Deck 15: Understanding Relationships - Numerical Data Part 2

Question 1

Use the scatterplot below to answer the following question: ​ When a linear model is fit to the data, which of the following can be stated with the greatest confidence?
​

A)The residual for point B is greater than the residual for point D.
B)The residual for point A is greater than the residual for point C.
C)The residual for point C is greater than the residual for point B.
D)The residual for point B is greater than the residual for point A.
E)The residual for point C is greater than the residual for point D.

Answer 1

The residual for point C is greater than the residual for point B.

Question 2

The point estimate a + bx* is an unbiased statistic that can be used to estimate the mean value of y when x = x*.

Answer 2

True

Question 3

In the simple linear regression, the standard deviation of y is the same as the standard deviation of the random deviation
e.

Answer 3

True

Question 4

A residual is a deviation from the population regression line.

Question 5

In the simple linear regression mode, σ_e depends on the value of x.

Question 6

The estimated standard deviation, s_e, has n − 1 degrees freedom.

Question 7

The estimated standard deviation, s_e, is roughly the magnitude of a typical deviation from the least squares regression line.

Question 8

If there is no evidence of a linear relationship between x and y, what decision will be made in a test of ? ​

A)Reject and conclude that there is no evidence that the linear model is useful
B)Reject and conclude that there is evidence that the linear model is useful
C)Fail to reject and conclude that there is no evidence that the linear model is useful
D)Fail to reject and conclude that there is evidence that the linear model is useful
E)Not enough information to say.

Question 9

A study was carried out to investigate the relationship between x = the number of components needing repair and y = the time of the service call (in minutes) for a computer repair company. The number of components and the service times for a random sample of 20 service calls were used to fit a simple linear regression model. Partial computer output is shown below. ______________________________________________________________
The regression equation is Time = 32.441 + 11.6205 Number ​
S=19/058, R- Sq=91.3%, R-Sq(adj)=90.0%
______________________________________________________________
One of the numbers in the above computer output indicates the number of additional minutes required per additional component serviced.
What is the value of this number?

A)32.441
B)11.6205
C)8.121
D)0.7988
E)19.058

Question 10

A standardized residual plot with spread increasing from left to right suggests that the variance of y is not the same at each x value.

Question 11

In the simple linear regression model, α and β are fixed numbers that are usually unknown.

Question 12

The simple linear regression model is a deterministic model.

Question 13

The estimated mean value of y is a + bx* when x has the value x*.

Question 14

In the simple linear regression model, the point estimate and the point prediction are identical for a particular value of x.

Question 15

A study was carried out to investigate the relationship between x = the number of components needing repair and y = the time of the service call (in minutes) for a computer repair company. The number of components and the service times for a random sample of 20 service calls were used to fit a simple linear regression model. Partial computer output is shown below. ______________________________________________________________
The regression equation is Time = 32.441 + 11.6205 Number ​ S=19.058, R-Sq=91.3%, R-Sq(adj)=90.0%
______________________________________________________________
One of the numbers in the above computer output indicates the magnitude of a typical difference between an observed service time and the service call time predicted by the linear model.
What is the value of this number?
​

A)8.121
B)0.7988
C)11.6205
D)12.31
E)19.058

Question 16

The general form of a probabilistic model is y = deterministic function + random deviation.

Question 17

The expected change in the value of y for one unit change in x is α.

Question 18

A study was carried out to investigate the relationship between x = the number of components needing repair and y = the time of the service call (in minutes) for a computer repair company. The number of components and the service times for a random sample of 20 service calls were used to fit a simple linear regression model. Partial computer output is shown below. ______________________________________________________________
The regression equation is Time = 32.441 + 11.6205 Number ​
S=19.058, R-Sq=91.3%, R-Sq(adj)=90.0%
______________________________________________________________
One of the numbers in the above computer output indicates the percentage of the observed variation in the service call times that can be explained by the linear relationship between service call time and number of components needing repair.
What is the value of this number?

