Deck 10: Correlation and Regression

Question 1

For each of 200 randomly selected cities, Pete recorded the number of churches in the city (x) and the number of homicides in the past decade (y). He calculated the linear correlation coefficient and was surprised to find a strong positive linear correlation for the two variables. Does this suggest that building new churches causes an increase in the number of homicides? Why do you think that a strong positive linear correlation coefficient was
obtained? Explain your answer with reference to the term lurking variable.

Answer 1

The positive linear correlation coefficient suggests that cities with a lot of churches also tend to have a high number of homicides. However, this does not imply causality. It does not imply that building new churches would cause an increase in the number of homicides. The correlation between the two variables is explained by their association with another variable (called a lurking variable), population. Larger cities tend to have both more churches and more homicides than small cities.

Question 2

Determine which plot shows the strongest linear correlation.

A)



B)



C)


D)

Answer 2

Question 3

A set of data consists of the number of years that applicants for foreign service jobs have studied German and the grades that they received on a proficiency test. The following regression equation is obtained: $\hat { y } = 31.6 + 10.9 x$ where x represents the number of years of study and y represents the grade on the test. Identify the predictor and response variables.

Answer 3

The predictor variable is the number of years of study, and the response variable is the grade on the test.

Question 4

Define rank. Explain how to find the rank for data which repeats (for example, the data set:
4, 5, 5, 5, 7, 8, 12, 12, 15, 18).

Question 5

A set of data consists of the number of years that applicants for foreign service jobs have studied German and the grades that they received on a proficiency test. The following regression equation is obtained: $\hat { \mathrm { y } } = 31.6 + 10.9 \mathrm { x }$ where x represents the number of years of study and y represents the grade on the test. What does the slope of the regression line represent in terms of grade on the test?

Question 6

Describe the standard error of estimate, se. How do smaller values of se relate to the dispersion of data points about the line determined by the linear regression equation?
What does it mean when se is 0?

Question 7

Given: There is no significant linear correlation between scores on a math test and scores on a verbal test.
Conclusion: There is no relationship between scores on the math test and scores on the verbal test.

Question 8

The following residual plot is obtained after a regression equation is determined for a set of data. Does the residual plot suggest that the regression equation is a bad model? Why or why not?

Question 9

Which shows the strongest linear correlation?

A)


B)


C)

Question 10

A regression equation is obtained for the following set of data. For what range of x-values
would it be reasonable to use the regression equation to predict the y-value? Why? $$\begin{array} { r | r r r r r r } \mathrm { x } & 2 & 4 & 6 & 9 & 10 & 12 \\\hline \mathrm { y } & 28 & 33 & 39 & 45 & 47 & 52\end{array}$$

Question 11

Describe the error in the stated conclusion

-Given: Each school in a state reports the average SAT score of its students. There is a significant linear correlation between the average SAT score of a school and the average annual income in the district in which the school is located.
Conclusion: There is a significant linear correlation between individual SAT scores and family income.

Question 12

Use the rank correlation coefficient to test for a correlation between the two variables.

-A placement test is required for students desiring to take a finite mathematics course at a university. The instructor of the course studies the relationship between students' placement test score and final course score. A random sample of eight students yields the following data.

$\begin{array}{ccc}\text { Placement Score }& \text { Final Course Score } \\\hline 38&63\\90&41\\95&54\\51&32\\86&93\\74&60\\60&61\\57&89\end{array}$
Compute the rank correlation coefficient, rs, of the data and test the claim of correlation between placement score and final course score. Use a significance level of 0.05.

Question 13

Describe the rank correlation test. What types of hypotheses is it used to test? How does the rank correlation coefficient rs differ from the Pearson correlation coefficient r?

Question 14

Determine which scatterplot shows the strongest linear correlation.

-Which shows the strongest linear correlation?

A)


B)



C)

Question 15

Explain what is meant by the coefficient of determination, $\mathrm { r } ^ { 2 }$ . Give an example to support your result.

Question 16

Given: There is a significant linear correlation between the number of homicides in a town and the number of movie theaters in a town.
Conclusion: Building more movie theaters will cause the homicide rate to rise.

Question 17

Define the terms predictor variable and response variable. Give examples for each.

