Deck 14: Inference for Regression

Question 1

Which of the following is not true about residuals? Assume the conditions of the linear model hold. If all the statements are true choose (d).

A)The residuals can be described as the distances the points are from the line due to randomness.
B)The residuals can be determined by finding the difference between the actual observed value and the predicted dependent variable.
C)The residuals are the result of natural variation in the dependent variable.
D)All of the above are true.

Answer 1

All of the above are true.

Question 2

A researcher wanted to see if there was a linear relationship between the heights of sons and the heights of their fathers. Suppose a regression analysis was run to predict the heights of the sons from the heights of their fathers. What would it mean if the intercept were 2 and the slope of the regression line 1?

A)On average, sons are 2 inches taller than their fathers.
B)On average, fathers are 2 inches taller than their sons.
C)On average, sons are 1 inch taller than their fathers.
D)On average, fathers are 1 inch taller than their sons.

Answer 2

On average, sons are 2 inches taller than their fathers.

Question 3

Which of the following statements is not true about the constant standard deviation condition of the linear regression model?

A)A residual plot can highlight the existence of a nonconstant standard deviation even when it is hard to see in the original scatterplot
B)A constant standard deviation means that the vertical spread of the y-values about the line is the same across the entire line.
C)A fan shape in a residual plot indicates that the constant standard deviation condition does not hold.
D)All of the above are true about the constant standard condition.

Answer 3

All of the above are true about the constant standard condition.

Question 4

Which of the following is not a condition of the linear regression model?

A)Linearity
B)Residuals must be fan-shaped
C)Constant Standard Deviation of errors
D)Normality of errors

Question 5

Which of the following statements is true about residuals? If all of the statements are true choose (d).

A)The residual can be computed by finding the difference between the deterministic component and the random component of the regression model.
B)The residual can be found by finding the difference between the actual observed value and the predicted value of the dependent variable.
C)The residual can be found by finding the difference between the actual observed value and the predicted value of the independent variable.
D)All of the above are true.

Question 6

Many universities use ACT test scores as one criterion for admission. Admission counselors believe that the deterministic part of a regression model predicting college GPA at the end of the first year from the ACT score is a straight line. What factors might contribute to the random component? In other words, why might a student's end of year GPA not fall exactly on this line?

Question 7

Consider the following statement: "When the conditions of the linear model hold, the estimators for slope and intercept are unbiased." What is meant by the word unbiased in this context?

Question 8

Use the following information for following questions:. A salesman says he can predict the sales
price of used Toyota Corollas by using the equation
Sale Price = 14300 - 960 Age
Using the information above, find the value of the residual for a used Toyota Corolla that is 7 years old and has an actual price of $6950.

A)$630
B)$0
C)-$630
D)-$6390

Question 9

Which of the following is not true about the coefficient of determination, r2?

A)The coefficient of determination, r2, ranges from 0% to 100% and represents the amount of variability in the response variable explained by the regression line.
B)In order to interpret the coefficient of determination, r2, the linearity condition of the linear regression model must be satisfied.
C)A hypothesis test should be conducted to verify that the coefficient of variation, r2, is large enough to conclude a linear relationship exists.
D)The coefficient of determination, r 2, is a statistic that will give some information about how well the data fit the model, but it should not be the only piece of information taken into consideration
When determining how useful a linear model might be.

Question 10

A researcher wanted to see if there was a linear relationship between the length of a person's leg and the length of a person's arm. Suppose a regression analysis was run to predict the lengths of legs from the lengths of arms. What would it mean if the intercept were 6 and the slope of the regression line 1?

A)On average, legs are 1 inch longer than arms.
B)On average, arms are 1 inch longer than legs.
C)On average, legs are 6 inches longer than arms.
D)On average, arms are 6 inches longer than legs.

Question 11

Choose the statement(s) that are not true about the estimators for slope and intercept of a regression line when the conditions of the linear model hold. If each statement is true choose (d).

