Question 1

The test scores of 6 randomly picked students and the numbers of hours they prepared are as follows: $\begin{array} { c | r r r r r r } 
Hours & 5 & 10 & 4 & 6 & 10 & 9 \
\hline Score & 64 & 86 & 69 & 86 & 59 & 87
\end{array}$
The equation of the regression line is $\hat { y } = 1.06604 \mathrm { x } + 67.3491$. Find the coefficient of determination.

Accepted Answer

A)  $0.2242$ 
B)  $0.6781$ 
C)  $0.0503$ 
D)  $- 0.2242$

Question 2

Find the best predicted value of y corresponding to the given value of x.
-Six pairs of data yield $\mathrm { r } = 0.789$ and the regression equation $\hat { \mathrm { y } } = 4 \mathrm { x } - 2$. Also, $\overline { \mathrm { y } } = 19.0$. What is the best predicted value of $y$ for $x = 5$ ?

Accepted Answer

A)  $18.5$ 
B)  $19.0$ 
C)  $22.0$ 
D)  $18.0$

Question 3

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 x$. Find the unexplained variation.
$\begin{array}{r|rrrrrrr}
\mathrm{x} & 9 & 7 & 2 & 3 & 4 & 22 & 17 \
\hline \mathrm{y} & 43 & 35 & 16 & 21 & 23 & 102 & 81
\end{array}$

Accepted Answer

A)  $18.923$ 
B)  $6,544.86$ 
C)  $6,531.37$ 
D)  $13.479$

Question 4

Use computer software to obtain the regression and identify RS1U1P12S1S1P0, adjusted RS1U1P12S1S1P0, and the P-value
-A visitor to Yellowstone National Park sat down one day and observed Old Faithful, which faithfully spurts throughout the day, day in and day out. He surmised that the height of a given spurt was caused by the pressure build-up during the interval between spurts and by the momentum build-up during the duration of the spurt. He wrote down the data to test his hypothesis, but he didn't know what to do with his data. Can you help him out with his theory? $\begin{array}{ccc}
\hline\text { HEIGHT } & \text { INTERVAL } & \text { DURATIO } \
\hline 150 & 86 & 240 \
154 & 86 & 237 \
140 & 62 & 122 \
140 & 104 & 267 \
160 & 62 & 113 \
140 & 95 & 258 \
150 & 79 & 232 \
150 & 62 & 105 \
160 & 94 & 276 \
155 & 79 & 248\125 & 86 & 243 \
136 & 85 & 241 \
140 & 86 & 214 \
155 & 58 & 114 \
130 & 89 & 272 \
125 & 79 & 227 \
125 & 83 & 237 \
139 & 82 & 238\125 & 84 & 203 \
140 & 82 & 270 \
140 & 82 & 270 \
140 & 78 & 218 \
135 & 87 & 270 \
140 & 70 & 241 \
100 & 56 & 102 \
105 & 81 & 271\
\hline
\end{array}$

Accepted Answer

A)  $ .213, .182, .213 $ 
B)  $ .025,-.060, .750 $ 
C)  $ .089, .032, .634 $ 
D)  $ .049,-.021, .123 $

Question 5

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 \mathrm { x }$. Find the standard error of estimate.
$\begin{array}{r|rrrrrrr}
\mathrm{x} & 9 & 7 & 2 & 3 & 4 & 22 & 17 \
\hline \mathrm{y} & 43 & 35 & 16 & 21 & 23 & 102 & 81
\end{array}$

Accepted Answer

A)  $3.1270$ 
B)  $2.6959$ 
C)  $0.8275$ 
D)  $1.6419$

Question 6

A regression equation is obtained for a collection of paired data. It is found that the total variation is 25.753, the explained variation is 18.658, and the unexplained variation is 7.095. Find the coefficient of determination.

