Exam 13: Simple Linear Regression and Correlation: Inferential Methods

Researchers study the relationship between x = Average salary (in thousands of dollars) and y = Percent of total income spent on recreation and culture. The x values and the corresponding standardized residuals from a simple linear regression are as follows: ​ ​ Construct a standardized residual plot. Based on this plot, does the simple linear regression model seems reasonable? ​

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.

A manager of the restaurant chain examines how waiters' hourly wages affect the average service rating from visitors. A scatterplot of the data, a residual plot, and a histogram of standardized residuals are shown below. Are the assumptions of the simple linear regression model reasonable? ​ ​

In which case is the confidence interval for β narrow? ​

The simple linear regression model is a deterministic model.

In a study of the bidding behavior when antique grandfather clocks are sold at auction, investigators analyzed the selling price (y) and the age in years of the grandfather clock (x). Summary quantities from this analysis yield: n = 32 S_xx = 23625.9 s_e = 273.03 ​ a) Calculate the estimated standard deviation of a + bx* for x* = 150. b) For what value x* is the estimated standard deviation of a + bx* smallest, and why?

Let x be the number of pages in a geometry 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 = 0.05 α = y intercept of population regression line = 30.0 ​ 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 geometry book of 300 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 the geometry book? e) What is the average change in price that is associated with a 20-page increase in size of the geometry book?

In a study of the bidding behavior when antique grandfather clocks are sold at auction, investigators analyzed the selling price (y) and the age in years of the grandfather clock (x). Summary quantities from this analysis yield: n = 32 S_xx = 23625.9 s_e = 273.03 ​ a) Obtain a 95% confidence interval for α + β(150), the true mean selling price when the age of the grandfather clock is 150. b) Obtain a 95% prediction interval for the price of a single grandfather clock when the age of the grandfather clock is 150.

In a study of gender typing of toys, fathers and mothers were asked to rate the gender appropriateness of toys on a scale of 1 (Definitely for girls) to 5 (Definitely for boys). The investigators did a regression analysis of the fathers' mean ratings vs. the mothers mean ratings for each toy. A computer-generated normal probability plot of the residuals is shown below. Write a brief paragraph describing the plot. Based on your description recommend to the investigators whether or not the appropriate assumption is met for simple linear regression.

A residual is a deviation from the population regression line.

An experiment was conducted to study the relationship between x = amount of new experimental fertilizer (kg/ha) and y = growth rate of the Madonna lilies stem (cm/week). Perform the Model Utility Test at the 0.05 level of significance using the Minitab output. ​ ​

The t-ratio for testing ρ = 0 is equivalent to the t-ratio for testing β = 0.

The scatter plot and residual plot shown below are from a regression analysis of Prehistoric Native American projectile points. The lengths of the projectiles were regressed on the width 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.

In a forensic study of the relation between time since death (hrs) and the thigh temperature in deer (F), a regression analysis of y = time since death on x = thigh temperature was performed. From their examination of the data the investigators felt that the assumptions of the linear regression model were satisfied. Summary quantities from this analysis yield: ​ 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 gender-typing of toys, parents and non-parents were asked to rate the gender appropriateness of toys on a scale of 1 (Definitely for girls) to 5 (Definitely for boys). The investigators did a regression analysis of the parents mean scores vs. the non-parents mean scores for each toy. A computer-generated normal probability plot of the residuals is shown below. Write a brief paragraph describing the plot. Based on your description recommend to the investigators whether or not the appropriate assumption is met for simple linear regression.

A scatterplot, residual plot, and a boxplot of residuals from the Minitab output are based on summary quantities given for x = Weight of a fully-grown green iguana (in kilograms) and y = Length of its tail (in centimeters). Does it appear that the assumptions of the simple linear regression model are plausible? ​ ​