Question 18

Nine adults were selected at random from among those working full time in the town of Workington.
Each person was asked the number of years of college education they had completed and was also asked to rate their job satisfaction on a scale of 1 to 10.
The pairs of data values area plotted in the scatterplot below.
The four points in the lower left corner correspond to employees from company A and the five points in the upper right corner correspond to employees from company B.
a. Using the pairs of values for all 9 points, find the equation of the regression line.
b. Using only the pairs of values for the four points in the lower left corner, find the equation of the regression line.
c. Using only the pairs of values for the five points in the upper right corner, find the equation of the regression line.
d. Compare the results from parts a, b, and c.

Question 19

The sample data below are the typing speeds (in words per minute) and reading speeds (in words per minute) of nine randomly selected secretaries. Here, x denotes typing speed, and y denotes reading speed.
$$\begin{array} { l l l l l l l l l l } \hline \mathrm { x } & 60 & 56 & 52 & 63 & 70 & 58 & 44 & 79 & 62 \\\hline \mathrm { y } & 370 & 551 & 528 & 348 & 645 & 454 & 503 & 618 & 500 \\\hline\end{array}$$
The regression equation $\hat { y } = 290.2 + 3.502 \mathrm { x }$ was obtained. Construct a residual plot for the data.

Question 20

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 of0.05.

$\begin{array}{c|ccccc}\mathrm{x} & 1.2 & 2.7 & 4.4 & 6.6 & 9.5 \\\hline \mathrm{y} & 1.6 & 4.7 & 8.9 & 9.5 & 12.0\end{array}$

Question 21

Ten trucks were ranked according to their comfort levels and their prices.
$$\begin{array} { c c c } \hline \text { Make } & \text { Comfort } & \text { Price } \\\hline \text { A } & \text { 1 } & 6 \\\text { B } & 6 & 2 \\\text { C } & 2 & 3 \\\text { D } & 8 & 1 \\\text { E } & 4 & 4 \\\text { F } & 7 & 8 \\\text { G } & 9 & 16 \\\text { H } & 1 Q & 9 \\\text { J } & \mathbf { 3 } & 5 \\\hline\end{array}$$
Find the rank correlation coefficient and test the claim of correlation between comfort and price. Use a significance level of 0.05.

Question 22

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.

Question 23

Use the rank correlation coefficient to test for a correlation between the two variables.

-Use the sample data below to find the rank correlation coefficient and test the claim of correlation between math and verbal scores. Use a significance level of 0.05.
$$\begin{array} { l l l l l l l l l l } \text { Mathematics } & 347 & 440 & 327 & 456 & 427 & 349 & 377 & 398 & 425 \\\hline \text { Verbal } & 285 & 378 & 243 & 371 & 340 & 271 & 294 & 322 & 385\end{array}$$

Question 24

Describe what scatterplots are and discuss the importance of creating scatterplots.

Question 25

Use the rank correlation coefficient to test for a correlation between the two variables.

-Given that the rank correlation coefficient, rs, for 73 pairs of data is -0.663, test the claim of correlation between the two variables. Use a significance level of 0.05.

Question 26

Explain why having a significant linear correlation does not imply causality. Give an example to support your answer.

Question 27

Use the rank correlation coefficient to test for a correlation between the two variables.

-The scores of twelve students on the midterm exam and the final exam were as follows. $$\begin{array} { l c c } \hline \text { Student } & \text { Midterm } & \text { Final } \\\hline \text { Navarro } & 93 & 91 \\\text { Reaves } & 89 & 85 \\\text { Hurlburt } & 71 & 73 \\\text { Knuth } & 65 & 77 \\\text { Lengyel } & 62 & 67 \\\text { Mcmeekan } & 74 & 79 \\\text { Bolker } & 77 & 65 \\\text { Ammatto } & 87 & 83 \\\text { Pothakos } & 82 & 89 \\\text { Sul1 ivan } & 81 & 71 \\\text { Hahl } & 91 & 81 \\\text { Zurfiuh } & 83 & 94 \\\hline\end{array}$$
Find the rank correlation coefficient and test the claim of correlation between midterm score and final exam score. Use a significance level of 0.05.

Question 28

The variables height and weight could reasonably be expected to have a positive linear correlation coefficient, since taller people tend to be heavier, on average, than shorter people. Give an example of a pair of variables which you would expect to have a negative linear correlation coefficient and explain why. Then give an example of a pair of variables whose linear correlation coefficient is likely to be close to zero.

