Question 1

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories =$\alpha$ + $\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. This assumption is probably dubious. The following results were obtained from software. r² = 0.846 S = 43.5747 The sample correlation between calories and fat in fast-food sandwich menu items is: A)0.716. B)0.92. C)-0.92. D)-0.716.

Accepted Answer

B

Question 2

Forced expiratory volume (FEV) is the volume of exhaled air and is related to lung size as well as lung function. It is typically lower in persons with impaired lung function due to disease. A regression of height (in inches) on FEV (in liters) showed a nonlinear pattern. Therefore, height was regressed on log(FEV) (natural log) and the residual plot showed a much improved fit. The intercept a and slope b from the regression of height on log(FEV) are a = -2.2 and b = 0.06. The correlation for this regression is:
A)positive.
B)negative.
C)zero.
D)There is not enough information to determine the answer.

Accepted Answer

The slope (b) of the regression line is positive (0.06), indicating a positive correlation between height and log(FEV).

Question 3

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories = $\alpha$ +$\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. This assumption is probably dubious. The following results were obtained from software. r² = 0.846 S = 43.5747 A 95% confidence interval for the mean number of calories per gram of fat in sandwich menu items at fast-food restaurants is: A)99.627 to 202.553. B)89.007 to 213.172. C)10.169 to 14.923. D)10.042 to 15.05.

Accepted Answer

D

Question 4

The following is a plot of the residuals versus fat for 26 menu items at a fast food restaurant.  Which of the following statements is supported by this plot?
A)A linear model is appropriate for explaining the relationship between the explanatory and response variables for this case.
B)There is evidence that the deviations described by the model are not Normal in distribution.
C)The abundance of outliers and influential observations in the plot means that the assumptions for regression are clearly violated.
D)None of the answer options is correct.

Accepted Answer

The answer of The following is a plot of the...

Question 5

Forced expiratory volume (FEV) is the volume of exhaled air and is related to lung size as well as lung function. It is typically lower in persons with impaired lung function due to disease. A regression of height (in inches) on FEV (in liters) showed a nonlinear pattern. Therefore, height was regressed on log(FEV) (natural log) and the residual plot showed a much improved fit. The intercept a and slope b from the regression of height on log(FEV) are a = -2.2 and b = 0.06. A 5-foot-tall person is predicted to have a lung volume of:
A)1.4 liters.
B)4.1 liters.
C)2.2 liters.
D)0.06 liters.

Accepted Answer

The answer of Forced expiratory volume (FEV) is the volume...

Question 6

The scatterplot below suggests a linear relationship between the age (in years) of an antique clock and its sale price (in euros) at auction. The data are age and sale price for 11 antique clocks sold at a recent auction. We fit the least-squares regression line to the model price = $\alpha$ + $\beta$ (age), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. A summary of the output is given. r² = 0.848 S = 33.1559 Suppose the researchers test the hypotheses H₀: $\beta$ = 0, H_a: $\beta$ $\neq$ 0. The value of the t statistic for this test is: A)-7.09. B)1.893. C)7.09. D)0.796.

Accepted Answer

The t statistic for testing the null hypothesis that the slope parameter is equal to 0 is given by t = (b1 - 0)/SE(b1), where b1 is the estimate of the slope parameter, and SE(b1) is the standard error of the estimate. From the output given, we see that b1 = 37.67 and SE(b1) = 5.3299. Thus, t = (37.67 - 0)/5.3299 = 7.09. Therefore, the correct answer is (C).

Question 7

Forced expiratory volume (FEV) is the volume of exhaled air, and it is related to lung size as well as to lung function. FEV is typically lower in persons with impaired lung function due to disease. The residual plots below are from two regressions: the first plot is from regressing height on FEV and the second is from regressing height on log(FEV).   Based on these plots, which statement is correct?
A)The residual plot for height on FEV shows a nonconstant variance.
B)The residual plot for height regressed on FEV shows a nonlinear pattern.
C)The residual plot for the regression of height on log(FEV)shows a linear pattern.
D)All of the answer options are correct.

Accepted Answer

The residual plot for height on FEV shows a nonconstant variance, indicating that the variability of the residuals changes across the range of FEV values. The residual plot for height regressed on FEV shows a nonlinear pattern, indicating that a linear model may not be appropriate for the data. The residual plot for the regression of height on log(FEV) shows a linear pattern, indicating that a linear model is appropriate for this regression. Therefore, all of the answer options are correct.

Question 8

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories =$\alpha$ + $\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. The explanatory variable in this study is:
A)number of calories.
B)fat content (grams of fat).
C)the slope, $\beta$.
D)12.546.

Accepted Answer

The explanatory variable in this study is the amount of fat in a menu item, measured in grams. This is indicated by the regression model: calories = F1F1F1S1F1F1F10 + F1F1F1S1F1F1F10 (fat).

