Exam 13: Simple Linear Regression and Correlation: Inferential Methods

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

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.

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.

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. c) The estimated standard deviation of a + bx* for x* = 30 is calculated as: sa_+b(30) = 0.2417. Calculate a 95% confidence interval for the true average vital tooth temperature when the non-vital tooth temperature is 30 degrees centigrade. d) In one individual, a single surface temperature for a non-vital tooth is found to be 30 degrees centigrade. Use a 95% prediction interval to predict the resulting temperature of a vital tooth.

In a forensic study of the relation between time since death (hrs) and the nostril temperature in deer (F) , a regression analysis of y = time since death on x = nostril temperature was performed. Summary quantities from this analysis yield: n = 384 S_xx = 49,999.464 s_e = 1.997 ​ a) Calculate the estimated standard deviation of a + bx* for x* = 80. b) For what value x* is the estimated standard deviation of a + bx* smallest, and why?

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. c) The estimated standard deviation of a + bx* for x* = 30 is calculated as: sa+b₍₃₀₎ = 0.1665. Calculate a 95% confidence interval for the true average non-vital tooth temperature when the vital tooth temperature is 30 degrees centigrade. d) In one individual, a single surface temperature for a vital tooth is found to be 30 degrees centigrade. Use a 95% prediction interval to predict the resulting temperature of a non-vital tooth.

The standard deviation of a + bx* grows larger the farther x* is from .

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

An experiment to study the relationship between x = head wind speed (m/s) and y = 100-m sprint time (seconds) resulted in the following summary statistics. ​ n = 20 ​ Calculate a 99% confidence interval for β, the average change in the 100-m finish time associated with a 0.1 (that is, 1 unit) increase in wind speed. Round your answers to four decimal places. ​

The population correlation ρ is always between 0 and 1.

The test of independence involves testing H₀ : ρ = 1 vs. H₀ : ρ ≠ 1.

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

A simple linear regression model was used to describe the relationship between concentration y (measured in the scores of a specially designed test) and the room temperature x (in °F). A random sample of 15 observations was used to produce the accompanying Minitab output. Calculate a 95% confidence interval for β, the average change in the test score associated with a 1 °F increase in room temperature, using the output. Round your answers to three decimal places. ​ ​

The Head of Special Projects and Planning of a network of shops develops a plan for opening new stores. He suggests that the annual revenue of a supermarket depends mainly on the size of the sales area. A scatterplot of the data, a residual plot, and a histogram of the residuals are shown below. What do the graphs suggest about the adequacy of the simple linear regression model? ​ ​

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?

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.

The mean value of the statistic b is β.

The standard deviation of the statistic b, σb, is small when σe and S_xx are both small.

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

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 slopw 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.