Question 1

Use the computer display to answer the question.
-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 $\text { Score }=31.55+10.90 \text { Years. }$

$\begin{array}{lcccc}
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}$

$\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{Adj})=82.7 \%$

$\text { Predicted values }$

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 R-Sq (Adjusted) value represents the percentage of the total variation in the response variable (test scores) that is explained by the linear relationship with the predictor variable (years of study). Therefore, the percentage of total variation unexplained by the linear relationship is 100% - R-Sq (Adjusted) = 17%.

Question 2

Construct the indicated prediction interval for an individual y.
-The equation of the regression line for the paired data below is $\hat { y } = 6.1829 + 4.3394 x$ and the standard error of estimate is $\mathrm { se } ^ { = } = 1.6419$. Find the $99 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 6$.
$\begin{array}{r|rrrrrrr}
\mathrm{x} & 9 & 7 & 2 & 3 & 4 & 22 & 17 \
\hline \mathrm{y} & 43 & 35 & 16 & 21 & 23 & 102 & 81
\end{array}$

Accepted Answer

The answer of Construct the indicated prediction interval for an...

Question 3

Find the explained variation for the paired data.
-The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { y } = 44.8447 + 3.52427 x$. 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

First, we need to find the total variation of the data set:
$$\sum (y-y_{avg})^2 = \sum y^2 - n(y_{avg})^2$$
where
$$\sum y = 337, \quad \sum y^2 = 25057, \quad n = 5$$
$$y_{avg} = \frac{\sum y}{n} = \frac{337}{5} = 67.4$$
So,
$$\sum (y-y_{avg})^2 = 25057 - 5(67.4)^2 = 1062.4$$
Next, we need to find the residual variation, which is the unexplained variation of the data around the regression line:
$$\sum (y-\hat y)^2 = \sum (y - (44.8447 + 3.52427x))^2 = 563.676$$
Finally, we can find the explained variation by subtracting the residual variation from the total variation:
$$\sum (\hat y - y_{avg})^2 = \sum y^2 - n(y_{avg})^2 - \sum (y-\hat y)^2 = 1062.4 - 563.676 = 498.724$$
Therefore, the answer is B.

Question 4

A regression equation is obtained for a collection of paired data. It is found that the total variation is 110.7, the explained variation is 93.3, and the unexplained variation is 17.4. Find the coefficient of determination.

Accepted Answer

The coefficient of determination is given by the ratio of the explained variation to the total variation.

Coefficient of determination, R^2 = Explained variation/Total variation 
R^2 = 93.3/110.7 
R^2 = 0.8428 (rounded to 3 decimal places)

Therefore, the answer is choice letter A (0.843).

Question 5

Find the unexplained variation for the paired data.
-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.
$\begin{array}{c|rrrrr}
\text { x Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline \text { y Test score } & 64 & 48 & 72 & 73 & 80
\end{array}$

Accepted Answer

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

Question 6

Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is 0.611.

Accepted Answer

The answer of Find the coefficient of determination, given that...

Question 7

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

Accepted Answer

The answer of The test scores of 6 randomly picked...

Question 8

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

Accepted Answer

The unexplained variation is the sum of the squared deviations of the y-values from the regression line. The regression line is y = 3x, so the predicted y-values are 6, 12, 15, and 18. The deviations of the y-values from the predicted y-values are 1, -1, -2, and 2. The squared deviations are 1, 1, 4, and 4. The sum of the squared deviations is 10.00.

Question 9

Find the explained variation for the paired data.
-The equation of the regression line for the paired data below is $y = 6.18286 + 4.33937 x$. 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 answer of Find the explained variation for the paired...

Question 10

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

Accepted Answer

The first step is to find the mean of x and y, which are $\bar {x} = 4.25$ and $\bar {y} = 12.75$. Then, we can find the total sum of squares (SST) by summing the squared differences between each y value and the mean of y:

$SST = (7-12.75)^2 + (11-12.75)^2 + (13-12.75)^2 + (20-12.75)^2 = 70.75$

Next, we can find the sum of squared errors (SSE) by summing the squared differences between each y value and its predicted value from the regression line:

$SSE = (7-6)^2 + (11-12)^2 + (13-15)^2 + (20-18)^2 = 16$

Finally, we can find the explained variation (SSR) by subtracting SSE from SST:

$SSR = SST - SSE = 70.75 - 16 = 54.75$

Therefore, the explained variation is 54.75, which corresponds to answer choice C.