A)91.3
B)90.0
C)19.058
D)32.441
E)0.7988

Question 19

In the simple linear regression model, the randomness of e implies that y itself is subject to uncertainty.

Question 20

The mean value of the statistic b is β.

Question 21

Let x be the number of pages in a statistics text book and y be the cost of the book (in dollars.) Suppose that x and y are related according to the simple linear regression model with
β
= slope of the population regression line = 0.075
α
= y intercept of population regression line = 25.0
​
a)What is the equation of the population regression line?
b)On the graph below, sketch the population regression line using the information above. Be sure to label and scale your axes correctly. c)For a statistics book of 900 pages, what is the predicted cost in dollars?
d)What is the average change in price that is associated with a one-page increase in size of a statistics book?
e)What is the average change in price that is associated with a 20-page increase in size of a statistics book?

Question 22

The scatter plot and residual plot shown below is from a regression analysis of Prehistoric Native American projectile points. The widths of the projectiles were regressed on the length of the projectile points. Using these plots categorize each of the Basic Assumptions of the simple linear regression model as one of the following:
a)Not checkable
b)Checked and met
c)Checked and not met
For all assumptions checked, explain what aspect of the graphs led you to your conclusion that the assumption was met or not met.

Question 23

What are the four "Basic Assumptions" of the simple linear regression model?

Question 24

As part of a study of the swimming speed of sharks, a random sample of 18 lemon sharks (Triakis semifasciata) were observed in a laboratory sea tunnel. Body lengths and maximum sustainable swimming speeds ("MSSS," reported in body lengths per second) were measured for each shark. The computer output from a regression with y = MSSS and x = body length is given below. ​
Linear Fit MSSS=1.8928955-0.0104278 * Length
Summary of Fit
RSquare 0.526395
RSquare Adj 0.496794
S 0)272031
Mean of Response 1.24
N Observations 18
​
Analysis of Variance
Source DF Sum of Squares Mean Square F Ratio
Model 1 1.3159870 1.31599 17.7834
Error 16 1.1840130 0.07400 Prob > F
Total 17 2.5000000 0.0007*
​
Parameter Estimates
Term Estimate Std Error t Ratio Prob > |t|
Intercept 1.8928955 0.167575 11.30 < 0.0001*
Length(cm) -0.010428 0.002473 -4.22 0.0007*
​
According to the above computer output, if body length increases by one unit, by how much will the swimming speed, in body lengths per second, change?
​

A)1.893
B)-0.0104
C)-1.893
D)1.184
E)-4.22

Question 25

In a study of the surface temperatures of vital (living) and non-vital (dead) teeth in a random sample of 32 adults seeking dental care at a university clinic, temperatures of vital and non-vital teeth were measured for each individual. The computer output shown below was generated. ​
a)What proportion of observed variation in the surface temperature of vital teeth can be explained by the surface temperature of non-vital teeth?
b)Does the simple linear regression model appear to be useful? Justify your response with specific references to the computer output.

Question 26

The computer output given below shows a regression of Social Studies Concepts vs. Reading Comprehension scores for a random sample of sophomores. a)What is the estimated regression line?
b)What is the calculated value of s_b?
b)What is the calculated value of s_e?
d)What is the calculated value of the t-ratio for the Model Utility Test?
e)Construct a 95% confidence interval for β.

Question 27

In a study of the surface temperatures of vital (living) and non-vital (dead) teeth in a random sample of 32 adults seeking dental care at a university clinic, temperatures of vital and non-vital teeth were measured for each individual. From their examination of the data the investigators felt that the assumptions of the linear regression model were satisfied. The computer output shown below was generated. ​
a)What proportion of observed variation in the surface temperature of non-vital teeth can be explained by the surface temperature of vital teeth?
b)Does the simple linear regression model appear to be useful? Justify your response with specific references to the computer output.

