, it can be concluded that the variability in the predicted ACT scores is less in comparison to the variability in the ACT scores.
(c)
To find: The SAT score for a student who is one standard deviation above the mean.
(c)
Answer to Problem 62E
Solution: The SAT score for a student who is one standard deviation above the mean is 1092.8.
Explanation of Solution
Calculation: With the help of Minitab, the mean of the SAT scores can be calculated as shown below:
Step 1: Enter the data into the worksheet of Minitab.
Step 2: Go to
Step 3: Go to statistics and select mean. Click on “OK” twice.
Hence, the required mean obtained is 912.7.
With the help of Minitab, the standard deviation of the SAT scores can be calculated as shown below:
Step 1: Enter the data into the worksheet of Minitab.
Step 2: Go to
Step 3: Go to statistics and select standard deviation. Click on “OK” twice.
Hence, the required standard deviation obtained is 180.1.
For
Interpretation: Therefore, it can be concluded that the predicted SAT score is 1092.8, which provides a standard score of about one by using the standard deviation of the predicted SAT scores.
To find: The predicted ACT score for a student who is one standard deviation above the mean and standardize this predicted ACT score.
Answer to Problem 62E
Solution: The required predicted ACT score is 25.02.
Explanation of Solution
Calculation: The predicted ACT score can be calculated as
Interpretation: Therefore, it can be concluded that the predicted ACT score is 25.02, which provides a standard score of about one by using the standard deviation of the predicted ACT scores.
(d)
To find: The SAT score for a student who is one standard deviation below the mean.
(d)
Answer to Problem 62E
Solution: The SAT score for a student who is one standard deviation below the mean is 732.6.
Explanation of Solution
Calculation: For
To find: The predicted ACT score for a student who is one standard deviation below the mean and standardize this predicted ACT score.
Answer to Problem 62E
Solution: The required predicted ACT score is 17.3.
Explanation of Solution
Calculation: For
The predicted ACT score is
(e)
To explain: The conclusions on the basis of part (c) and part (d).
(e)
Answer to Problem 62E
Solution: The standard score of the predicted value is same as the explanatory variable’s standard score for the parts (c) and (d).
Explanation of Solution
Therefore, it can be said that the standard score of the predicted value is the same as the explanatory variable’s standard score.
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Chapter 10 Solutions
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