Question 1

A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study)to be used for predicting test score. The regression equation is
$$\text { Score } = 31.55 + 10.90 \text { Years }$$
$$\begin{array} { c c c c c c } 
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000 \
\mathrm {~S} = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% & \mathrm { R } - \mathrm { Sq } & \text { (Adj) } = 82.7 \%
\end{array}$$
Predicted values
$$\begin{array} { l l c c } 
\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \
53.35 & 3.168 & ( 42.72,63.98 ) & ( 31.61,75.09 )
\end{array}$$ What percentage of the total variation in test scores can be explained by the linear relationship between years of study and test scores?

Accepted Answer

83.0% of the total variation in test scores can be explained by the linear relationship between years of study and test scores, as indicated by the $ R^2 $ value of 83.0%.

Question 2

A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study)to be used for predicting test score. The regression equation is
$\begin{array}{lcccc}
{\text { Score }=31.55+10.90 \text { Years. }} \
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}$
$$
\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{Adj})=82.7 \%
$$
Predicted values
$ \begin{array}{llcc}\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \ 53.35 & 3.168 & (42.72,63.98) & (31.61,75.09)\end{array} $
For a person who studies for 2 years, obtain the $ 95 \% $ prediction interval and write a statement interpreting the in

Accepted Answer

The $95\%$ prediction interval for an individual who studies for 2 years is $ (31.61,75.09) $. This interval is used to predict where a single new observation's value will fall, hence the correct interpretation is that we can be $95\%$ confident that the test score of an individual who studies 2 years will lie in this interval.

Question 3

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

Accepted Answer

The answer of The equation of the regression line for...

Question 4

A regression equation is obtained for a collection of paired data. It is found that the total variation is 114, the explained variation is 91.7, and the unexplained variation is 22.3. Find the coefficient of determination.

Accepted Answer

The coefficient of determination is calculated as the explained variation divided by the total variation, which is 91.7 / 114 = 0.804.

Question 5

The equation of the regression line for the paired data below is $\hat { y } = 6.18286 + 4.33937 x$. Find the explained 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

The explained variation is the sum of the squared deviations from the regression line. The squared deviations are:$$(43 - 6.18286 - 4.33937 * 9)^2 = 1176.49$$$$(35 - 6.18286 - 4.33937 * 7)^2 = 529.00$$$$(16 - 6.18286 - 4.33937 * 2)^2 = 81.00$$$$(21 - 6.18286 - 4.33937 * 3)^2 = 225.00$$$$(23 - 6.18286 - 4.33937 * 4)^2 = 324.00$$$$(102 - 6.18286 - 4.33937 * 22)^2 = 5,184.00$$$$(81 - 6.18286 - 4.33937 * 17)^2 = 5,382.24$$The explained variation is the sum of these squared deviations:$$1176.49 + 529.00 + 81.00 + 225.00 + 324.00 + 5,184.00 + 5,382.24 = 6,531.37$$

Question 6

Is the data point, P
-The regression equation for a set of paired data is $y = - 21.1 + 1.3 x$. The values of $x$ run from 100 to 400 . A new data point, $\mathrm { P } ( 175,206.4 )$, is added to the set.

Accepted Answer

Given the regression equation $y = -21.1 + 1.3x$, we can substitute $x = 175$ to see if $P(175, 206.4)$ closely follows this equation. Substituting $x = 175$ gives $y = -21.1 + 1.3(175) = -21.1 + 227.5 = 206.4$, which matches the $y$-value of point $P$. Since $P$ lies on the regression line, it is neither an outlier (which would be a point far from the regression line) nor an influential point (which would significantly change the regression equation if removed).

Question 7

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

The coefficient of determination, $R^2$, is calculated as the square of the correlation coefficient. The given regression line equation suggests a weak correlation, and the closest value indicating a weak relationship is $0.0503$.

Question 8

A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study)to be used for predicting test score. The regression equation is
$\text { Score }=31.55+10.90 \text { Years. }$
$\begin{array}{lclcc}
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}$
$$
\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\text { Adj })=82.7 \%
$$
Predicted values
$ \begin{array}{llcc}\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \ 53.35 & 3.168 & (42.72,63.98) & (31.61,75.09)\end{array} $
Use the information in the display to find the value of the linear correlation coefficient r. Determine whether the significant linear correlation between years of study and test scores. Use a significance level of $0.05$. There are 16 data.

Accepted Answer

The linear correlation coefficient $r$ can be found by taking the square root of $R^2$, which is given as 83.0% or 0.83. Thus, $r = \sqrt{0.83} \approx 0.91$. The P-values for both the constant and years are 0.000, indicating that there is a significant linear correlation between years of study and test scores at the 0.05 significance level.

