Chapter 5 AP Stats Review

All but one of the following statements contains an error. Which statement could be correct?

• There is a correlation of 0.54 between the position a football player plays and his weight.

• We found a correlation of r = –0.63 between gender and political party preference.

• The correlation between the distance travelled by a hiker and the time spent hiking is r = 0.9 meters per second.

• We found a high correlation between the height and age of children: r = 1.12.

• The correlation between mid-August soil moisture and the per-acre yield of tomatoes is r = 0.53.

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.

Mr. Nerdly asked the students in his AP Statistics class to report their overall grade point averages and their SAT Math scores. The scatterplot below provides information about his students’ data. The dark line is the least-squares regression line for the data, and its equation isŷ = 410.54 + 67.3x
Which of the following statements about the circled point is true? (click on the picture to enlarge)

• The standard score for this student’s GPA is positive.

• If we used the least-squares line to predict this student’s SAT Math score, we would make a prediction that is too low.

• This student’s residual is positive.

• Removing this data point would not change the correlation between SAT math score and GPA.

• Removing this student’s data point would decrease the slope of the least-squares line

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

The correlation between the heights of fathers and the heights of their (fully grown) sons is r = 0.52. This value was based on both variables being measured in inches. If fathers' heights were measured in feet (one foot equals 12 inches), and sons' heights were measured in furlongs (one furlong equals 7920 inches), the correlation between heights of fathers and heights of sons would be

• much smaller than 0.52

• slightly smaller than 0.52

• unchanged: equal to 0.52

• slightly larger than 0.52

• much larger than 0.52

Which statements below about least-squares regression are correct?

• Only I

• Only II

• Only III

• Both II and III

• I, II, and III

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.

For children between the ages of 18 months and 29 months, there is an approximately linear relationship between height and age. The relationship can be represented by ŷ = 64.93 + 0.63x, where y represents height (in centimeters) and x represents age (in months). Loretta is 20 months old and is 80 cm tall . What is her residual?

• -2.47

• 2.47

• -12.6

• 12.6

• 77.53

A LSRL for predicting weights of basketball players on the basis of their heights produced the residual plot below. What does the residual plot tell you about the linear model? (click on the picture to enlarge)

• A residual plot is not an appropriate means for evaluating a linear model.

• The curved pattern in the residual plot suggests that there is no association between the weight and height of basketball players.

• The curved pattern in the residual plot suggests that the linear model is not appropriate.

• There are not enough data points to draw any conclusions from the residual plot.

• The linear model is appropriate, because there are approximately the same number of points above and below the horizontal line in the residual plot.