Question 28

As part of a study of the swimming speed of sharks, a random sample of 18 lemon sharks (Triakis semifasciata) were observed in a laboratory sea tunnel. Body lengths and maximum sustainable swimming speeds ("MSSS," reported in body lengths per second) were measured for each shark. The computer output from a regression with y = MSSS and x = body length is given below. ​
Linear Fit ​
Summary of Fit
RSquare 0.526395
RSquare Adj 0.496794
S 0)272031
Mean of Response 1.24
N Observations 18
​
Analysis of Variance
Source DF Sum of Squares Mean Square F Ratio
Model 1 1.3159870 1.31599 17.7834
Error 16 1.1840130 0.07400 Prob > F
Total 17 2.5000000 0.0007*
​
Parameter Estimates
Term Estimate Std Error t Ratio Prob > |t|
Intercept 1.8928955 0.167575 11.30 < 0.0001*
Length(cm) -0.010428 0.002473 -4.22 0.0007*
​
The null hypothesis is that there is no linear relationship between body length and maximum sustainable swimming rate. At the 0.05 level of significance, should the null hypothesis be rejected? Assume all the proper conditions are met for applying a linear regression analysis.
​

A)Yes
B)No
C)There is not enough information to tell.

Question 29

The Iowa Tests of Educational Development are given each year to students around the country. Let x = the previous year's science scores and y = the current year's science scores for a sample of current sophomores who took the test both years.
b = estimated slope of the regression line = 0.93
a = estimated y intercept of the regression line = 30.36
​
a)What is the equation of the estimated regression line?
b)On the graph below, sketch the estimated regression line using the information above. Be sure to label and scale your axes correctly.
c)For students with a score of 250 the previous year, what is the predicted score for the current year?
d)What is the average difference in science score that is associated with a 10-point difference in the previous year's science score?
e)The value of r² is 0.71; interpret r² in the context of this problem.
f)The value of s_e is 25.5. Interpret s_e in the context of this problem.

Question 30

The computer output given below shows a regression of Science Knowledge vs. Reading Comprehension scores for a random sample of sophomores. ​
a)What is the estimated regression line?
b)What is the calculated value of s_b?
b)What is the calculated value of s_e?
d)What is the calculated value of the t-ratio for the Model Utility Test?
e)Construct a 95% confidence interval for β.

Question 31

A study investigating the relationship between the major league batting average of non-pitching rookies (y) and their career minor league batting average (x) yields the following data: ​
a)Write the estimated regression line below.
b)On the graph below, sketch the sample regression line using the information above. Be sure to label and scale your axes correctly. c)What is your estimate of the average change in major-league batting average associated with an increase of 0.050 in the career minor-league batting average?
d)What major-league batting average would you predict for a randomly selected individual with a career minor-league batting average of 0.300?
e)The value of s_e is 0.015. Interpret s_e in the context of this problem.

Question 32

The computer output given below shows a regression of Body Mass (g) vs. Wing Length (mm) for sphinx moths. a)What is the estimated regression line?
b)What is the calculated value of s_b?
c)What is the calculated value of s_e?
d)In a few sentences, interpret s_e in the context of this problem.
e)Using the computer output above, perform the Model Utility Test at the .05 level of significance.
f)Construct a 95% confidence interval for β.

Question 33

The Iowa Tests of Educational Development are given each year to students around the country. Let x = the previous year's math scores and y = the current year's math scores for a sample of current sophomores who took the test both years.
b
= estimated slope of the regression line = 0.9
a
= estimated y intercept of the regression line = 35.0
​
a)What is the equation of the estimated regression line?
b)On the graph below, sketch the estimated regression line using the information above. Be sure to label and scale your axes correctly. c)For a students with a score of 250 the previous year, what is the predicted score for the current year?
d)What is the average difference in math score that is associated with a 10-point difference in the previous year's math score?
e)The value of r² is 0.74; interpret r² in the context of this problem.
f)The value of s_e is 20.5. Interpret s_e in the context of this problem.