Accepted Answer

A) 0.724 
B) 1.380 
C) 0.380 
D) 0.276

Question 7

Below are performance and attitude ratings of employees. $\begin{array}{l|l|l|l|l|l|l|l|l|l}
\text {Performance  }&59 & 63 & 65 & 69 & 58 & 77 & 76 & 69 & 70 & 64 \
\hline\text {Attitude }& 72 & 67 & 78 & 82 & 75 & 87 & 92 & 83 & 87 & 78
\end{array}$
Managers also rate the same employees according to adaptability, and below are the results that correspond to th given above.
Adaptability: $\begin{array} { l l l l l l l l l l l } 50 & 52 & 54 & 60 & 46 & 67 & 66 & 59 & 62 & 55 \end{array}$
Find the multiple regression equation that expresses performance in terms of attitude and adaptability.

Accepted Answer

A)  $\hat { \mathrm { P } } = 14.09 + 0.907$ (Att.) $+ 0.014$ (Adapt.) 
B)  $\hat { \mathrm { P } } = 14.09 + 0.895$ (Att.) $+ 0.213$ (Adapt.) 
C)  $\hat { \mathrm { P } } = 14.09 + 0.213$ (Att.) $+ 0.895$ (Adapt.) 
D)  $\hat { \mathrm { P } } = 14.09 + 0.014$ (Att.) $+ 0.907$ (Adapt.)

Question 8

Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several
models and compare the values of RS1U1P12S1S1P0
-$\begin{array}{l|ccccc}
\mathrm{x} & 1 & 2 & 3 & 4 & 5 \
\hline \mathrm{y} & 7 & 17 & 20 & 25 & 28
\end{array}$

Accepted Answer

A)  $y = 7.19 + 12.8 \ln x$ 
B)  $\mathrm { y } = 4.40 + 5.00 \mathrm { x }$ 
C)  $y = 7.82 x ^ { 0.844 }$ 
D)  $y = 6.81 e ^ { 0.316 x }$

Question 9

Construct a scatter diagram for the given data
-$\begin{array}{r|r|r|r|r|r|r|r|r|r}
\mathrm{x} & -4 & 3 & 7 & 5 & 11 & 10 & 6 & -1 & -1 \
\hline \mathrm{y} & 2 & 7 & 7 & 9 & 9 & 6 & 7 & 2 & 3
\end{array}$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 10

The equation of the regression line for the paired data below is $\hat{y} = 3 x$. Find the total variation.
$\begin{array}{r|rrrr}
\mathrm{x} & 2 & 4 & 5 & 6 \
\hline \mathrm{y} & 7 & 11 & 13 & 20
\end{array}$

Accepted Answer

A)  $10.00$ 
B)  $88.75$ 
C)  $92.25$ 
D)  $78.75$

Question 11

A fitness rating was obtained for 9 randomly selected adult women. Each person was also asked her age, weight, and the number of hours she spent exercising each week. The results are shown below. Age 39 27 41 48 56 59 22 64 35 Weight 140 129 137 125 162 152 118 142 126 Hours of exercise per week 2 6 4 9 0 3 11 3 4 Fitness rating 72 88 63 84 47 52 90 31 64 Identify the multiple regression equation that expresses fitness in terms of age, weight, and hours of exercise per week.

Accepted Answer

A) ^F = 36.07 - 1.02A + 0.429W + 3.30E 
B) ^F = 52.46 - 2.14A + 1.39W + 2.48E 
C) ^F = 23.79 - 1.36A + 0.241W + 1.43E 
D) ^F = -45.12 + 0.918A + 0.461W + 3.30E

Question 12

In the context of regression, determine whether the following statement is true or false: If there is a very strong correlation between x and y, the amount of unexplained variation should be relatively large.

