Question 1

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?

Accepted Answer

The answer of Let x be the number of pages...

Question 2

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.

Accepted Answer

The answer of The scatter plot and residual plot shown...

Question 3

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

Accepted Answer

The answer of What are the four "Basic Assumptions" of...

Question 4

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

Accepted Answer

The coefficient for body length in the regression equation is -0.0104278, indicating that for each one-unit increase in body length, the maximum sustainable swimming speed decreases by approximately 0.0104 body lengths per second.

Question 5

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.

Accepted Answer

The answer of In a study of the surface temperatures...

Question 6

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.

Accepted Answer

The answer of The computer output given below shows a...

Question 7

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.

Accepted Answer

The answer of In a study of the surface temperatures...

Question 8

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.

Accepted Answer

The answer of As part of a study of the...

Question 9

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.

Accepted Answer

The answer of The Iowa Tests of Educational Development are...

Question 10

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.

Accepted Answer

The answer of The computer output given below shows a...

Question 11

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.

Accepted Answer

The answer of A study investigating the relationship between the...

Question 12

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.

Accepted Answer

The answer of The computer output given below shows a...

Question 13

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.

Accepted Answer

The answer of The Iowa Tests of Educational Development are...

Question 14

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.

Accepted Answer

The answer of In a random sample of large cities...

Question 15

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, ?

Accepted Answer

The answer of When the four basic assumptions for a...

Question 16

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.

Accepted Answer

The answer of The residual plot below indicates that one...

Question 17

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.

Accepted Answer

The answer of The computer output given below shows a...

Question 18

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.

Accepted Answer

The answer of A veterinary graduate student is studying the...

Question 19

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.

Accepted Answer

The answer of A random sample of steelhead trout was...

Question 20

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)

Accepted Answer

The answer of If there is a negative linear relationship...

Quiz 15: Understanding Relationships - Numerical Data Part 2

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

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

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? ​

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? ​

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, ?

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?

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?