Question 1

Use the given information to find the coefficient of determination. 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.
A) 1.114
B) 0.102
C) 0.898
D) 0.114

Accepted Answer

The coefficient of determination, denoted by r², is the ratio of the explained variation to the total variation. Thus, 
r² = Explained variation / Total variation 
Plugging in the given values, 
r² = 18.592 / 20.711 
r² = 0.898 
Therefore, the answer is choice C.

Question 2

A(n)___________________________ is a point lying far away from other data points on a scatterplot.
A) outlier
B) marginal point
C) residual point
D) least-squares point

Accepted Answer

An outlier is a data point that lies far away from other data points on a scatterplot. This can occur due to measurement error, data entry errors, or other factors. Outliers can have a significant impact on statistical analyses and should be carefully considered when interpreting data. The other options listed (marginal point, residual point, least-squares point) do not accurately describe an outlier.

Question 3

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

A) $13.479$
B) $6,531.37$
C) $6,544.86$
D) $6,421.83$

Accepted Answer

B

Question 4

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

A) $88.75$
B) $78.75$
C) $14.25$
D) $10.00$

Accepted Answer

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

Question 5

Use computer software to find the best multiple regression equation to explain the variation in the dependent variable, Y, in terms of the independent variables, $X _ { 1 } , X _ { 2 } , X _ { 3 }$\(\begin{array} { c c c c l } Y & X _ { 1 } & X _ { 2 } & X _ { 3 } & \ 456 & 9896 & 29.1 & 1 & \ 421 & 9680 & 42.3 & 2 & \ 653 & 10449 & 29.8 & 3 & \ 573 & 10811 & 26.0 & 4 & \text { CORRELATION COEFFICIENTS } \ 546 & 10014 & 34.3 & 5 & \ 499 & 10293 & 22.7 & 6 & Y / X _ { 1 } = 0.509 \ 504 & 9413 & 24.2 & 7 & Y / X _ { 2 } = 0.280 \ 611 & 9860 & 31.6 & 8 & Y / X _ { 3 } = 0.930 \ 646 & 9782 & 25.6 & 9 & \ 789 & 12139 & 37.9 & 10 & \text { COEFFICIENTS OF DETERMINATION } \ 773 & 12166 & 33.9 & 11 & \ 753 & 9976 & 37.4 & 12 & \text { YI } X _ { 1 } = 0.259 \ 852 & 10645 & 27.0 & 13 & Y / X _ { 2 } = 0.079 \ 755 & 9738 & 31.5 & 14 & Y / X _ { 3 } = 0.864 \ 815 & 9933 & 39.9 & 15 & Y / X _ { 1 } , X _ { 3 } = 0.880 \ 902 & 10132 & 25.3 & 16 & Y / X _ { 1 } , X _ { 2 } , X _ { 3 } = 0.884 \ 986 & 11145 & 30.4 & 17 & \ 909 & 9775 & 32.7 & 18 & \ 945 & 9549 & 35.0 & 19 & \ 866 & 10077 & 33.8 & 20 & \ 1178 & 11550 & 29.4 & 21 & \ 1230 & 10600 & 37.1 & 22 & \ 1207 & 11280 & 42.9 & 23 & \ 968 & 12100 & 32.2 & 24 & \ 1118 & 12420 & 30.5 & 25 & \end{array}\)

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

Accepted Answer

The answer of Use computer software to find the best...

Question 6

Find the value of the linear correlation coefficient r. The paired data below consist of the temperatures on randomly chosen days and the amount a certain kind of plant grew (in Millimeters). $\begin{array}{l|lllllllll}
\text { Temperature } & 62 & 76 & 50 & 51 & 71 & 46 & 51 & 44 & 79 \
\hline \text { Growth } & 36 & 39 & 50 & 13 & 33 & 33 & 17 & 6 & 16
\end{array}$

A) 0
B) $0.196$
C) $- 0.210$
D) $0.256$

Accepted Answer

The answer of Find the value of the linear correlation...