Question 29

Use the rank correlation coefficient to test for a correlation between the two variables.

-A college administrator collected information on first-semester night-school students. A random sample taken of 12 students yielded the following data on age and GPA during the first semester.
$\begin{array}{llcc} \text { Age} & \text { GPA} \\ \underline{\text {x }} & \underline{\text {y}}\\18&1.2\\26&3.8\\27&2.0\\37&3.3\\33&2.5\\47&1.6\\20&1.4\\48&3.6\\50&3.7\\38&3.4\\34&2.7\\22&2.8\end{array}$
Do the data provide sufficient evidence to conclude that the variables age, $x$ , and GPA, $y$ , are correlated? Apply a rank-correlation test. Use $\alpha = 0.05$ .

Question 30

A regression equation is obtained for a set of data. After examining a scatter diagram, the researcher notices a data point that is potentially an influential point. How could she confirm that this data point is indeed an influential point?

Question 31

Suppose paired data are collected consisting of, for each person, their weight in pounds and the number of calories burned in 30 minutes of walking on a treadmill at 3.5 mph.
How would the value of the correlation coefficient, r, change if all of the weights were converted to kilograms?

Question 32

Use the rank correlation coefficient to test for a correlation between the two variables.

-Ten luxury cars were ranked according to their comfort levels and their prices. $$\begin{array} { c c c } \hline \text { Make } & \text { Comfort } & \text { Price } \\\hline \text { A } & \mathbf { 5 } & 1 \\\text { B } & 8 & 7 \\\text { C } & 9 & 3 \\\text { D } & 10 & 5 \\\text { E } & 4 & 4 \\\text { F } & 3 & 2 \\\text { G } & 2 & 10 \\\text { H } & 1 & 9 \\\text { I } & 7 & 6 \\\mathbf { J } & 6 & 8 \\\hline\end{array}$$
Find the rank correlation coefficient and test the claim of correlation between comfort and price. Use a significance level of 0.05.

Question 33

Given: The linear correlation coefficient between scores on a math test and scores on a test of athletic ability is negative and close to zero.

Conclusion: People who score high on the math test tend to score lower on the test of athletic ability.

Question 34

Create a scatterplot that shows a perfect positive correlation between x and y. How would the scatterplot change if the correlation showed a) a strong positive correlation, b) a weak positive correlation, and c) no correlation?

Question 35

Given that the rank correlation coefficien $\mathrm { r } _ { \mathrm { S } },$ for 15 pairs of data is -0.636, test the claim of correlation between the two variables. Use a significance level of 0.01.

Question 36

Use the rank correlation coefficient to test for a correlation between the two variables.

-Given that the rank correlation coefficient, r $\mathrm { r } _ { \mathrm { S } },$ , for 33 pairs of data is 0.338, test the claim of
correlation between the two variables. Use a significance level of 0.01.

Question 37

The following residual plot is obtained after a regression equation is determined for a set of data. Does the residual plot suggest that the regression equation is a bad model? Why or why not?

Question 38

What is the relationship between the linear correlation coefficient and the usefulness of the regression equation for making predictions?

Question 39

Discuss the guidelines under which the linear regression equation should be used for prediction. Refer to the correlation coefficient and the value of the predictor variable.

Question 40

Applicants for a particular job, which involves extensive travel in Spanish speaking countries, must take a proficiency test in Spanish. The sample data below were obtained in a study of the relationship between the numbers of years applicants have studied Spanish (x) and their score on the test (y).
$$\begin{array}{l}\begin{array} { l c c c c c c c c c c } \hline \mathrm { x } & 3 & 4 & 4 & 2 & 5 & 3 & 4 & 5 & 3 & 2 \\\hline \mathrm { y } & 57 & 78 & 72 & 58 & 89 & 63 & 73 & 84 & 75 & 48 \\\hline\end{array}\\\text { The regression equation } \hat { y } = 31.55 + 10.90 x \text { was obtained. Construct a residual plot for the }\end{array}$$

Question 41

Is the data point, P, an outlier, an influential point, both, or neither?

A) Influential point
B) Neither
C) Outlier
D) Both

Question 42

Is the data point, P, an outlier, an influential point, both, or neither?