Question 11

Use the computer display to answer the question.
-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 Score $= 31.55 + 10.90$ Years.
$\begin{array} { l c l c c } 
\text { Predictor } & \text { Coef } & \text { StDev } & \text { T } & \text { P } \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}$

$\mathrm { S } = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { Adj } ) = 82.7 \%$

Predicted values

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

The answer of Use the computer display to answer the...

Question 12

Use the computer display to answer the question.
-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  $$\mathrm { S } = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { Adj } ) = 82.7 \%$$
Predicted values
$\begin{array}{|lccc|}
\hline \text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \
53.35 & 3.168 & (42.72,63.98) & (31.61,75.09) \
\hline
\end{array}$
 Use the information in the display to find the value of the linear correlation coefficient r. Determine whether
There is significant linear correlation between years of study and test scores. Use a significance level of 0.05.
There are 10 pairs of data.

Accepted Answer

The answer of Use the computer display to answer the...

Question 13

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

Accepted Answer

The coefficient of determination, denoted as R², is the square of the correlation coefficient (r). It represents the proportion of the variance in the dependent variable that is predictable from the independent variable. In this case, the correct R² value is 0.5009, indicating that approximately 50.09% of the variance in the number of products sold can be explained by the advertising cost.

Question 14

Use the computer display to answer the question.
-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 $\text { Score }=31.55+10.90 \text { Years. }$

$\begin{array}{lcccc}
\text { Predictor } & \text { Coef } & \text { StDev } & \mathrm{T} & \mathrm{P} \
\text { Constant } & 31.55 & 6.360 & 4.96 & 0.000 \
\text { Years } & 10.90 & 1.744 & 6.25 & 0.000
\end{array}$

$\mathrm{S}=5.651 \quad \mathrm{R}-\mathrm{Sq}=83.0 \% \quad \mathrm{R}-\mathrm{Sq}(\mathrm{Adj})=82.7 \%$

$\text { Predicted values }$

$\begin{array}{lccc}
\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \
53.35 & 3.168 & (42.72,63.98) & (31.61,75.09) \
\hline
\end{array}$
 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

The percentage of the total variation in test scores that can be explained by the linear relationship between years of study and test scores is given by the coefficient of determination, which is the square of the correlation coefficient (r) between the two variables. In this case, the value of R-Sq is 83.0%, indicating that 83.0% of the total variation in test scores can be explained by the linear relationship with years of study. Therefore, option D is the correct answer.

Question 15

A regression equation is obtained for a collection of paired data. It is found that the total variation is 20.711, the explained variation is 18.592, and the unexplained variation is 2.119. Find the coefficient of determination.

Accepted Answer

The coefficient of determination is the ratio of explained variation to total variation. Therefore, coefficient of determination = explained variation / total variation = 18.592 / 20.711 = 0.898.

Question 16

Find the unexplained variation for the paired data.
-The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { \mathrm { y } } = - 181.342 + 144.46 \mathrm { x }$. Find the unexplained variation.
$\begin{array}{c|rrrrrr}
\text { x Height (meters) } & 1.61 & 1.72 & 1.78 & 1.80 & 1.67 & 1.88 \
\hline \text { y Weight }(\mathrm{kg}) & 54 & 62 & 70 & 84 & 61 & 92
\end{array}$

Accepted Answer

The unexplained variation is the sum of the squared residuals. The residual for each data point is the difference between the observed value and the predicted value. The squared residual is the square of the residual. The sum of the squared residuals is the unexplained variation.The predicted values for the data points are:$\hat { \mathrm { y } } = - 181.342 + 144.46 \mathrm { x }$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.61 ) = 53.98$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.72 ) = 63.02$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.78 ) = 72.06$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.80 ) = 74.10$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.67 ) = 60.94$$\hat { \mathrm { y } } = - 181.342 + 144.46 ( 1.88 ) = 83.08$The residuals are:$54 - 53.98 = 0.02$$62 - 63.02 = -1.02$$70 - 72.06 = -2.06$$84 - 74.10 = 9.90$$61 - 60.94 = 0.06$$92 - 83.08 = 8.92$The squared residuals are:$0.02^2 = 0.0004$$(-1.02)^2 = 1.0404$$(-2.06)^2 = 4.2436$$9.90^2 = 98.01$$0.06^2 = 0.0036$$8.92^2 = 79.5824$The sum of the squared residuals is the unexplained variation:$0.0004 + 1.0404 + 4.2436 + 98.01 + 0.0036 + 79.5824 = 182.8804$The unexplained variation is 182.8804.