Accepted Answer

A) True 
 B)False

Question 13

A confidence interval for the $y$-intercept $\beta _ { 0 }$ for a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ can be found by evaluating the limits in the interval below:
$$\mathrm { b } _ { 0 } - \mathrm { E } < \beta _ { 0 } < \mathrm { b } _ { 0 } + \mathrm { E } ,$$
where $\mathrm { E } = \left( \mathrm { t } _ { \alpha / 2 } \right) \mathrm { se } \sqrt { \frac { 1 } { \mathrm { n } } + \overline { \mathrm { x } } 2 / \left[ \sum \mathrm { x } ^ { 2 } - \left( \sum \mathrm { x } \right) ^ { 2 } / \mathrm { n } \right] }$.
The critical value $t _ { \alpha / 2 }$ is found from the $t$-table using $n - 2$ degrees of freedom and $b _ { 0 }$ is calculated in the usual v from the sample data.
Use the data below to obtain a $95 \%$ confidence interval estimate of $\beta _ { 0 }$.
$\begin{array}{c|ccccc}
\mathrm{x} \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \
\hline \mathrm{y} \text { (score on test) } & 66 & 70 & 60 & 83 & 93
\end{array}$

Accepted Answer

A)  $31.10 < \beta _ { 0 } < 74.66$ 
B)  $37.83 < \beta _ { 0 } < 67.93$ 
C)  $33.50 < \beta _ { 0 } < 72.26$ 
D)  $28.10 < \beta _ { 0 } < 77.66$

Question 14

Find the best predicted value of y corresponding to the given value of x.
-The regression equation relating dexterity scores $( \mathrm { x } )$ and productivity scores ( $\mathrm { y } )$ for the employees of a company $\hat { y } = 5.50 + 1.91 \mathrm { x }$. Ten pairs of data were used to obtain the equation. The same data yield $\mathrm { r } = 0.986$ and $\bar { y } = 56.3$. What is the best predicted productivity score for a person whose dexterity score is 33 ?

Accepted Answer

A)  $56.30$ 
B)  $58.20$ 
C)  $68.53$ 
D)  $183.41$

Question 15

Find the value of the linear correlation coefficient r.
-$\begin{array}{r|rrrrr}
\mathrm{x} & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \
\hline \mathrm{y} & 54 & 53 & 55 & 54 & 56
\end{array}$

Accepted Answer

A)  $\hat { y }= 50 + 3 x$ 
B)  $\hat { y } = 55.3 + 2.40 \mathrm { x }$ 
C)  $\hat { y } = 54$ 
D)  $\hat { y } = 50.4 + 2.50 x$

Question 16

Below are the results of two separate tests designed to measure a student's ability to solve problems. $\begin{array}{c|ccccccccc}
\text { Test A } & 48 & 52 & 58 & 44 & 43 & 43 & 40 & 51 & 59 \
\hline \text { Test B } & 73 & 67 & 73 & 59 & 58 & 56 & 58 & 64 & 74
\end{array}$
In addition to these results, a third test was designed to measure the same problem-solving ability, and the follo' results correspond to the same students.
Test C: $\begin{array} { l l l l l l l l l } 48 & 41 & 59 & 45 & 42 & 44 & 40 & 58 & 60 \end{array}$
Find the multiple regression equation that expresses results from Test $\mathrm { C }$ in terms of Test $\mathrm { A }$ and Test $\mathrm { B }$.

Accepted Answer

A)  $\hat { C } = 3.40 - 0.145 \mathrm {~A} + 1.12 \mathrm {~B}$ 
B)  $\hat { C } = 1.60 + 2.02 \mathrm {~A} - 1.93 \mathrm {~B}$ 
C)  $\hat { C } = 3.40 + 1.12 \mathrm {~A} - 0.145 \mathrm {~B}$ 
D)  $\hat { C } = 8.16 + 1.12 \mathrm {~A} + 3.49 \mathrm {~B}$

Question 17

Find the best predicted value of y corresponding to the given value of x.
-Ten pairs of data yield $\mathrm { r } = 0.003$ and the regression equation $\hat { \mathrm { y } } = 2 + 3 \mathrm { x }$. Also, $\overline { \mathrm { y } } = 5.0$. What is the best predicted value of $y$ for $x = 2$ ?