Question 9

Frequent food questionnaires (FFQs) are often given to large groups of people to obtain information on their dietary habits. Study participants are asked about the frequency with which they consume certain goods. Another method to obtain information on foods consumed is a food diary. People are asked to record every type of food and amount consumed for a few days. Food diaries are more difficult to obtain and response rates are lower than for FFQs. A study was conducted to see how well FFQs predict food consumed based on food diaries. In this study, the explanatory variable is:
A)the amount of food consumed.
B)the amount of food consumed according to the food diary.
C)the amount of food consumed according to the FFQ.
D)dependent on the response.

Accepted Answer

The explanatory variable in the study is the amount of food consumed according to the FFQ. This is because the study is assessing how well the FFQ predicts the food consumed based on food diaries. The FFQ is the variable being tested for its ability to predict food consumption, while the food diary is the reference or comparison measure of actual food consumption. The amount of food consumed (A) may be a potential dependent variable or outcome measure, but it is not the explanatory variable in this study. The amount of food consumed according to the diary (B) is the comparison or reference measure, but it is not the variable being tested as the predictor. The response rate (D) may be a potential confounding variable, but it is not the explanatory variable that is being manipulated or measured in this study.

Question 10

Frequent food questionnaires (FFQs) are often given to large groups of people to obtain information on their dietary habits. Study participants are asked about the frequency with which they consume certain goods. Another method to obtain information on foods consumed is a food diary. People are asked to record every type of food and amount consumed for a few days. Food diaries are more difficult to obtain and response rates are lower than for FFQs. A study was conducted to see how well frequent food questionnaires predict food consumed based on food diaries. The regression of alcohol consumption FFQ on alcohol consumption food diary had an intercept of a = 2.96 and a slope of b = 0.67. The predicted number of drinks consumed in the food diary, if the FFQ states 5 drinks, is:
A)6.31.
B)0.67.
C)2.96.
D)5.

Accepted Answer

To predict the number of drinks consumed in the food diary, we use the equation: predicted number of drinks consumed = a + b * FFQ reported number of drinks consumed. Substituting the values given in the problem, we get:
predicted number of drinks consumed = 2.96 + 0.67 * 5
predicted number of drinks consumed = 6.31
Therefore, the predicted number of drinks consumed in the food diary, if the FFQ states 5 drinks, is 6.31, which is option A.

Question 11

A random sample of 19 companies from the Forbes 500 list was selected, and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear regression model was used: profits = $\alpha$ + $\beta$ (sales), where the deviations were assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. This model was fit to the data using the method of least squares. The following results were obtained from statistical software. r² = 0.662 S = 466.2 Is there evidence of a straight-line relationship between sales and profits? A)Yes, because the slope of the least-squares line is positive. B)Yes, because the P-value for testing if the slope is 0 is quite large. C)Yes, because the P-value for testing if the slope is 0 is quite small. D)It is impossible to say, because we are not given the actual value of the correlation.

Accepted Answer

The P-value for testing if the slope is 0 is quite small, indicating that there is evidence of a significant linear relationship between sales and profits. Additionally, the positive correlation coefficient (r) also suggests a positive linear relationship between sales and profits.

Question 12

The scatterplot below suggests a linear relationship between the age (in years) of an antique clock and its sale price (in euros) at auction. The data are age and sale price for 11 antique clocks sold at a recent auction. We fit the least-squares regression line to the model price = $\alpha$ + $\beta$ (age), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. A summary of the output is given. r² = 0.848 S = 33.1559 An approximate 95% confidence interval for the slope $\beta$in the simple linear regression model is: A)1.289 to 2.497 euros per year. B)1.289 to 2.497 euros. C)1.289 to 2.497 years per euro. D)None of the answer options is correct.

Accepted Answer

The answer of The scatterplot below suggests a linear relationship...

Question 13

A random sample of 19 companies from the Forbes 500 list was selected, and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear regression model was used: profits = $\alpha$+ $\beta$(sales), where the deviations were assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. This model was fit to the data using the method of least squares. The following results were obtained from statistical software. r² = 0.662 S = 466.2 Suppose the researchers test the hypotheses H₀: $\beta$₁= 0, H_a: $\beta$₁> 0. The P-value of the test is: A)greater than 0.10. B)between 0.10 and 0.05. C)between 0.05 and 0.01. D)less than 0.01.

Accepted Answer

The answer of A random sample of 19 companies from...

Question 14

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories = $\alpha$ + $\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. This assumption is probably dubious. The following results were obtained from software. r² = 0.846 S = 43.5747 The slope of the least-squares regression line is: A)151.09. B)12.546. C)0.846. D)None of the answer options is correct.

Accepted Answer

According to the given regression model, calories = 12.546 x (fat) + 0. The slope of the regression line represents the change in y (calories) for each unit increase in x (fat), and the coefficient of fat (12.546) is the slope of the line. Therefore, the slope of the least-squares regression line is 12.546.