Question 7

A 0.05 significance level is being used to test a correlation between two variables. If th? linear correlation coefficient r is found to be 0.591 and the critical values are $r = \pm 0.878$ what can you conclude?
A) There is not sufficient evidence to support the claim of a linear correlation between the two variables.
B) There is sufficient evidence to support the claim of a linear correlation between the two variables.

Accepted Answer

The correlation coefficient $r = 0.591$ does not exceed the critical values of $r = \pm 0.878$, indicating that there is not sufficient evidence to support the claim of a linear correlation between the two variables at the 0.05 significance level.

Question 8

Find the value of the linear correlation coefficient r. The paired data below consist of the costs of advertising (in thousands of dollars)and the number of products sold (in thousands). $\begin{array}{l|rrrrrrrr}
\text { Cost } & 9 & 2 & 3 & 4 & 2 & 5 & 9 & 10 \
\hline \text { Number } & 85 & 52 & 55 & 68 & 67 & 86 & 83 & 73
\end{array}$

A) $0.708$
B) $0.246$
C) $- 0.071$
D) $0.235$

Accepted Answer

The linear correlation coefficient $ r $ is calculated using the formula for Pearson's correlation coefficient. After computing the necessary sums and products, the value of $ r $ is approximately $ 0.708 $, indicating a moderate positive correlation between advertising costs and the number of products sold.

Question 9

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. Two different tests are designed to measure employee
Productivity and dexterity. Several employees are randomly selected and tested with these
Results. $\begin{array}{l|l|l|l|l|l|l|l|l|l|l}
\text { Productivity (x) } & 23 & 25 & 28 & 21 & 21 & 25 & 26 & 30 & 34 & 36 \
\hline \text { Dexterity (y) } & 49 & 53 & 59 & 42 & 47 & 53 & 55 & 63 & 67 & 75
\end{array}$

A) $\hat { y } = 2.36 + 2.03 x$
B) $\hat { y } = 10.7 + 1.53 x$
C) $\hat { y } = 75.3 - 0329 x$
D) $\hat { y } = 5.05 + 1.91 x$

Accepted Answer

The answer of Use the given data to find the...

Question 10

Use computer software to find the multiple regression equation. Can the equation be used for prediction? A wildlife analyst gathered the data in the table to develop an equation to predict
The weights of bears. He used WEIGHT as the dependent variable and CHEST, LENGTH,
And SEX as the independent variables. For SEX, he used male=1 and female=2.
 $\begin{array}{cccc}
\hline \text { WEIGHT } & \text { CHEST } & \text { LENGTH } & \text { SEX } \
\hline 344 & 45.0 & 67.5 & 1 \
416 & 54.0 & 72.0 & 1 \
220 & 41.0 & 70.0 & 2 \
360 & 49.0 & 68.5 & 1 \
332 & 44.0 & 73.0 & 1 \
140 & 32.0 & 63.0 & 2 \
436 & 48.0 & 72.0 & 1 \
132 & 33.0 & 61.0 & 2 \
356 & 48.0 & 64.0 & 2 \
150 & 35.0 & 59.0 & 1 \
202 & 40.0 & 63.0 & 2 \
365 & 50.0 & 70.5 & 1 \
\hline
\end{array}$

A) WEIGHT $= 196 + 2.35$ CHEST $+ 3.40 \mathrm { LENGTH } + 25 \mathrm { SEX }$; Yes, because the $R ^ { 2 }$ is high.
B) WEIGHT $= - 320 + 10.6$ CHEST + 7.3LENGTH - 10.7SEX; Yes, because the $P$-value is high.
C) WEIGHT $= - 442.6 + 12.1$ CHEST + 3.6LENGTH $- 23.8 \mathrm { SEX }$; Yes, because the adjusted $R ^ { 2 }$ is high.
D) WEIGHT $= 442.6 + 12.1$ CHEST $+ 4.2 \mathrm { LENGTH } - 21 \mathrm { SEX }$; Yes, because the $P$-value is low.

Accepted Answer

The equation in option C has a high adjusted $R^2$, indicating a good fit for the data, making it suitable for prediction.

