12. Regression

In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:

b. Compute the standard error, the point estimate for σ.

Intervals constructed about the predicted value of y, at a given level of x, that are used to measure the accuracy of a single individual’s prediction are called___________intervals for a(n)_________response.

The output shown was obtained from Minitab.

a. The least-squares regression equation is y^ = 1.3962x + 12.396. What is the predicted value of y at x = 10?

The output shown was obtained from Minitab.

c. The standard error, se, is 2.167. What is an estimate of the standard deviation of y at x=10?

[DATA] Height versus Head Circumference [See Problem 13 in Section 12.3] A pediatrician wants to determine the relation that may exist between a child’s height and head circumference. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data:

d. State your conclusion to the hypotheses from part (b).

[DATA] Hurricanes [See Problem 14 in Section 12.3] The following data represent the maximum wind speed (in knots) and atmospheric pressure (in millibars) for a random sample of hurricanes that originated in the Atlantic Ocean.

c. Use the “Randomization test for slope” applet in StatCrunch to build the null model to generate at least 2000 repetitions of this study to obtain a P-value. Interpret the P-value.

[DATA] Concrete [See Problem 15 in Section 12.3] As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

b. Suppose a researcher wanted to determine if there is a linear relation between 7-day strength and 28-day strength. What would be the null and alternative hypotheses?

[DATA] Concrete [See Problem 15 in Section 12.3] As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

d. State your conclusion to the hypotheses from part (b).

[DATA] CEO Performance [See Problem 19 in Section 12.3] The following data represent the total compensation for 12 randomly selected chief executive officers (CEOs) and the company’s stock performance in 2017.

b. Suppose a researcher wanted to determine if there is a linear relation between compensation and stock return. What would be the null and alternative hypotheses?

[DATA] Bear Markets [See Problem 20 in Section 12.3] A bear market is a market condition in which the price of the security falls. A bear market in the stock market is defined as a condition in which market declines by 20% or more over the course of at least two months. The following data represent the number of months and percentage change in the S&P500 (a group of 500 stocks).

b. Suppose a researcher wanted to determine if there is a linear relation between months and percent change. What would be the null and alternative hypotheses?

[DATA] Home Runs (Refer to Problem 31, Section 4.2) The following data represent the speed at which a ball was hit (in miles per hour) and the distance it traveled (in feet) for a random sample of home runs in a Major League baseball game in 2018.

a. Treating speed at which the ball was hit as the explanatory variable and distance the ball traveled as the response variable, determine estimates for β0 and β1.

In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:

c. Determine s_{b₁}.

In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:

d. Assuming the residuals are normally distributed, test H₀: β₁ = 0 versus H₁: β₁ ≠ 0 at the α = 0.05 level of significance.

[DATA] Putting It Together: Predicting Intelligence Can a photograph of an individual be used to predict their intelligence? Researchers at Charles University in Prague, Czech Republic, had 160 raters analyze the photos of 80 students and asked each rater to rate the intelligence and attractiveness of the individual in the photo on a scale from one to seven. To eliminate individual bias in ratings, each rater’s scores were converted to z-scores using each individual’s mean rating. The perceived intelligence and attractiveness of each photo was calculated as the mean z-score. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_4_19 using the file format of your choice. The following explains the variables in the data:

sex: Gender of the individual in the photo

age: Age of the individual in the photo

perceived intelligence (ALL): Mean z-score of the perceived intelligence of all 160 raters

perceived intelligence (WOMEN): Mean z-score of the perceived intelligence of the female raters

perceived intelligence (MEN): Mean z-score of the perceived intelligence of the male raters

attractiveness (ALL): Mean z-score of the attractiveness rating of all 160 raters

attractiveness (MEN): Mean z-score of the attractiveness rating of the male raters

attractiveness (WOMEN): Mean z-score of the attractiveness rating of the female raters

IQ: Intelligence quotient based on the Czech version of Intelligence Structure Test

d. Provide an interpretation of the intercept.

[DATA] Putting It Together: Predicting Intelligence Can a photograph of an individual be used to predict their intelligence? Researchers at Charles University in Prague, Czech Republic, had 160 raters analyze the photos of 80 students and asked each rater to rate the intelligence and attractiveness of the individual in the photo on a scale from one to seven. To eliminate individual bias in ratings, each rater’s scores were converted to z-scores using each individual’s mean rating. The perceived intelligence and attractiveness of each photo was calculated as the mean z-score. Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_4_19 using the file format of your choice. The following explains the variables in the data:

sex: Gender of the individual in the photo

age: Age of the individual in the photo

perceived intelligence (ALL): Mean z-score of the perceived intelligence of all 160 raters

perceived intelligence (WOMEN): Mean z-score of the perceived intelligence of the female raters

perceived intelligence (MEN): Mean z-score of the perceived intelligence of the male raters

attractiveness (ALL): Mean z-score of the attractiveness rating of all 160 raters

attractiveness (MEN): Mean z-score of the attractiveness rating of the male raters

attractiveness (WOMEN): Mean z-score of the attractiveness rating of the female raters

IQ: Intelligence quotient based on the Czech version of Intelligence Structure Test

e. A normal probability plot confirms the residuals are normally distributed. Test whether a positive linear relation exists between perceived intelligence and attractiveness.