Deck 13: Correlation and Linear Regression

Question 1

The confidence interval estimate of the expected value of y for a given value of x, when compared to the prediction interval of y for the same given value of x at the same confidence level, will be:

A) wider.
B) narrower.
C) the same.
D) impossible to know.

Answer 1

narrower.

Question 2

Factors A and B show a correlation of 0.95.This means that:

A) B causes A.
B) A causes B.
C) either A causes B or B causes A.
D) nothing; correlation does not mean the factors are actually linked.

Answer 2

nothing; correlation does not mean the factors are actually linked.

Question 3

Medical researchers conducted a study designed to evaluate the relationship between blood pressure (y), age (x1), and cholesterol (x2).They proposed a linear model.A total of 126 individuals were randomly sampled, and blood pressure, age, and cholesterol were recorded for each.The regression summary appears here: ​

Which predictor (if any) does not appear to be linearly correlated with blood pressure? Which reason supports your answer?

A) All of the predictors are significant because the p-value from the global utility test is essentially zero.
B) All of the predictors are significant because the R-Sq value is quite high (85%).
C) Cholesterol does not appear to be linearly correlated with blood pressure because its p- value is 0.937.
D) Age does not appear to be linearly correlated with blood pressure because its p-value is essentially zero.

Answer 3

Cholesterol does not appear to be linearly correlated with blood pressure because its p- value is 0.937.

Question 4

Ten used cars from a rental fleet are randomly selected.Each vehicle's annual maintenance cost from the previous calendar year (y) is recorded as well as the vehicle's current mileage (x).A simple linear regression is conducted on the data. ​

​
What is a 98% confidence interval for the mean annual repair cost of all used vehicles with 22,500 miles.=> (Given: Sxx= 8,036,677,024 and = $43,627.700.)

A) $(120.00, 759.20)
B) $(-431.51, 1,310.72)
C) $(22,199.25, 22,800.75)
D) $(128.29, 750.91)

Question 5

Which value of the correlation coefficient r indicates a stronger correlation than 0.72?

A) -0.89
B) -0.69
C) 0.55
D) 0.70

Question 6

The following ANOVA table describes a simple linear regression on two arbitrary variables x and y. ​

​
Based on the output, does there appear to be a significant linear relationship between x and y? Select the appropriate critical value (CV), test statistic (TS), and decision.(Use α = 0.01.)

A) CV = 240.54; TS = 52.618; no significant linear relationship.
B) CV = 5.12; TS = 6.578; there appears to be a significant linear relationship.
C) CV = 5.32; TS = 0.152; no significant linear relationship.
D) CV = 240.54; TS = 0.019; there appears to be a significant linear relationship.

Question 7

According to the spurious correlations website (https://www.tylervigen.com/spurious-correlations), there is a 0.985065 correlation from 2000 to 2009 between revenue from arcade games and the number of computer science doctorates awarded in the United States.Assuming this correlation reflects a real cause-effect relationship, what is a possible link between these two factors?

A) This is complete nonsense and should be ignored.
B) Kids who play lots of arcade games all go on to study computer science.
C) People who go on to get their degrees in computer science are attracted to computer-based games.
D) Arcades give discounts to PhD students.

Question 8

Consider the following regression summary: ​

Perform a global utility test for the model.State the correct decision and reasoning.(Use α = 0.01.)

A) The model appears to be useful because p-value = 0.000 < 0.01.
B) The model does not appear to be useful because p-value = 0.014 > 0.01.
C) The model appears to be useful because R-Sq (adj) is more than 0.01 less than R-Sq.
D) The model does not appear to be useful because the p-value = 0.018 > 0.01.

Question 9

If all the values of an independent variable x are equal, then performing a simple linear regression on a dependent variable y will result in a coefficient of determination that is equal to:

A) infinity.
B) 1.0.
C) 0.5.
D) 0.

Question 10

According to the spurious correlations website (https://www.tylervigen.com/spurious-correlations), there is a 0.952407 correlation from 1999 to 2010 between people who drowned after falling from a fishing boat and the marriage rate in Kentucky.This shows that:

A) the data are somehow wrong.
B) survivors of drowning victims like to get remarried in Kentucky.
C) people married in Kentucky do not know how to swim.
D) A strong correlation may be meaningless and just a coincidence.

Question 11

In simple linear regression, most often we perform a two-tailed test of the population slope to determine whether there is sufficient evidence to infer that a linear relationship exists.The null hypothesis is stated as:

A) H₀: = 0.
B) H₀: = b1.
C) H₀: = 1.
D) H₀: ≠ 0.