Question 34

In a random sample of large cities around the world, the ozone level (in parts per million) and the population (in millions) were measured. Fitting the simple linear regression model gave the estimated regression equation .
a)How would you interpret b = 16.6?
b)Substituting x = 3 gives = 58.69. Give two different interpretations of this number.
c)Do you think this equation should be used for predicting ozone in small towns in Iowa? Explain your response in a few sentences.
d)The value of r² is not presented here, but explain how you would interpret r² in the context of this problem.

Question 35

When the four basic assumptions for a simple linear regression model are satisfied,
a)What is μ_b equal to?
b)What is σ_b equal to?
c)What is the shape of the sampling distribution of b?
d)What is the probability distribution of the standardized variable, ?

Question 36

The residual plot below indicates that one or more of assumptions of the linear regression model may not be met. Which of the following is a reasonable conclusion based on this residual plot? ​

A)The residual plot clearly indicates a non-linear model would be more appropriate.
B)There is evidence that the residuals are not normally distributed.
C)The slope of the regression line is non-zero.
D)The correlation between x and y is non-zero.
E)There is evidence the residuals do not have the same variance for all x values.

Question 37

The computer output given below shows a regression of Wing Length (mm) vs. Body Mass (g) for sphinx moths. ​
a)What is the estimated regression line?
b)What is the calculated value of s_b?
c)What is the calculated value of s_e?
d)In a few sentences, interpret s_e in the context of this problem.
e)Using the computer output above, perform the Model Utility Test at the .05 level of significance.
f)Construct a 95% confidence interval for β.

Question 38

A veterinary graduate student is studying the relationship between the weight of one year-old golden retrievers in pounds (y) and the amount of dog food the dog is fed each day in pounds (x). A random sample of 10 golden retrievers yielded the following data and summary statistics. ​
a)Write the estimated regression line below.
b)On the graph below, sketch the sample regression line using the information above. Be sure to label and scale your axes correctly. c)What is your estimate of the average change in weight associated with a 1-pound increase in daily intake of dog food?
d)What weight would you predict for a randomly selected golden retriever who is fed 1.2 pounds of dog food per day?
e)The value of s_e is 4.860. Interpret s_e in the context of this problem.

Question 39

A random sample of steelhead trout was selected; the length of the trout in inches and the weight of the trout in pounds were measured. Fitting the simple linear regression model gave the estimated regression equation .
a)How would you interpret b = 0.976?
b)Substituting x = 30 gives = 9.98. Give two different interpretations of this number.
c)What happens if you predict the weight of an 18-inch trout? In a few sentences, explain why do you think this has occurred.
d)The value of r² is not presented here, but explain how you would interpret r² in the context of this problem.

Question 40

If there is a negative linear relationship between two variables x and y, which of the following must be true of β, the slope of the population regression line? ​

A)
B)
C)
D)
E)

Question 41

In a study of gender-typing of toys, parents and non-parents were asked to rate the gender appropriateness of toys on a scale of 1 (Definitely for girls) to 5 (Definitely for boys). The investigators did a regression analysis of the parents mean scores vs. the non-parents mean scores for each toy. A computer-generated normal probability plot of the residuals is shown below.
​ ​
Write a brief paragraph describing the plot. Based on your description recommend to the investigators whether or not the appropriate assumption is met for simple linear regression.

Question 42

The scatter plot and residual plot shown below are from a regression analysis of Prehistoric Native American projectile points. The lengths of the projectiles were regressed on the width of the projectile points. Using these plots categorize each of the Basic Assumptions of the simple linear regression model as one of the following:
a)Not checkable
b)Checked and met
c)Checked and not met
For all assumptions checked, explain what aspect of the graphs led you to your conclusion that the assumption was met or not met.

Question 43

In a study of gender typing of toys, fathers and mothers were asked to rate the gender appropriateness of toys on a scale of 1 (Definitely for girls) to 5 (Definitely for boys). The investigators did a regression analysis of the fathers' mean ratings vs. the mothers mean ratings for each toy. A computer-generated normal probability plot of the residuals is shown below.
​ ​
Write a brief paragraph describing the plot. Based on your description recommend to the investigators whether or not the appropriate assumption is met for simple linear regression.