Question 11

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of $r$ given the number of pairs of data $n$ and the significance level $\alpha$. 
$n = 12 , \alpha . = 0.01$
A) $r = 0.708$
B) $r = \pm 0.708$
C) $r = \pm 0.576$
D) $r = 0.735$

Accepted Answer

For $n = 12$ and $\alpha = 0.01$, the critical values for $r$ are $\pm 0.708$ based on the correlation coefficient table for a two-tailed test.

Question 12

The residual is the difference between the ____________________ and the ___________________.
A) predicted value of y; observed value of y
B) observed value of y; predicted value of y
C) predicted value of x; observed value of x
D) observed value of x; predicted value of y

Accepted Answer

The residual is defined as the difference between the observed value of the response variable (y) and the predicted value of the response variable (ŷ), so the correct answer is B where the observed value of y is subtracted from the predicted value of y.

Question 13

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary. $\begin{array}{l|l|l|l|l|l|l|l|l|l|l}
\text { Productivity (x) } & 23 & 25 & 28 & 21 & 21 & 25 & 26 & 30 & 34 & 36 \
\hline \text { Dexterity }(\boldsymbol{y}) & 49 & 53 & 59 & 42 & 47 & 53 & 55 & 63 & 67 & 75
\end{array}$

A) $\hat { y } = 2.36 + 2.03 x$
B) $\hat { y } = 10.7 + 1.53 x$
C) $\hat { y } = 75.3 - 0329 x$
D) $\hat { y } = 5.05 + 1.91 x$

Accepted Answer

The answer of Use the given data to find the...

Question 14

Use computer software to find the best multiple regression equation to explain the variation in the dependent variable, $Y$, in terms of the independent variables, $X _ { 1 } , X _ { 2 } , X _ { 3 }$. $\begin{array} { c c c c } Y & X _ { 1 } & X _ { 2 } & \ 98.6 & 87.4 & 108.5 & \ 101.2 & 97.6 & 110.1 & \ 102.4 & 96.7 & 110.4 & \text { CORRELATION COEFFICIENTS } \ 100.9 & 98.2 & 104.3 & \ 102.3 & 99.8 & 107.2 & \text { Y/ } X _ { 1 } = 0.850 \ 101.5 & 100.5 & 105.8 & \text { YI } X _ { 2 } = 0.742 \ 101.6 & 103.2 & 107.8 & \ 101.6 & 107.8 & 103.4 & \ 99.8 & 96.6 & 102.7 & \text { COEFFICIENT OF DETERMINATION } \ 100.3 & 88.9 & 104.1 & \text { Y/ } X _ { 1 } = 0.723 \ 97.6 & 75.1 & 99.2 & \text { Y/ } X _ { 2 } = 0.550 \ 97.2 & 76.9 & 99.7 & \text { Yl } X _ { 1 } , X _ { 2 } = 0.867 \ 97.3 & 84.6 & 102.0 & \ 96.0 & 90.6 & 94.3 & \ 99.2 & 103.1 & 97.7 & \ 100.3 & 105.1 & 101.1 & \ 100.3 & 96.4 & 102.3 & \ 104.1 & 104.4 & 104.4 & \ 105.3 & 110.7 & 108.5 & \ 107.6 & 127.1 & 111.3 & \end{array}$

A) $\hat { Y } = 57.6 + 0.153 X _ { 1 } + 0.270 X _ { 2 }$
B) $\hat { Y } = 58.9 + 0.612 X _ { 1 }$
C) $\hat { Y } = 52.6 + 0.462 X _ { 2 }$
D) $\hat { Y } = 48.0 + 0.398 X _ { 1 } + 0.228 X _ { 2 }$

Accepted Answer

The coefficient of determination for $Y$ with both $X_1$ and $X_2$ is 0.867, which is the highest, indicating that the model with both variables explains the most variation in $Y$. Option A includes both $X_1$ and $X_2$.

Question 15

For the data below, determine the value of the linear correlation coefficient $r$ between $y$ and $x ^ { 2 }$.
$\begin{array}{r|rrrrr}
\boldsymbol{x} & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \
\hline \boldsymbol{y} & 1.6 & 4.7 & 9.9 & 24.5 & 39.0
\end{array}$

A) $0.913$
B) $0.985$
C) $0.990$
D) $0.873$

Accepted Answer

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. When calculating $r$ for $y$ and $x^2$, the value is $0.990$, indicating a strong positive linear relationship.

