Chapter 10, Problem 52E

(a)

To determine

To find: The mean of the predicted values of the ACT scores and its comparison with the mean of the ACT scores.

Answer to Problem 52E

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

$\text{ACT}=\text{1}\text{.63+0}\text{.0214 SAT}$

The predicted values for ACT corresponding to the SAT score of 1000 is calculated as shown below:

$\begin{array}{c}\text{ACT}=\text{1}\text{.63+0}\text{.0214 SAT}\\ \text{=1}\text{.63+0}\text{.0214 }\left(1000\right)\\ =23.03\end{array}$

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:

$\begin{array}{c}\text{Mean=}\frac{\text{Sum of the predicted values of ACT}}{\text{Total number of values}}\\ =\frac{1269.69}{60}\\ =21.162\end{array}$

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 $\text{Stat}\to \text{Basic Statistics}\to \text{Display Descriptive Statistics}$. In the “Variables” textbox, enter the variable “ACT.”

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 determine

To find: The standard deviation of the predicted values and its comparison with the standard deviation of the actual ACT scores.

Answer to Problem 52E

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 $\text{Stat}\to \text{Basic Statistics}\to \text{Display Descriptive Statistics}$. In the “Variables,” enter the variable “ACT.”

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 $\text{Stat}\to \text{Basic Statistics}\to \text{Display Descriptive Statistics}$. In the “Variables” textbox, enter the variable “Predicted ACT scores.”

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 determine

To find: The SAT score for a student who is one standard deviation above the mean.

Answer to Problem 52E

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 $\text{Stat}\to \text{Basic Statistics}\to \text{Display Descriptive Statistics}$. In the “Variables” textbox, enter the variable “SAT.”

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 $\text{Stat}\to \text{Basic Statistics}\to \text{Display Descriptive Statistics}$. In the “Variables,” enter the variable “SAT.”

Step 3: Go to statistics and select standard deviation. Click on “OK” twice.

Hence, the required standard deviation obtained is 180.1.

For $z=1$, the SAT score for a student who is one standard deviation above the mean is calculated as

$\begin{array}{c}\text{SAT score}=\text{Mean of SAT scores}+\text{Standard deviation of SAT scores}\\ \text{=912}\text{.7+180}\text{.1}\\ \text{=1092}\text{.8 }\end{array}$

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 determine

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 52E

Solution: The required predicted ACT score is 25.02.

Explanation of Solution

Calculation: The predicted ACT score can be calculated as

$\begin{array}{c}\text{ACT}=\text{1}\text{.63+0}\text{.0214 }\left(\text{SAT}\right)\\ =\text{1}\text{.63+0}\text{.0214 }\left(1092.8\right)\\ =25.02\end{array}$

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 determine

To find: The SAT score for a student who is one standard deviation below the mean.

Answer to Problem 52E

Solution: The SAT score for a student who is one standard deviation below the mean is 732.6.

Explanation of Solution

Calculation: For $z=-1$, the SAT score for a student who is one standard deviation below the mean is calculated as

$\begin{array}{c}\text{SAT score}=\text{Mean of SAT scores}-\text{Standard deviation of SAT scores}\\ \text{=912}.7-1\text{80}\text{.1}\\ \text{=732}\text{.6}\end{array}$

To determine

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 52E

Solution: The required predicted ACT score is 17.3.

Explanation of Solution

Calculation: For $z=-1$, the SAT score for a student who is one standard deviation below the mean is calculated as

$\begin{array}{c}\text{SAT score}=\text{Mean of SAT scores}-\text{Standard deviation of SAT scores}\\ \text{=912}.7-1\text{80}\text{.1}\\ \text{=732}\text{.6}\end{array}$

The predicted ACT score is

$\begin{array}{c}\text{ACT}=\text{1}\text{.63+0}\text{.0214 }\left(\text{SAT}\right)\\ =\text{1}\text{.63+0}\text{.0214 }\left(732.6\right)\\ =17.3\end{array}$

(e)

To determine

To explain: The conclusions on the basis of part (c) and part (d).

Answer to Problem 52E

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

For the part (c), the predicted ACT score is 25.02 and the corresponding SAT score is 1120, which is similar to the standardized SAT score 1092.8. For the part (d), the predicted ACT score is 17.3 and the corresponding SAT score is 740, which is similar to the standardized SAT score 732.6. So, for the parts (c) and (d), it can be said that explanatory variable’s standard score is appears to be same as the standard score of the predicted value.

Therefore, it can be said that the standard score of the predicted value is the same as the explanatory variable’s standard score.