AP Stats Linear Regression and Correlation Review

In a statistics course, a linear regression equation was computed to predict the final-exam score from the score on the first test. The equation was  ŷ = 10 + 0.9x where y is the final exam score and x is the score on the first test. Carla scored 95 on the first test. What is the predicted value of her score on the final exam?

• 85.5

• 90

• 95

• 95.5

• none of these

A set of data describes the relationship between the size of annual salary raises and the performance ratings for employees of a certain company. The least squares regression equation is ŷ = 1400 + 2000x where y is the raise amount (in dollars) and x is the performance rating. Which of the following statements must be true?

• For each one-point increase in performance rating, the raise will increase on average by$1400.

• The actual relationship between salary raises and performance rating is linear.

• The residuals for half the observations in the dataset will be positive.

• The correlation between salary raise and performance rating is negative.

• If the mean performance rating is 1.2, then the mean raise is $3800.

You are interested in predicting the cost of heating houses on the basis of how many rooms the house has. A scatterplot of 25 houses reveals a strong linear relationship between these variables, so you calculate a least-squares regression line. “Least-squares” refers to

• Minimizing the sum of the squares of the 25 houses’ heating costs.

• Minimizing the sum of the squares of the number of rooms in each of the 25 houses.

• Minimizing the sum of the products of each house’s actual heating costs and the predicted heating cost based on the regression equation.

• Minimizing the sum of the squares of the difference between each house’s heating costs and number of rooms.

• Minimizing the sum of the squares of the residuals.

Leonardo da Vinci, the renowned painter, speculated that an ideal human would have an armspan (distance from the outstretched fingertip of the left hand to the outstretched fingertip of the right hand) that was equal to his height. Is it possible to predict armspan from height? The following computer regression printout shows the results of a least-squares regression of armspan on height, both in inches, for a sample of 18 high school students.

The students’ armspans ranged from 62 to 76 inches. Which of the following statements is true? (click on the picture to enlarge)

• If one of the students in the sample had a height of 70 inches and an armspan of 68 inches, then the residual for this student would be about –2.36 inches.

• The correlation between height and armspan is .871.

• Contrary to da Vinci’s speculation, the regression model suggests that, for these students at least, height is about 84% of armspan.

• For every one-inch increase in armspan, the regression model predicts about a 0.84-inch increase in height.

• For a student 66 inches tall, this model would predict an armspan of about 68 inches.

• A

• B

• C

• D

• E

Answer explanation

A person travels by car.  They record their miles driven in a data table.  Calculate the linear regression equation of this data.

• y = 61.93x - 1.79

• y = -1.79x + 61.93

• y = 0.016x + 0.03

• y = 0.03x + 0.016

• 82.4%

• 90.5%

• 98.0%

• 95.1%

• 95.3%

Is there a relationship between reaction time and age? A statistics student measures the reaction time of 12 subjects by measuring how far a yardstick drops, in inches, before a subject catches it between his or her thumb and forefinger (a high number thus indicates a slow reaction time). The student plans to use linear regression to see if reaction time can be predicted by the age of the subject. Output from a computer regression analysis is given below. The equation of the least-squares regression line is

• 
• 
• 

• The slope decreased and the r² increased.
• The slope increased and the r² decreased.
• The slope decreased but r² remained the same.
• Both slope and r² remained the same.

A restaurant sells pizza for the prices in the data table.  Calculate the linear regression equation of the data.

• y = 1.5x + 12

• y = 12x + 1.5

• y = 0.67x - 8

• y = -8x + 0.67