Question 16

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical value(s) of $r$ given that $n = 10$ and $\alpha = 0.05$.
A) $r = \pm 0.765$
B) $r = \pm 0.632$
C) $r = \pm 0.666$
D) $r = 0.632$

Accepted Answer

The answer of Suppose you will perform a test to...

Question 17

Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear
Correlation. Use a significance level of 0.05.
R = 0.543, n = 25 
A) Critical values: $r = \pm 0.487$, no significant linear correlation
B) Critical values: $r = \pm 0.396$, no significant linear correlation
C) Critical values: $r = \pm 0.487$, significant linear correlation
D) Critical values: $r = \pm 0.396$, significant linear correlation

Accepted Answer

The answer of Given the linear correlation coefficient r and...

Question 18

Determine which plot shows the strongest linear correlation. 
A)

B)

C)

D)

Accepted Answer

The answer of Determine which plot shows the strongest linear...

Question 19

Based on the scatterplot, select the most likely value of the linear correlation coefficient r. 
A) 0
B) 1
C) $- 1$
D) $- 0.5$

Accepted Answer

The answer of Based on the scatterplot, select the most...

Question 20

Use computer software to find the multiple regression equation. Can the equation be used for prediction? An anti-smoking group used data in the table to relate the carbon monoxide( CO)
Of various brands of cigarettes to their tar and nicotine (NIC)content. $\begin{array}{ccc}
\hline \text { CO } & \text { TAR } & \text { NIC } \
\hline 15 & 1.2 & 16 \
15 & 1.2 & 16 \
17 & 1.0 & 16 \
6 & 0.8 & 9 \
1 & 0.1 & 1 \
8 & 0.8 & 8 \
10 & 0.8 & 10 \
17 & 1.0 & 16 \
15 & 1.2 & 15 \
11 & 0.7 & 9 \
18 & 1.4 & 18 \
16 & 1.0 & 15 \
10 & 0.8 & 9 \
7 & 0.5 & 5 \
18 & 1.1 & 16 \
\hline
\end{array}$

A) $\mathrm { CO } = 1.37 + 5.50 \mathrm { TAR } - 1.38 \mathrm { NIC }$; Yes, because the $P$-value is high.
B) $\mathrm { CO } = 1.37 - 5.53 \mathrm { TAR } + 1.33 \mathrm { NIC }$; Yes, because the $R ^ { 2 }$ is high.
C) $\mathrm { CO } = 1.25 + 1.55 \mathrm { TAR } - 5.79 \mathrm { NIC }$; Yes, because the $P$-value is too low.
D) $\mathrm { CO } = 1.3 + 5.5 \mathrm { TAR } - 1.3 \mathrm { NIC }$; Yes, because the adjusted $R ^ { 2 }$ is high.

Accepted Answer

The answer of Use computer software to find the multiple...

Quiz 10: Correlation and Regression

Use the given information to find the coefficient of determination. 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.

A(n) ___________________________ is a point lying far away from other data points on a scatterplot.

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

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

A 0.05 significance level is being used to test a correlation between two variables. If th? linear correlation coefficient r is found to be 0.591 and the critical values are $r = \pm 0.878$ what can you conclude?

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical values of $r$ given the number of pairs of data $n$ and the significance level $\alpha$ . $n = 12 , \alpha . = 0.01$

The residual is the difference between the __ and the _.

For the data below, determine the value of the linear correlation coefficient $r$ between $y$ and $x ^ { 2 }$ . $\begin{array}{r|rrrrr}\boldsymbol{x} & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \\\hline \boldsymbol{y} & 1.6 & 4.7 & 9.9 & 24.5 & 39.0\end{array}$

Suppose you will perform a test to determine whether there is sufficient evidence to support a claim of a linear correlation between two variables. Find the critical value(s) of $r$ given that $n = 10$ and $\alpha = 0.05$ .

Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear Correlation. Use a significance level of 0.05. R = 0.543, n = 25

Determine which plot shows the strongest linear correlation.

Based on the scatterplot, select the most likely value of the linear correlation coefficient r.