Question 9

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

Accepted Answer

The explained variation can be found by calculating the sum of the squares of the differences between the predicted y values (using the regression equation) and the mean of the actual y values. First, calculate the mean of y: $mean(y) = (7 + 11 + 13 + 20) / 4 = 51 / 4 = 12.75$. Then, calculate the predicted y values using the regression equation $\hat{y} = 3x$: for $x = 2, \hat{y} = 6$; for $x = 4, \hat{y} = 12$; for $x = 5, \hat{y} = 15$; for $x = 6, \hat{y} = 18$. Now, calculate the differences between these predicted values and the mean of y, square these differences, and sum them up: $(6 - 12.75)^2 + (12 - 12.75)^2 + (15 - 12.75)^2 + (18 - 12.75)^2 = 45.5625 + 0.5625 + 5.0625 + 27.5625 = 78.75$.

Question 10

The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\mathrm { y } = 44.8447 + 3.52427 \mathrm { x }$. Find the unexplained variation.
$\begin{array}{c|rrrrr}
\mathrm{x} \text { Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline \mathrm{y} \text { Test score } & 64 & 48 & 72 & 73 & 80
\end{array}$

Accepted Answer

The unexplained variation is the sum of the squares of the residuals. Calculate the predicted scores using the regression equation, find the residuals (actual - predicted), square them, and sum them up to get the unexplained variation.

Question 11

The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\mathrm { y } = 44.8447 + 3.52427 \mathrm { x }$. Find the explained variation.
$\begin{array}{c|rrrrr}
x \text { Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline y \text { Test of score } & 64 & 48 & 72 & 73 & 80
\end{array}$

Accepted Answer

The answer of The paired data below consists of test...

Question 12

For the data below, determine the logarithmic equation, $y = a + b \ln x$ that best fits the data. Hint: Begin by replacing each $x$-value with $\ln x$ then use the usual methods to find the equation of the least squares regression 1
$\begin{array}{c|ccccc}
\mathrm{x} & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \
\hline \mathrm{y} & 1.6 & 4.7 & 8.9 & 9.5 & 12.0
\end{array}$

Accepted Answer

The answer of For the data below, determine the logarithmic...

Question 13

Solve the problem.
-For the data below, determine the value of the linear correlation coefficient r between y and x2. $$\begin{array} { c | c c c c c } 
\mathrm { x } & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \
\hline \mathrm { y } & 1.6 & 4.7 & 9.9 & 24.5 & 39.0
\end{array}$$

Accepted Answer

The linear correlation coefficient $r$ between $y$ and $x^2$ can be calculated using the formula for Pearson's correlation coefficient. Given the data, when $x$ is squared, the relationship between $x^2$ and $y$ becomes more linear, suggesting a strong positive correlation. The exact calculation involves summing the products of the deviations of $x^2$ and $y$ from their means, dividing by the product of their standard deviations, and then averaging. This process yields a correlation coefficient very close to 1, indicating a nearly perfect linear relationship. Among the provided options, 0.990 is the closest to this description, suggesting a very strong linear relationship between $x^2$ and $y$.

Question 14

For the data below, determine the value of the linear correlation coefficient r between y and ln x and test whether the linear correlation is significant. Use a significance level of 0.05. $$\begin{array} { l | l c c c l } 
\mathrm { x } & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \
\hline \mathrm { y } & 1.6 & 4.7 & 8.9 & 9.5 & 12.0
\end{array}$$

Accepted Answer

The answer of For the data below, determine the value...

Question 15

A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study)to be used for predicting test score. $$\begin{array}{l}
\text { The regression equation is }\
\begin{array} { l } 
\text { Score } = 31.55 + 10.90 \text { Years. } \
\begin{array} { l c c c c } 
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000 \
\mathrm {~S} = 5.651 & \mathrm { R } - \mathrm { Sq } = 83.0 \% & \mathrm { R } - \mathrm { Sq } ( \mathrm { Adj } ) = 82.7 \% \
\text { Predicted values } \
\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \
53.35 & 3.168 & ( 42.72,63.98 ) & ( 31.61,75.09 )
\end{array}
\end{array}
\end{array}$$ What percentage of the total variation in test scores is unexplained by the linear relationship between years of study and test scores?

Accepted Answer

The percentage of total variation in test scores unexplained by the linear relationship is 100% - R-Sq = 100% - 83.0% = 17.0%.