Question 12

Consider the following regression analysis for predicting sales per week from intelligence and extroversion scores: ​

What proportion of the variability in sales per week is explained by the model?

A) 0.1066
B) 0.224
C) 0.3526
D) 0.021

Question 13

We record the amount of time (x in minutes) per day that a group of 12 high school students spend using their cell phone (texting, talking, and web use).We also record the overall GPA (y) of each student.Following is a regression summary for these data: ​

​
What is a 95% confidence interval for the average GPA change associated with a 1-min increase in average daily cell use (β1)?

A) (-0.0225, -0.0082)
B) (3.4545, 4.5623)
C) (-0.0212, -0.0095)
D) (3.5578, 4.4590)

Question 14

In a simple linear regression analysis, the standard error of the slope is:

A) a measure of the variation in the regression slope from sample to sample.
B) a measure of the amount of change in the dependent variable y for a one-unit change in the independent variable x.
C) the square root of the standard error of the estimate.
D) the square of the standard error of the estimate.

Question 15

A simple linear regression between y (weight in pounds) and x (height in inches) results in the following least-squares line: y = 128 + 6x.This tells us that if the height is increased by 1 in, then.on average.the weight is predicted to:

A) decrease by 1 lb.
B) decrease by 6 lb.
C) increase by 6 lb.
D) increase by 128 lb.

Question 16

Consider the following regression summary: ​

What proportion of the Y3 variability is explained by the model?

A) 0.081
B) 0.014
C) 0.018
D) 0.193

Question 17

From a random sample of n = 6 college students, a school counselor obtained the students' high school GPAs (x) and their college GPAs (y).Using these data, she wishes to develop a model that will allow her to predict a student's college GPA from his or her high school GPA. ​ ​
What is the fitted least-squares regression equation?

A) y = 0.326 + 0.8660x
B) y = 3.360 + 0.0114x
C) y = 0.887 + 0.769x
D) y = 76.985 - 248.344x

Question 18

What is the difference between a general additive probabilistic model and a general deterministic model?

A) The deterministic model includes random deviation, but the probabilistic model does not.
B) The probabilistic model includes random deviation, but the deterministic model does not.
C) The two models differ only in terms of the type of deterministic functions appropriate for the models.
D) They are essentially the same (both names apply to the same model).

Question 19

The regression equation is y1 = 0.33 + 0.866 x1
​

​
Based on this regression summary, what is r2?

A) 0.8158
B) 0.6656
C) 0.3344
D) 0.5025

Question 20

In the late 1890s, scientists first noted that the frequency of a cricket's chirps is related to air temperature.For each of 30 crickets, the frequency of chirping (in chirps per minute) and the air temperature (in degrees Fahrenheit) were recorded, and a regression analysis was performed. ​

​
Given that Sxx= 14,497.9 and 067, what is a 90% prediction interval for the air temperature when a cricket is chirping at a frequency of 120 chirps per minute?

A) (66.539, 67.301)
B) (63.818, 68.022)
C) (65.462, 66.379)
D) (62.389, 67.451)

Question 21

In simple linear regression, the test statistic t is:

A) the square root of the corresponding F statistic.
B) twice the value of the corresponding F statistic.
C) the absolute value of the corresponding F statistic.
D) unrelated to any other statistic.

Question 22

Consider these five x, y pairs. ​ ​
The regression equation is What is ?

A) 1.19
B) 2.39
C) -2.39
D) -1.19

Question 23

Is it always possible to do a linear best fit to a data set?

A) Yes; the fact that all data are affected by random error makes them linear.
B) Yes, but the fit may be meaningless if the data are not really linear.
C) No; the data may not be linear to begin with.
D) No; the value may be impossible to calculate.

Question 24

The equation (where E is a random variable) represents:

A) a general additive probabilistic model.
B) a general additive deterministic model.
C) a fixed, one-to-one relationship between x and y.
D) a deterministic model with x as the independent variable and y as the dependent variable.

Question 25

We record the amount of time (xin minutes) per day that a group of 12 high school students spend using their cell phone (texting, talking, and web use).We also record the overall GPA (y) of each student.Following is a regression summary for these data: ​

​
What is a 90% confidence interval for β0, the mean GPA of high school students who spent no time using their cell phone?

A) (-0.0225, -0.0082)
B) (3.4545, 4.5623)
C) (-0.0212, -0.0095)
D) (3.5578, 4.4590)

Question 26

We record the amount of time (x in minutes) per day that a group of 12 high school students spend using their cell phone (texting, talking, and web use).We also record the overall GPA (y) of each student.Following a regression summary for these data: ​

​
Does there appear to be a significant linear relationship between the amount of cell phone use and GPA? (Use α = 0.01.)