Accepted Answer

A)  $5.0$ 
B)  $8.0$ 
C)  $17.0$ 
D)  $7.0$

Question 18

Use computer software to find the best regression equation to explain the variation in the dependent variable, Y, in terms of the independent variables, X1, X2, X3
-$\begin{array}{cccc}
Y & X_{1} & X_{2} & X_{3} \
456 & 9896 & 29.1 & 1 \
421 & 9680 & 42.3 & 2 \
653 & 10449 & 29.8 & 3 \
573 & 10811 & 26.0 & 4 \
546 & 10014 & 34.3 & 5 \
499 & 10293 & 22.7 & 6\
504 & 9413 & 24.2 & 7 \
611 & 9860 & 31.6 & 8 \
646 & 9782 & 25.6 & 9 \
789 & 12139 & 37.9 & 10 \
773 & 12166 & 33.9 & 11 \
753 & 9976 & 37.4 & 12 \
852 & 10645 & 27.0 & 13\
755 & 9738 & 31.5 & 14 \
815 & 9933 & 39.9 & 15 \
902 & 10132 & 25.3 & 16 \
986 & 11145 & 30.4 & 17 \
909 & 9775 & 32.7 & 18\945 & 9549 & 35.0 & 19 \
866 & 10077 & 33.8 & 20 \
1178 & 11550 & 29.4 & 21 \
1230 & 10600 & 37.1 & 22 \
1207 & 11280 & 42.9 & 23 \
968 & 12100 & 32.2 & 24 \
1118 & 12420 & 30.5 & 25
\end{array}$
CORRELATION COEFFICIENTS
$
\begin{array}{l}
\mathrm{Y} / \mathrm{X}_{1}=.509 \
\mathrm{Y} / \mathrm{X}_{2}=.280 \
\mathrm{Y} / \mathrm{X}_{3}=.930
\end{array}
$
COEFFICIENTS OF DETERMINATION
$
\begin{array}{c}
\mathrm{Y} / \mathrm{X}_{1}=.259 \
\mathrm{Y} / \mathrm{X}_{2}=.079 \
\mathrm{Y} / \mathrm{X}_{3}=.864 \
\mathrm{Y}_{1} / \mathrm{X}_{1}, \mathrm{X}_{3}=.880 \
\mathrm{Y} / \mathrm{X}_{1}, \mathrm{X}_{2}, \mathrm{X}_{3}=.884
\end{array}
$

Accepted Answer

A)  $\hat { \mathrm { Y } } = 308.6 + 29.9 \mathrm { X } _ { 3 }$ 
B)  $\hat { \mathrm { Y } } = 57.8 + 0.036 \mathrm { X } _ { 1 } + 28.1 \mathrm { X } _ { 3 }$ 
C)  $\hat { Y } = - 21.1 + 0.36 X _ { 1 } + 2.62 X _ { 2 } + 27.6 X _ { 3 }$ 
D)  $\hat { Y } = 201.7 + 0.40 \mathrm { X } _ { 1 } + 22.3 \mathrm { X } _ { 3 }$

Question 19

Provide an appropriate response.
-Suppose there is significant correlation between two variables. Describe two cases under which it might be inappropriate to use the linear regression equation for prediction. Give examples to support these cases.

Accepted Answer

The answer of Provide an appropriate response.
-Suppose there is significant...

Question 20

Find the best predicted value of y corresponding to the given value of x.
-Six pairs of data yield $\mathrm { r } = 0.444$ and the regression equation $\hat { \mathrm { y } } = 5 \mathrm { x } + 2$. Also, $\overline { \mathrm { y } } = 18.3$. What is the best predicted value of $y$ for $x = 5$ ?

Accepted Answer

A)  $93.5$ 
B)  $18.3$ 
C)  $4.22$ 
D)  27

Exam 10: Correlation and Regression

Find the best predicted value of y corresponding to the given value of x. -Six pairs of data yield $\mathrm { r } = 0.789$ and the regression equation $\hat { \mathrm { y } } = 4 \mathrm { x } - 2$ . Also, $\overline { \mathrm { y } } = 19.0$ . What is the best predicted value of $y$ for $x = 5$ ?

