Exam 14: Inference About Simple Regression

Shown below is a scatterplot of y versus x. Which choice is most likely to be the approximate value of r², the proportion of variation in y explained by the linear relationship with x?

Use the following information for questions: A sample of 19 female bears was measured for chest girth (y) and neck girth (x), both in inches. The least squares regression equation relating the two variables is $\hat{y}$ = 5.32 + 1.53 x. The standard deviation of the regression is s = 3.692. -What is a 95% confidence interval for the average (or mean) chest girth for bears with a neck girth of 18 inches? The standard error of the fit, s.e.(fit), = 0.860.

Use the following information for questions: A representative sample of n = 12 male college students is used to find a regression equation for y = weight (lbs) and x = height (inches). The least squares regression equation is $\hat{y}$ = 30 + 2 x. The standard error of the estimated slope is 1. The following hypotheses will be tested: H₀: $β_1 = 0$ H_a: $β_1 ≠ 0$ -What is the value of the t-statistic for testing these hypotheses?

Use the following information for questions: A car salesman is curious if he can predict the fuel efficiency of a car (in MPG) if he knows the fuel capacity of the car (in gallons). He collects data on a variety of makes and models of cars. The scatterplot shows that a linear model is appropriate. SPSS output is provided below. -What is the numerical value of the correlation coefficient between fuel efficiency and fuel capacity?

Use the following information for questions: A car salesman is curious if he can predict the fuel efficiency of a car (in MPG) if he knows the fuel capacity of the car (in gallons). He collects data on a variety of makes and models of cars. The scatterplot shows that a linear model is appropriate. SPSS output is provided below. -To assess if there is a significant linear relationship between fuel efficiency and fuel capacity, what are the hypotheses to be tested?

Use the following information for questions: A representative sample of n = 12 male college students is used to find a regression equation for y = weight (lbs) and x = height (inches). The least squares regression equation is $\hat{y}$ = 30 + 2 x. The standard error of the estimated slope is 1. The following hypotheses will be tested: H₀: $β_1 = 0$ H_a: $β_1 ≠ 0$ -What is the p-value for the test? (Table A.3 or its equivalent needed.)

Use the following information for questions: Data has been obtained on the house size (in square feet) and the selling price (in dollars) for a sample of 100 homes in your town. Your friend is saving to buy a house and she asks you to investigate the relationship between house size and selling price and to develop a model to predict the price from size. -Predict the price for the 60^th observed house, which had a home size of 2430 square feet.

Use the following information for questions: Grades for a random sample of students who have taken statistics from a certain professor over the past 20 year were used to estimate the relationship between y = grade on the final exam and x = average exam score (for the three exams given during the term). -The "Fit"information shown at the end of the output is for ExamAvg = 75. From this, we can conclude that

Use the following information for questions: Based on a representative sample of college men, a regression line relating y = ideal weight tox = actual weight, for men, is given by Ideal weight = 53 + 0.7 *actual weight -For a man with actual weight = 200 pounds, his ideal weight is predicted to be

Use the following information for questions: A consumer agency employee has been researching and testing food products for months. He is convinced that the dimensions of the packaging used for dry goods (crackers, cookies, cereal, etc) all satisfy a certain ratio. He will try to model the relationship between the minimum width of a box and the height of the box, to see if he can predict the height of the box from the minimum width (both measured in inches). Data is collected on 24 randomly selected products. The scatterplot shows that a linear model is appropriate. (Partial) SPSS output is provided below. -The residual plot is supplied below. Based on the residual plot, is there any evidence that one of the assumptions for a linear regression model does not hold? Explain.

Which choice is not an appropriate description of $\hat{y}$ in a regression equation?

Use the following information for questions: A car salesman is curious if he can predict the fuel efficiency of a car (in MPG) if he knows the fuel capacity of the car (in gallons). He collects data on a variety of makes and models of cars. The scatterplot shows that a linear model is appropriate. SPSS output is provided below. -Calculate a 98% confidence interval for the average fuel efficiency for all cars that hold 18 gallons of fuel. Descriptive statistics for the two variables are provided below. Use the mean to obtain $\overline{x}$ and the variance to calculate . Use these two values then to calculate s.e.(fit).

Use the following information for questions: The heights (in inches) and foot lengths (in centimeters) of 32 college men were used to develop a model for the relationship between height and foot length. The scatterplot shows that a linear model is appropriate. SPSS output is provided below. -How much would you expect foot length to increase for each 1-inch increase in height? Include the units.

Use the following information for questions: The relation between y = ideal weight (lbs) and x =actual weight (lbs), based on data fromn = 119 women, resulted in the regression line $\hat{y}$ = 44 + 0.60 x -The intercept of the regression line is ______

Shown below are four residual plots (residuals versus predicted values). Which one of the four does not show any evidence that one of the assumptions for a linear regression model is violated?

Use the following information for questions: A regression line relating y = hours of sleep the previous day to x =hours studied the previous day is estimated using data from n = 10 students. The estimated slope b₁ = -0.30. The standard error of the slope is s.e.(b₁) = 0.20. -What is a 90% confidence interval for $\beta$ ₁ , the population slope?

In a statistics class at Penn State University, a group working on a project recorded the time it took each of 20 students to drink a 12-ounce beverage and also recorded body weights for the students. Which of these statistical techniques would be the most appropriate for determining if there is a statistically significant relationship between drinking time and body weight? (Assume that the necessary conditions for the correct procedure are met.)

Use the following information for questions: Can an athlete's cardiovascular fitness (as measured by time to exhaustion on a treadmill) help to predict the athlete's performance in a 20k ski race? To address this question the time to exhaustion on a treadmill (minutes) and the time to complete a 20k ski race (minutes) were recorded for a sample of 11 athletes. The researcher hypothesizes that there would be an inverse linear relationship, that is, the longer the time to exhaustion, the faster the athlete's time in the 20k ski race, on average. The data were used to provide the following output. -Below are two intervals, one is the 95% prediction interval for the ski time for the 7^th skier and the other is the 95% confidence interval for the mean ski time for all skiers with a treadmill exhaustion time of 10 minutes. Which of the following is the prediction interval? Interval 1: (63.9, 67.0) Interval 2: (60.3, 70.7)

Use the following information for questions: A regression line relating y = hours of sleep the previous day to x =hours studied the previous day is estimated using data from n = 10 students. The estimated slope b₁ = -0.30. The standard error of the slope is s.e.(b₁) = 0.20. -What is the value of the test statistic for the following hypothesis test about $?_1$ , the population slope? H₀: $?_1 = 0$ H_a: $?_1 ? 0$

Use the following information for questions: The relation between y = ideal weight (lbs) and x =actual weight (lbs), based on data fromn = 119 women, resulted in the regression line $\hat{y}$ = 44 + 0.60 x -What is the interpretation of the value 44 in the regression for this question?