Question 17

Find the explained variation for the paired data.
-The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { y } = - 181.342 + 144.46 x$. Find the explained variation.
$\begin{array}{c|rrrrrr}
x \text { Height (meters) } & 1.61 & 1.72 & 1.78 & 1.80 & 1.67 & 1.88 \
\hline y \text { Weight }(\mathrm{kg}) & 54 & 62 & 70 & 84 & 61 & 92
\end{array}$

Accepted Answer

The formula for total variation is:
$$	ext{Total Variation (SS}_{	ext{Total}}	ext{)}=\sum (y_i-\bar{y})^2$$
where $\bar{y}$ is the mean of the $y$-values.

Using a calculator or software, we get:

$$\bar{y}=68.5$$
$$	ext{Total Variation}=2536$$

The formula for explained variation (SS$_{	ext{Explained}}$) is:
$$	ext{SS}_{	ext{Explained}}=n(\hat{y}-\bar{y})^2$$
where $n$ is the sample size, and $\hat{y}$ is the predicted $y$-value from the regression equation.

Using the regression equation, we get:

\begin{align*}
\hat{y}_1&=-181.342+144.46(1.61)=54.01\
\hat{y}_2&=-181.342+144.46(1.72)=61.35\
\hat{y}_3&=-181.342+144.46(1.78)=64.32\
\hat{y}_4&=-181.342+144.46(1.80)=65.61\
\hat{y}_5&=-181.342+144.46(1.67)=57.71\
\hat{y}_6&=-181.342+144.46(1.88)=72.69\
\end{align*}

Using these predicted values, we can calculate the explained variation:

\begin{align*}
	ext{SS}_{	ext{Explained}}&=6(\hat{y}-\bar{y})^2\
&=6((54.01-68.5)^2+(61.35-68.5)^2+(64.32-68.5)^2\
&\qquad+(65.61-68.5)^2+(57.71-68.5)^2+(72.69-68.5)^2)\
&=979.44
\end{align*}

Therefore, the best choice is B, with explained variation of 979.44.

Question 18

Use the computer display to answer the question.
-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 Score $= 31.55 + 10.90$ Years.

$\mathrm { S } = 5.651 \quad \mathrm { R } - \mathrm { Sq } = 83.0 \% \quad \mathrm { R } - \mathrm { Sq } ($ Adj $) = 82.7 \%$

Predicted values

$\begin{array} { l c c c } 
\text { Fit } & \text { StDev Fit } & 95.0 \% \text { CI } & 95.0 \% \text { PI } \
53.35 & 3.168 & ( 42.72,63.98 ) & ( 31.61,75.09 )
\end{array}$
 For a person who studies for 2 years, obtain the 95% prediction interval and write a statement interpreting the
Interval.

Accepted Answer

The answer of Use the computer display to answer the...

Question 19

Construct the indicated prediction interval for an individual y.
-The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { \mathrm { y } } = 44.845 + 3.524 \mathrm { x }$ and the standard error of estimate is se $= 5.40$. Find the $99 \%$ prediction interval for the test score of a person who spent 7 hours preparing for the test.
$\begin{array}{c|rrrrr}
\text { x Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline \mathrm{y} \text { Test score } & 64 & 48 & 72 & 73 & 80
\end{array}$

Accepted Answer

The prediction interval is calculated using the formula $\hat{y} \pm t \cdot se \cdot \sqrt{1 + \frac{1}{n} + \frac{(x - \bar{x})^2}{\sum (x_i - \bar{x})^2}}$. For 7 hours, $\hat{y} = 44.845 + 3.524 \times 7 = 69.513$. Using a t-value for 99% confidence with 3 degrees of freedom (approximately 4.541), the interval is approximately $35 < y < 104$.

Question 20

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

Accepted Answer

The answer of Find the unexplained variation for the paired...

Quiz 10: Correlation and Regression

A regression equation is obtained for a collection of paired data. It is found that the total variation is 110.7, the explained variation is 93.3, and the unexplained variation is 17.4. Find the coefficient of determination.

Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is 0.611.

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

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

A regression equation is obtained for a collection of paired data. It is found that the total variation is 20.711, the explained variation is 18.592, and the unexplained variation is 2.119. Find the coefficient of determination.