-The regression equation for a set of paired data is $\hat { \mathrm { y } } = 53.5 + 0.6 \mathrm { x }$ . The values of x run from 100 to 400. A new data point, P(176, 159.1), is added to the set.

A) Neither
B) Both
C) Outlier
D) Influential point

Question 43

The following table gives the US domestic oil production rates (excluding Alaska) over the past few years. A regression equation was fit to the data and the residual plot is shown below. $\begin{array} { l } \begin{array}{c|c}\text { Year } & \text { Millions of barrels per day } \\\hline 1987 & 6.39 \\1988 & 6.12 \\1989 & 5.74 \\1990 & 5.58 \\1991 & 5.62 \\1992 & 5.46 \\1993 & 5.26 \\1994 & 5.10\end{array}&\begin{array}{c|c}\text { Year } & \text { Millions of barrels per day } \\\hline 1995 & 5.08 \\1996 & 5.07 \\1997 & 5.16 \\1998 & 5.08 \\1999 & 4.83 \\2000 & 4.85 \\2001 & 4.84 \\2002 & 4.83\end{array}\end{array}$


Does the residual plot suggest that the regression equation is a bad model? Why or why
not?

Question 44

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation
$$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the $\mathrm { t }$ -score from the $t$ Distribution table corresponding to $\mathrm { n } - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 11$ and $\alpha = 0.01$ .

A) $\pm 0.726$
B) $\pm 0.685$
C) $\pm 0.735$
D) $\pm 0.411$

Question 45

Use the rank correlation coefficient to test for a correlation between the two variables.

-Given that the rank correlation coefficient, r $\mathrm { r } _ { \mathrm { S } },$ for 20 pairs of data is 0.827, test the claim of correlation between the two variables. Use a significance level of 0.05.

Question 46

The following scatterplot shows the percentage of the vote a candidate received in the 2004 senatorial elections according to the voter's income level based on an exit poll of voters conducted by CNN. The income levels 1-8 correspond to the following income classes: $1 =$ Under $\$ 15,000 ; 2 = \$ 15 - 30,000 ; 3 = \$ 30 - 50,000 ; 4 = \$ 50 - 75,000 ; 5 = \$ 75 - 100,000 ; 6 = \$ 100 - 150,000$ ; $7 = \$ 150 - 200,000 ; 8 = \$ 200,000$ or more.



-Use the election scatterplot to determine whether there is a correlation between percentage of vote and income level at the 0.01 significance level with a null hypothesis of $\mathrm { Q } _ { \mathrm { s } } = 0 .$

A) The test statistic is between the critical values, so we fail to reject the null hypothesis . There is no evidence to support a claim of correlation between percentage of vote and income level.
B) The test statistic is between the critical values, so we reject the null hypothesis . There is sufficient evidence to support a claim of correlation between percentage of vote and income level.
C) The test statistic is not between the critical values, so we fail to reject the null hypothesis . There is no evidence to support a claim of correlation between percentage of vote and income level.
D) The test statistic is not between the critical values, so we reject the null hypothesis . There is sufficient evidence to support a claim of correlation between percentage of vote and income level.

Question 47

Is the data point, P, an outlier, an influential point, both, or neither?

-

A) Influential point
B) Neither
C) Outlier
D) Both

Question 48

The following scatterplot shows the percentage of the vote a candidate received in the 2004 senatorial elections according to the voter's income level based on an exit poll of voters conducted by CNN. The income levels 1-8 correspond to the following income classes: $1 =$ Under $\$ 15,000 ; 2 = \$ 15 - 30,000 ; 3 = \$ 30 - 50,000 ; 4 = \$ 50 - 75,000 ; 5 = \$ 75 - 100,000 ; 6 = \$ 100 - 150,000$ ; $7 = \$ 150 - 200,000 ; 8 = \$ 200,000$ or more.


-Use the election scatterplot to the find the critical values corresponding to a $0.01$ significance level used to test the null hypothesis of $\varrho _ { \mathrm { S } } = 0$ .