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 x$ . Find the unexplained variation. $\begin{array}{r|rrrrrrr}\mathrm{x} & 9 & 7 & 2 & 3 & 4 & 22 & 17 \\\hline \mathrm{y} & 43 & 35 & 16 & 21 & 23 & 102 & 81\end{array}$

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 \mathrm { x }$ . Find the standard error of estimate. $\begin{array}{r|rrrrrrr}\mathrm{x} & 9 & 7 & 2 & 3 & 4 & 22 & 17 \\\hline \mathrm{y} & 43 & 35 & 16 & 21 & 23 & 102 & 81\end{array}$

A regression equation is obtained for a collection of paired data. It is found that the total variation is 25.753, the explained variation is 18.658, and the unexplained variation is 7.095. Find the coefficient of determination.

The equation of the regression line for the paired data below is $\hat{y} = 3 x$ . Find the total variation. $\begin{array}{r|rrrr}\mathrm{x} & 2 & 4 & 5 & 6 \\\hline \mathrm{y} & 7 & 11 & 13 & 20\end{array}$

In the context of regression, determine whether the following statement is true or false: If there is a very strong correlation between x and y, the amount of unexplained variation should be relatively large.

Find the value of the linear correlation coefficient r. - $\begin{array}{r|rrrrr}\mathrm{x} & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline \mathrm{y} & 54 & 53 & 55 & 54 & 56\end{array}$

Find the best predicted value of y corresponding to the given value of x. -Ten pairs of data yield $\mathrm { r } = 0.003$ and the regression equation $\hat { \mathrm { y } } = 2 + 3 \mathrm { x }$ . Also, $\overline { \mathrm { y } } = 5.0$ . What is the best predicted value of $y$ for $x = 2$ ?

Provide an appropriate response. -Suppose there is significant correlation between two variables. Describe two cases under which it might be inappropriate to use the linear regression equation for prediction. Give examples to support these cases.

Find the best predicted value of y corresponding to the given value of x. -Six pairs of data yield $\mathrm { r } = 0.444$ and the regression equation $\hat { \mathrm { y } } = 5 \mathrm { x } + 2$ . Also, $\overline { \mathrm { y } } = 18.3$ . What is the best predicted value of $y$ for $x = 5$ ?

Exam 10: Correlation and Regression

A regression equation is obtained for a collection of paired data. It is found that the total variation is 25.753, the explained variation is 18.658, and the unexplained variation is 7.095. Find the coefficient of determination.

Construct a scatter diagram for the given data - x−437511106−1−1y277996723\begin{array}{r|r|r|r|r|r|r|r|r|r}\mathrm{x} & -4 & 3 & 7 & 5 & 11 & 10 & 6 & -1 & -1 \\\hline \mathrm{y} & 2 & 7 & 7 & 9 & 9 & 6 & 7 & 2 & 3\end{array}xy​−42​37​77​59​119​106​67​−12​−13​​

The equation of the regression line for the paired data below is y^=3x\hat{y} = 3 xy^​=3x . Find the total variation. x2456y7111320\begin{array}{r|rrrr}\mathrm{x} & 2 & 4 & 5 & 6 \\\hline \mathrm{y} & 7 & 11 & 13 & 20\end{array}xy​27​411​513​620​​

In the context of regression, determine whether the following statement is true or false: If there is a very strong correlation between x and y, the amount of unexplained variation should be relatively large.

Find the value of the linear correlation coefficient r. - x1.21.41.61.82.0y5453555456\begin{array}{r|rrrrr}\mathrm{x} & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline \mathrm{y} & 54 & 53 & 55 & 54 & 56\end{array}xy​1.254​1.453​1.655​1.854​2.056​​

Provide an appropriate response. -Suppose there is significant correlation between two variables. Describe two cases under which it might be inappropriate to use the linear regression equation for prediction. Give examples to support these cases.

The equation of the regression line for the paired data below is $\hat{y} = 3 x$ . Find the total variation. $\begin{array}{r|rrrr}\mathrm{x} & 2 & 4 & 5 & 6 \\\hline \mathrm{y} & 7 & 11 & 13 & 20\end{array}$

Find the value of the linear correlation coefficient r. - $\begin{array}{r|rrrrr}\mathrm{x} & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline \mathrm{y} & 54 & 53 & 55 & 54 & 56\end{array}$