Question 15

A random sample of 19 companies from the Forbes 500 list was selected, and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear regression model was used: profits = $\alpha$+ $\beta$ (sales), where the deviations were assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. This model was fit to the data using the method of least squares. The following results were obtained from statistical software. r² = 0.662 S = 466.2 The approximate slope of the least-squares regression line is: A)0.09. B)0.0075. C)-176.64. D)61.16.

Accepted Answer

The answer of A random sample of 19 companies from...

Question 16

A random sample of 19 companies from the Forbes 500 list was selected, and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated by regression. The following simple linear regression model was used: profits =$\alpha$+ $\beta$ (sales), where the deviations were assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. This model was fit to the data using the method of least squares. The following results were obtained from statistical software. r² = 0.662 S = 466.2 An approximate 90% confidence interval for the slope $\beta$ in the simple linear regression model is: A)-176.66 to -176.63. B)0.079 to 0.106. C)0.071 to 0.114. D)None of the answer options is correct.

Accepted Answer

The answer of A random sample of 19 companies from...

Question 17

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories = $\alpha$ + $\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. This assumption is probably dubious. The following results were obtained from software. r² = 0.846 S = 43.5747 Suppose the researchers test the hypotheses H₀: $\beta$ = 0, H_a: $\beta$ > 0. The value of the t statistic for this test is: A)10.343. B)5.023. C)-5.023. D)-10.343.

Accepted Answer

Since the alternative hypothesis is one-tailed (F1F1F1S1F1F1F10 > 0), we need to calculate the one-tailed p-value for the t-statistic.

Using the formula for the t-statistic:

t = (b1 - 0) / (SE(b1))

where b1 is the estimate of F1F1F1S1F1F1F10 from the regression output, SE(b1) is the standard error of the estimate of b1, and "0" is the null value we are testing against.

From the output, we have b1 = 16.0711 and SE(b1) = 1.55567.

Thus,

t = (16.0711 - 0) / (1.55567) = 10.343

The degrees of freedom for this test is n - 2 = 24.

Using a t-distribution table, the one-tailed p-value for a t-statistic with 24 degrees of freedom and a value of 10.343 is less than 0.001.

Therefore, we reject the null hypothesis and conclude that there is evidence to suggest that the true slope of the regression line is greater than zero.

Question 18

You can visit the official website of any large restaurant chain to examine the nutritional data for menu items. For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than from carbohydrates or protein). Here we investigate the relationship between the amount of fat in a menu item (in grams) and the number of calories. To predict the number of calories in a menu item given its fat content, we use the simple linear regression model calories = $\alpha$+ $\beta$ (fat), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model to the data using the method of least squares. We treat these 26 menu items (which came from one restaurant) as a sample from the population of all sandwich items at all fast-food restaurants. Suppose the researchers test the hypotheses H₀: $\beta$ = 0, H_a:$\beta$ > 0. The value of the t statistic for this test is greater than 5. The P-value corresponding to the test of the hypotheses is: A)more than 0.02. B)between 0.001 and 0.02. C)between 0.0005 and 0.001. D)less than 0.0005.

Accepted Answer

A t statistic greater than 5 corresponds to a very small p-value, much smaller than 0.0005. Therefore, the correct answer is D.

Question 19

The scatterplot below suggests a linear relationship between the age (in years) of an antique clock and its sale price (in euros) at auction. The data are age and sale price for 11 antique clocks sold at a recent auction. We fit the least-squares regression line to the model price = $\alpha$ + $\beta$ (age), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. A summary of the output is given. r² = 0.848 S = 33.1559 The correlation between the age of the antique clock and the auction price of the antique clock is: A)-0.921. B)0.921. C)-0.719. D)0.719.

Accepted Answer

The summary output gives the correlation coefficient (r) as 0.848, which is positive and indicates a strong positive linear relationship between the age and sale price of the antique clocks. The options A and C both have negative correlation coefficients, which is inconsistent with the given summary output. Therefore, the best choice is B, which is a positive correlation coefficient of 0.921.

Question 20

The scatterplot below suggests a linear relationship between the age (in years) of an antique clock and its sale price (in euros) at auction. The data are age and sale price for 11 antique clocks sold at a recent auction. We fit the least-squares regression line to the model price = $\alpha$ + $\beta$ (age), where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation $\sigma$. A summary of the output is given. R² = 0.848 S = 33.1559 The approximate intercept of the least-squares regression line is: A)27.73. B)1.893. C)34.84. D)0.267.

Accepted Answer

The answer of The scatterplot below suggests a linear relationship...

Quiz 13: Inference for Regression

The following is a plot of the residuals versus fat for 26 menu items at a fast food restaurant. Which of the following statements is supported by this plot?