A) No, because the r2 value is too low.
B) Yes, because the regression equation is defined.
C) Yes, because the p-value from the test for significance is 0.000.
D) Yes, because the p-value from the test for significance is 0.001.

Question 27

From the following residual versus fit plot, it appears that:

A) the mean error is not zero.
B) there may be a problem with nonconstant variance.
C) the errors do not violate any regression assumptions.
D) The regression equation is not shown, so it is not possible to make a determination.

Question 28

Consider the following regression analysis for predicting sales per week from intelligence and extroversion scores: ​

What is a 90% confidence interval for the average sales per week for all people with an intelligence score of 103 and an extroversion score of 19?

A) (2594.34, 3176.91)
B) (2609.35, 2960.72)
C) (2181.25, 3388.83)
D) (3072.19, 3687.78)

Question 29

Assume a least-squares regression line is found to be y = -5.73 + 0.282x.The coefficient of determination for these data is 0.64.What is the correlation coefficient?

A) 0.411
B) -0.64
C) 0.80
D) -0.80

Question 30

Ten used cars from a rental fleet are randomly selected.Each vehicle's annual maintenance cost from the previous calendar year (y) is recorded as well as the vehicle's current mileage (x).A simple linear regression is conducted on the data. ​

What is an 80% prediction interval for the annual repair cost of a used vehicle with 40,000 miles? (Given: Sxx= 8,036,677,024 and = $43,627.700.)

A) $(144.72, 956.96)
B) $(140.67, 961.01)
C) $(426.25, 675.43)
D) $(427.48, 674.20)

Question 31

In simple linear regression, the plot of residuals versus fitted values can be used to check for:

A) normality.
B) independence.
C) a constant variance independent of x.
D) All of the above.

Question 32

Based on the following scatter plot of age versus blood pressure, which of the regression assumptions may be violated?

A) Mean error = 0.
B) Error variance is constant.
C) Errors are normally distributed.
D) Errors are independent.

Question 33

If the sample correlation coefficient is -0.95, what can we conclude?

A) The two variables are not correlated.
B) The two variables are weakly correlated.
C) An increase in the independent variable is associated with a decrease in the dependent variable.
D) An increase in the independent variable is associated with an increase in the dependent variable.

Question 34

In the late 1890s, scientists first noted that the frequency of a cricket's chirps is related to air temperature.For each of 30 crickets, the frequency of chirping (in chirps per minute) and the air temperature (in degrees Fahrenheit) were recorded, and a regression analysis was performed. ​

Given that Sxx= 14,497.9 and 117.067, what is a 95% confidence interval for the mean air temperature when a cricket is chirping at a frequency of 120 chirps per minute?

A) (66.539, 67.301)
B) (63.818, 68.022)
C) (65.462, 66.379)
D) (62.389, 67.451)

Question 35

In a regression analysis, the residuals represent:

A) the difference between the actual x and y values.
B) the difference between the actual y values and the predicted values.
C) the difference between the actual x values and the predicted values.
D) the change in y per a one-unit change in x.

Question 36

In simple linear regression, the plot of residuals versus fitted values should:

A) be free of any patterns.
B) appear to be a random scatter of points about 0 on the vertical axis.
C) have approximately the same spread for all fitted values.
D) All of the above.

Question 37

The appropriate pattern for a residual plot is:

A) an overall parabolic shape.
B) a funnel shape with the tapered end at the lower values.
C) a funnel shape with the tapered end at the higher values.
D) a random scatter centered at zero.

Question 38

If the correlation coefficient r is 0.81, what percentage of variation in the dependent variable is explained by a linear regression with the independent variable?

A) 90%
B) 81%
C) 66%
D) 10%

Question 39

In a simple linear regression, if the coefficient of determination is 0.94, then:

A) 94% of the x-values are equal.
B) 98% of the total variation in x can be explained by the regression line.
C) 94% of the total variation in y can be explained by the regression line.
D) 98% of the total variation in y can be explained by the regression line.

Question 40

Suppose , where .This is:

A) a simple linear regression model.
B) a deterministic model.
C) a generalized linear model.
D) a deterministic relationship.

Question 41

Ten used cars from a rental fleet are randomly selected.Each vehicle's annual maintenance cost from the previous calendar year (y) is recorded as well as the vehicle's current mileage (x).A simple linear regression is conducted on the data. ​

Conduct a test of hypothesis to see if there is sufficient evidence to conclude that used vehicles with 22,500 miles will have a mean annual maintenance cost that exceeds $400.Select the appropriate critical value (CV), test statistic (TS), and decision.(Use α = 0.05).(Given: Sxx= 803,667,7024 and = $43,627.700.)