A) $- 0.881$
B) $- 0.738$ and $0.738$
C) $0.881$
D) $- 0.881$ and $0.881$

Question 49

Two different tests are designed to measure employee productivity and dexterity. Several employees are randomly selected and tested with these results.
$\begin{array} {l l | l | l | l | l | l | l | l | l | l } \text { Productivity}&23 & 25 & 28 & 21 & 21 & 25 & 26 & 30 & 34 & 36 \\\hline \text { Dexterity }&49 & 53 & 59 & 42 & 47 & 53 & 55 & 63 & 67 & 75\end{array}$

A) 0.115
B) 0.986
C) 0.471
D) - 0.280

Question 50

The following scatterplot shows the percentage of the vote a candidate received in the 2004 senatorial elections according to the voter's income level based on an exit poll of voters conducted by CNN. The income levels 1-8 correspond to the following income classes: $1 =$ Under $\$ 15,000 ; 2 = \$ 15 - 30,000 ; 3 = \$ 30 - 50,000 ; 4 = \$ 50 - 75,000 ; 5 = \$ 75 - 100,000 ; 6 = \$ 100 - 150,000$ ; $7 = \$ 150 - 200,000 ; 8 = \$ 200,000$ or more.



-Use the election scatterplot to the find the value of the rank correlation coefficient $\mathrm { r } _ { \mathrm { S } }$ .

A) $r _ { S } = - 1$
B) $r _ { \mathrm { S } } = 1.9762$
C) $\mathrm { r } _ { \mathrm { S } } = - 0.9762$
D) $r _ { \mathrm { S } } = 0.9762$

Question 51

Construct a scatterplot for the given data.

- $$\begin{array} { l l | l | l | r | r | r | r | r | r | r } \mathrm { x } & 2 & - 7 & - 2 & - 8 & - 9 & 2 & - 4 & 8 & - 4 & - 1 \\\hline \mathrm { y } & - 3 & - 7 & - 2 & 4 & 3 & 2 & 1 & 5 & - 7 & - 5\end{array}$$

A)


B)


C)


D)

Question 52

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

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

Question 53

Find the value of the linear correlation coefficient r.

- $$\begin{array} { r | r r r r r r r } \mathrm { x } & 57 & 53 & 59 & 61 & 53 & 56 & 60 \\\hline \mathrm { y } & 156 & 164 & 163 & 177 & 159 & 175 & 151\end{array}$$

A) 0.214
B) -0.078
C) -0.054
D) 0.109

Question 54

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.

- $r = - 0.466 , n = 15$

A) Critical values: $\mathrm { r } = \pm 0.514$ , no significant linear correlation
B) Critical values: $r = \pm 0.532$ , no significant linear correlation
C) Critical values: $r = \pm 0.514$ , significant linear correlation
D) Critical values: $\mathrm { r } = 0.514$ , no significant linear correlation

Question 55

For which of the following sets of data points can you reasonably determine a regression line?

A) II, III, and IV
B) All of these
C) None of these
D) II and III

Question 56

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation:
$$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the $t$ -score from the $t$ Distribution table corresponding to $n - 2$ degrees of freedom. Use this approximation to find critical values of $r _ { \mathrm { S } }$ for the case where $n = 40$ and $\alpha = 0.10$ .

A) $\pm 0.312$
B) $\pm 0.304$
C) $\pm 0.264$
D) $\pm 0.202$

Question 57

Find the value of the linear correlation coefficient r.

-Two separate tests are designed to measure a student's ability to solve problems. Several students are randomly selected to take both tests and the results are shown below. $$\begin{array} { c | l | l | l | l | l | l | l | l | l } \text { Test A } & 48 & 52 & 58 & 44 & 43 & 43 & 40 & 51 & 59 \\\hline \text { Test B } & 73 & 67 & 73 & 59 & 58 & 56 & 58 & 64 & 74\end{array}$$

A) 0.109
B) 0.714
C) 0.548
D) 0.867

Question 58

Use the given data to find the best predicted value of the response variable.

-Eight pairs of data yield $\mathrm { r } = 0.708$ and the regression equation $\hat { y } = 55.8 + 2.79 x$ . Also, $\bar { y } = 71.125$ What is the best predicted value of $y$ for $x = 9.9 ?$

A) 71.13
B) 83.42
C) 555.21
D) 57.80

Question 59

The paired data below consist of the test scores of 6 randomly selected students and the number of hours they studied for the test. $$\begin{array} { c | r r r r r r } \text { Hours } & 5 & 10 & 4 & 6 & 10 & 9 \\\hline \text { Score } & 64 & 86 & 69 & 86 & 59 & 87\end{array}$$

A) 0.224
B) -0.224
C) -0.678
D) 0.678

Question 60

The regression equation for a set of paired data is $\hat { y } = 9 + 2 x$ . The correlation coefficient for the data is 0.87. A new data point, P(12, 53), is added to the set.