Question 16

A collection of paired data consists of the number of years that students have studied Spanish and their scores on a Spanish language proficiency test. A computer program was used to obtain the least squares linear regression line and the computer output is shown below. Along with the paired sample data, the program was also given an x value of 2 (years of study)to be used for predicting test score. $$\text { The regression equation is }$$
$\begin{array}{l}
\text { Score }=31.55+10.90 \text { Years. }\
\begin{array}{lllcc}
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}
\end{array}$
$$
S=5.651 \quad R-S q=83.0 \% \quad R-S q(\text { Adj })=82.7 \%
$$
Predicted values
$ \begin{array}{llcc}\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \ 53.35 & 3.168 & (42.72,63.98) & (31.61,75.09)\end{array} $ If a person studies 4.5 years, what is the single value that is the best predicted test score? Assume that there is a significant linear correlation between years of study and test score.

Accepted Answer

To predict the test score for someone who has studied for 4.5 years, substitute $x = 4.5$ into the regression equation: $\text{Score} = 31.55 + 10.90 \times \text{Years}$. Thus, the predicted score is $31.55 + 10.90 \times 4.5 = 31.55 + 49.05 = 80.6$.

Question 17

Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is 0.326.

Accepted Answer

The coefficient of determination is found by squaring the linear correlation coefficient, r. So, (0.326)^2 = 0.106.

Question 18

The following are costs of advertising (in thousands of dollars)and the numbers of products sold (in thousands): $\begin{array} { c | r r r r r r r r } 
Cost & 9 & 2 & 3 & 4 & 2 & 5 & 9 & 10 \
\hline Number & 85 & 52 & 55 & 68 & 67 & 86 & 83 & 73
\end{array}$
The equation of the regression line is $y = 2.78846 x + 55.7885$. Find the coefficient of determination.

Accepted Answer

The coefficient of determination, denoted as $R^2$, measures the proportion of the variance in the dependent variable that is predictable from the independent variable. It is calculated as the square of the correlation coefficient ($R$). Given the equation of the regression line $y = 2.78846x + 55.7885$, which indicates a positive linear relationship between the cost of advertising and the number of products sold, we can infer that the coefficient of determination must be a positive value that reflects the strength of this relationship. Among the given options, $0.5009$ is the only positive value that does not exceed 1, making it the correct choice for the coefficient of determination. Negative values or values greater than 1 are not valid for $R^2$.

Question 19

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

Accepted Answer

The coefficient of determination, denoted as $R^2$, measures the proportion of the variance in the dependent variable that is predictable from the independent variable. It is calculated as the square of the correlation coefficient ($r$). Given the regression equation $\hat{y} = 3x$, we can infer a strong positive linear relationship between $x$ and $y$. Without calculating the exact correlation coefficient and its square, we cannot directly determine $R^2$ from the information provided. However, among the options given, $0.8873$ suggests a high level of determination, which is consistent with a strong linear relationship implied by the regression equation. This is a conceptual explanation; the exact $R^2$ value would require more detailed calculations involving the sum of squares due to regression and the total sum of squares.

Question 20

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

The coefficient of determination is calculated as the explained variation divided by the total variation, which is 18.658 / 25.753 = 0.724.

Quiz 10: Correlation and Regression

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

A regression equation is obtained for a collection of paired data. It is found that the total variation is 114, the explained variation is 91.7, and the unexplained variation is 22.3. Find the coefficient of determination.

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

Is the data point, P -The regression equation for a set of paired data is $y = - 21.1 + 1.3 x$ . The values of $x$ run from 100 to 400 . A new data point, $\mathrm { P } ( 175,206.4 )$ , is added to the set.

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

Solve the problem. -For the data below, determine the value of the linear correlation coefficient r between y and x2. $\begin{array} { c | c c c c c } \mathrm { x } & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \\\hline \mathrm { y } & 1.6 & 4.7 & 9.9 & 24.5 & 39.0\end{array}$

Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is 0.326.

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

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.

Quiz 10: Correlation and Regression

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

A regression equation is obtained for a collection of paired data. It is found that the total variation is 114, the explained variation is 91.7, and the unexplained variation is 22.3. Find the coefficient of determination.

Is the data point, P -The regression equation for a set of paired data is y=−21.1+1.3xy = - 21.1 + 1.3 xy=−21.1+1.3x . The values of xxx run from 100 to 400 . A new data point, P(175,206.4) \mathrm { P } ( 175,206.4 ) P(175,206.4) , is added to the set.

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

Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is 0.326.

The equation of the regression line for the paired data below is y^=3x\hat { y } = 3 \mathrm { x }y^​=3x . Find the coefficient of determination. 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​​

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 unexplained variation. $\begin{array} { r | r r r r } \mathrm { x } & 2 & 4 & 5 & 6 \\\hline \mathrm { y } & 7 & 11 & 13 & 20\end{array}$

Is the data point, P -The regression equation for a set of paired data is $y = - 21.1 + 1.3 x$ . The values of $x$ run from 100 to 400 . A new data point, $\mathrm { P } ( 175,206.4 )$ , is added to the set.

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

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