A)The sampling distribution of the estimators will follow the Normal model.
B)The estimators for the slope and intercept of a regression line are unbiased.
C)The sampling distributions of the estimators follow the chi-square model.
D)All of the above are true.

Question 12

Environmental biologists studying the relationship between the number of owls in a forested region and the number of field mice in the region believe that the deterministic component of the relationship is a straight line. A scatterplot shows that even though the general trend is linear, the points do not fall exactly on a straight line. Which of the following factors might account for the random component of this regression model?

A)Different size owls might affect the number of mice eaten.
B)Variability might appear in the instrument used to count mice.
C)Variation in the size of the mice might cause variation in the amount of mice consumed by owls.
D)All of the above are possible factors that could account for the random component of the regression model.

Question 13

Which of the following statements is true about checking for the constant standard deviation condition?

A)If the points of the QQ plot are close to the straight line, then we can conclude the constant standard deviation condition is met.
B)If the plot of the residuals against time shows no trend, then we can conclude the constant standard deviation condition is met.
C)If the residual plot shows a fan shape, then we can conclude the constant standard deviation condition is met.
D)If the residual plot shows no features such as a trend or a fan shape, then we can conclude the constant standard deviation condition is met.

Question 14

Explain how a residual plot can be useful in determining whether the condition for linearity and the condition for a constant standard deviation have been satisfied.

Question 15

Choose the statement that is true about the estimators for the slope and intercept of a regression line. Assume the conditions of the linear model hold. If all statements are true, then choose (d).

A)The estimators of the slope and intercept of a regression line are unbiased.
B)The sampling distributions of the estimators will follow the Normal distribution if the Normal condition of the error terms is met.
C)The sampling distributions of the estimators will be approximately Normal if the Normal condition of the error terms is not met but the sample size is large.
D)All of the above are true.

Question 16

Explain the difference between confidence intervals and prediction intervals. Be sure to include the type of situation in which each type of interval would be used. Which interval is likely to be wider and why?

Question 17

Use the following information for following questions:. A salesman says he can predict the sales
price of used Toyota Corollas by using the equation
Sale Price = 14300 - 960 Age
Using the equation above, find the predicted value for a used Toyota Corolla that is 6 years old.

A)$8540
B)$6950
C)$13340
D)Cannot be determined from the information given.

Question 18

Which of the following statements is true about prediction intervals?

A)Prediction intervals are wider than confidence intervals because there is more uncertainty in predicting an individual's value.
B)A prediction interval is concerned with predicting values for individuals.
C)The width of the prediction interval is affected by the size of the standard deviation of the population distribution
D)All of the above statements are true

Question 19

If we wanted to estimate the average price of all bottles of wine with a taste score of 18, we would use a

A)Confidence interval
B)Prediction interval

Question 20

State the two conditions that must be satisfied, without exception, to make inferences using a linear model.

Question 21

What statement about prediction intervals and confidence intervals is true. If none of the statements is true, select (d).

A)Prediction intervals are always smaller than confidence intervals because confidence intervals are predicting the mean of the population and prediction intervals are predicting the value of only one
Item in the population.
B)Prediction intervals are always smaller than confidence intervals because prediction intervals are predicting the mean of all the items in the population and confidence intervals are predicting the
Value of only one item in the population.
C)Sometimes prediction intervals are smaller than confidence intervals and sometimes prediction intervals are larger than confidence intervals.
D)None of the statements is true.

Question 22

Suppose we know that a particular bottle of wine had a taste score of 15. What can be said about the price of this bottle of wine? Select the statement that is true about the price of this bottle of wine.

A)The expected price is $6.18.We can be confident that this bottle of wine will cost less than $20.
B)The expected price is $25.61.We can be confident that the actual price of this bottle of wine will be between $21.25 and $29.97.
C)The expected price is $25.61.We can be confident that the actual price of this bottle of wine will be between $24.45 and $29.97.
D)The expected price is $25.61.We can be confident that the average price of all bottles of wine with a taste score of 15 will be between $21.25 and $29.97.