A) Outlier
B) Influential point
C) Both
D) Neither

Question 61

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.

- $\mathrm { r } = 0.893 , \mathrm { n } = 9$

A) Critical values: $r = \pm 0.666$ , significant linear correlation
B) Critical values: $r = 0.666$ , no significant linear correlation
C) Critical values: $r = \pm 0.666$ , no significant linear correlation
D) Critical values: $\mathrm { r } = - 0.666$ , no significant linear correlation

Question 62

Construct a scatterplot for the given data

- $\begin{array}{r|r|r|r|r|r|r|r|r}\mathrm{x} & 0.55 & 0.33 & 0.24 & 0.2 & -0.34 & 0.57 & 0.41 & 0.14 \\\hline \mathrm{y} & 0.68 & 0.78 & 0.66 & 0.34 & -0.13 & 0.95 & 0.8 & -0.12\end{array}$

A)


B)


C)


D)

Question 63

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): $$\begin{array} { c | r r r r r r r r r } \text { Temp } & 62 & 76 & 50 & 51 & 71 & 46 & 51 & 44 & 79 \\\hline \text { Growth } & 36 & 39 & 50 & 13 & 33 & 33 & 17 & 6 & 16\end{array}$$

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

Question 64

A study was conducted to compare the average time spent in the lab each week versus course grade for computer programming students. The results are recorded in the table below. $\begin{array}{cc}\text { Number of hours spent in lab } & \text { Grade (percent) } \\\hline 10 & 96 \\11 & 51 \\16 & 62 \\9 & 58 \\7 & 89 \\15 & 81 \\16 & 46 \\10 & 51\end{array}$

A) $- 0.284$
B) $0.462$
C) $0.017$
D) $- 0.335$

Question 65

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data.

- $\mathrm { n } = 16 , \alpha = 0.01$

A) $- 0.635$
B) $\pm 0.635$
C) $\pm 0.503$
D) $0.635$

Question 66

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximation:
$$r _ { S } = \pm \sqrt { \frac { t ^ { 2 } } { t ^ { 2 } + n - 2 } }$$
where $t$ is the $\mathrm { t }$ -score from the $t$ Distribution table corresponding to $\mathrm { n } - 2$ degrees of freedom. Use this approximation to find critical values of $\mathrm { r } _ { \mathrm { S } }$ for the case where $\mathrm { n } = 7$ and $\alpha = 0.05$ .

A) $\pm 0.669$
B) $\pm 0.569$
C) $\pm 0.755$
D) $\pm 0.448$

Question 67

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data.

- $\mathrm { n } = 10 , \alpha = 0.05$

A) $\pm 0.648$
B) $- 0.648$
C) $0.648$
D) $\pm 0.564$

Question 68

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary.

- $\begin{array}{l|lllll}\mathrm{x} & 24 & 26 & 28 & 30 & 32 \\\hline \mathrm{y} & 15 & 13 & 20 & 16 & 24\end{array}$

A) $\hat { y } = - 11.8 + 1.05 x$
B) $\hat { y } = 11.8 + 1.05 x$
C) $\hat { y } = 11.8 + 0.950 \mathrm { x }$
D) $\hat { y } = - 11.8 + 0.950 x$

Question 69

Use the given data to find the best predicted value of the response variable.

-Four pairs of data yield $\mathrm { r } = 0.942$ and the regression equation $\hat { y } = 3 x$ . Also, $\bar { y } = 12.75$ . What is the best predicted value of $y$ for $x = 5.8$ ?

A) 0.942
B) 2.826
C) 12.75
D) 17.4

Question 70

Which of the following statements concerning the linear correlation coefficient are true? I: If the linear correlation coefficient for two variables is zero, then there is no relationship between The variables.
II: If the slope of the regression line is negative, then the linear correlation coefficient is negative.
III: The value of the linear correlation coefficient always lies between -1 and 1.
IV: A linear correlation coefficient of 0.62 suggests a stronger linear relationship than a linear Correlation coefficient of -0.82.