A) CV = 1.8331; TS = 0.1415; fail to reject H₀
B) CV = 1.8595; TS = 0.02611; fail to reject H₀
C) CV = 1.8331; TS = 0.2986; fail to reject H₀
D) CV = 1.8595; TS = 0.3589; fail to reject H₀

Question 42

A study is designed to evaluate the relationship between a person's height and his or her salary.A random sample of 800 workers at a large corporation is drawn, and each individual's height (x) and salary (y) are recorded.If the coefficient of determination from the sample is 0.126, we can conclude that:

A) 12.6% of the salaries are correlated with height.
B) 12.6% of the variability in the salaries is explained by the model.
C) 0.126% of the variability in the salaries is explained by the model.
D) there is a strong positive linear correlation between salary and height.

Question 43

In the linear regression model, the homogeneity of variance assumption basically means:

A) the variance in y at any given x-value is random.
B) the variance in x at any given y-value is constant.
C) the y variance at any given x within the range of the model is constant.
D) the amount of variance in y is smallest when x is the smallest value considered in the model.

Question 44

The statement that "correlation does not imply causality" indicates that:

A) a strong correlation coefficient does not mean there is a cause-effect relationship between factors.
B) a weak correlation coefficient means there is weak link between factors.
C) there is an excuse to ignore the meaning of a correlation.
D) None of the above.

Question 45

Which of the following is NOT an assumption for the simple linear regression model?

A) The error terms are independent of each other.
B) The error terms have equal variances for all values of the independent variable.
C) The distribution of the error terms will be skewed to the left or right depending on the values of the dependent variable.
D) The mean of the dependent variable for all levels of the independent variable can be connected by a straight line.

Question 46

The amount of variability explained by the regression model is:

A) the coefficient of regression.
B) the coefficient of explanation.
C) the coefficient of correlation.
D) the coefficient of determination.

Question 47

The least-squares method for determining the best fit minimizes the:

A) total variation in the dependent variable.
B) sum of squares for regression.
C) sum of squares for error.
D) sum of squares for the x-values.

Question 48

An instructor's records show a 0.956 correlation between grades and attendance.If the correlation implies causality, what might this imply?

A) The instructor could get away with giving grades based solely on attendance.
B) Effort should be expended to make sure all students attend all classes.
C) There should be many questions on exams that can be answered only if the student attends class.
D) All the above.

Question 49

In a deterministic relationship between two variables x and y of the form it is understood that:

A) completely determines
B) x and y are not related.
C) y is completely determined by x.
D) x and y have similar properties.

Question 50

In the following residual plot:

A) the plot appears to be random, so there are no major concerns.
B) the plot clearly shows that the residuals are not normally distributed.
C) there appears to be at least one highly influential observation.
D) the mean of the error distribution appears to differ from zero.

Question 51

The linear regression model assumes the error random variable is:

A) a constant value.
B) normally distributed.
C) centered at some positive value.
D) a latent variable with an unknown distribution.

Question 52

The linear relationship (where and are constants) is an example of:

A) an indeterminate relationship.
B) a linear deterministic relationship.
C) a linear probabilistic relationship.
D) a nonlinear deterministic relationship.

Question 53

This histogram shows residuals from a simple linear regression: It appears that:

A) there is a dependence among the errors.
B) the mean of the error distribution is not zero.
C) the error distribution is not normal.
D) the error variance is not constant.

Question 54

If the sum of squares for error (SSE) is equal to zero, then the coefficient of determination, r2, must be:

A) 1.5.
B) 1.0.
C) 0.5.
D) 0

Question 55

The multiple linear regression model assumes:

A) the mean error is zero, error variance is nonconstant, and the errors are independent and normally distributed.
B) the mean error is not zero, error variance is constant, and the errors are independent and normally distributed.
C) the sample size is large (at least 30), and errors are independent and normally distributed.
D) the mean error is zero, error variance is constant, and the errors are independent and normally distributed.

Question 56

Concerning the simple linear regression model, SST can be expressed as:

A) SSA + SSB + SSE.
B) SSR + SSE.
C) SST + SSR.
D) SST + SSR + SSE.

Question 57

Given a specific value of x and a confidence level, which of the following statements is correct?

A) The confidence interval estimate of the expected value of y can be calculated, but the prediction interval of y for the given value of x cannot be calculated.
B) The confidence interval estimate of the expected value of y will be narrower than the prediction interval.
C) The prediction interval of y for the given value of x can be calculated, but the confidence interval estimate of the expected value of y cannot be calculated.
D) The confidence interval estimate of the expected value of y will be wider than the prediction interval.