A) III and IV
B) I and II
C) II and III
D) I and IV

Question 71

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary.

- $\begin{array}{l|lllll}\mathrm{x} & 0 & 3 & 4 & 5 & 12 \\\hline \mathrm{y} & 8 & 2 & 6 & 9 & 12\end{array}$

A) $\hat { y } = 4.98 + 0.725 x$
B) $\hat { y } = 4.88 + 0.625 x$
C) $\hat { y } = 4.98 + 0.425 x$
D) $\hat { y } = 4.88 + 0.525 x$

Question 72

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary.

- $\begin{array}{r|rrrrr}\mathrm{x} & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline \mathrm{y} & 54 & 53 & 55 & 54 & 56\end{array}$

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

Question 73

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.

- $\mathrm { r } = - 0.285 , \mathrm { n } = 90$

A) Critical values: $r = \pm 0.217$ , no significant linear correlation
B) Critical values: $r = \pm 0.207$ , no significant linear correlation
C) Critical values: $\mathrm { r } = \pm 0.207$ , significant linear correlation
D) Critical values: $\mathrm { r } = 0.217$ , significant linear correlation

Question 74

$$\begin{array} { r | r | r | r | r | r| r | r | r | r } \mathrm { x } & - 1 & 3 & 8 & 5 & 11 & 10 & - 1 & - 3 & - 1 \\\hline \mathrm { y } & 2 & 7 & 9 & 11 & 9 & 11 & 5 & 4 & 2\end{array}$$

A)


B)



C)

Question 75

Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they entered the program were between 3.5 and 4.0. The following data were obtained regarding their GPAs on entering the program versus their current GPAs. $\begin{array}{cc}\text { Entering GPA } & \text { Current GPA } \\\hline 3.5 & 3.6 \\3.8 & 3.7 \\3.6 & 3.9 \\3.6 & 3.6 \\3.5 & 3.9 \\3.9 & 3.8 \\4.0 & 3.7 \\3.9 & 3.9 \\3.5 & 3.8 \\3.7 & 4.0\end{array}$

A) $\hat { y } = 3.67 + 0.0313 x$
B) $\hat { y } = 2.51 + 0.329 x$
C) $\hat { y } = 4.91 + 0.0212 x$
D) $\hat { y } = 5.81 + 0.497 x$

Question 76

Use the given data to find the best predicted value of the response variable.

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

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

Question 77

Find the value of the linear correlation coefficient r.

-Managers rate employees according to job performance and attitude. The results for several randomly selected employees are given below. $$\begin{array} { c c | c | c | c | c | c | c | c | c | c } \text { Performance } & 59 & 63 & 65 & 69 & 58 & 77 & 76 & 69 & 70 & 64 \\\hline \text { Attitude } & 72 & 67 & 78 & 82 & 75 & 87 & 92 & 83 & 87 & 78\end{array}$$

A) 0.610
B) 0.863
C) 0.729
D) 0.916

Question 78

Use the given data to find the equation of the regression line. Round the final values to three significant digits, if necessary.

-Managers rate employees according to job performance and attitude. The results for several randomly selected employees are given below.
$\begin{array}{cc|c|c|c|c|c|c|c|c|c}\text { Performance } & 59 & 63 & 65 & 69 & 58 & 77 & 76 & 69 & 70 & 64 \\\hline \text { Attitude }& 72 & 67 & 78 & 82 & 75 & 87 & 92 & 83 & 87 & 78\end{array}$

A) $\hat { y } = 2.81 + 1.35 x$
B) $\hat { y } = 92.3 - 0.669 \mathrm { x }$
C) $\hat { y } = 11.7 + 1.02 x$
D) $\hat { y }= - 47.3 + 2.02 \mathrm { x }$

Question 79

Find the critical value. Assume that the test is two-tailed and that n denotes the number of pairs of data.

- $\mathrm { n } = 50 , \alpha = 0.05$

A) $\pm 0.280$
B) $\pm 0.277$
C) $- 0.280$
D) $0.280$

Question 80

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 $\alpha$

- $\mathrm { n } = 6 , \alpha = 0.05$

A) $r = \pm 0.811$
B) $\mathrm { r } = \pm 0.917$
C) $r = 0.811$
D) $r = 0.878$