AP Stats Random Variables Practice

Two variables, x and y, have a correlation of 0.75. If x has a mean of 25 and a standard deviation of 3, and y has a mean of 12 and a standard deviation of 6, which of the following is the least-squares regression line for the two variables?

• ŷ = −25.5 + 1.5x

• ŷ = 12 + 1.5x

• ŷ = 12 + 0.75x

• ŷ = 16 + 0.75x

• Not enough information

Data are collected on the amount of fat (in grams) and calories in the french fry orders at nine fast food restaurants. The least-squares regression line for the data is y = 274.34 + 9.55x, where y is the predicted number of calories and x is grams of fat. Which of the following is the correct interpretation of the slope of the least-squares regression line?

• The calories increase by 9.55, on average.

• For every increase in fat, the calories increase as well.

• Every increase of 1 gram of fat causes an increase of 9.55 calories.

• For every increase of 1 gram of fat, the predicted calories increase by 9.55.

• For every increase of 1 calorie, the predicted grams of fat increase by 9.55.

A random sample of households is taken. For each household, the number of hours spent watching television and the power consumption (in kWh) during a day are recorded. The table below shows computer output from a linear regression analysis on the data. Which of the following is the equation of the least-squares regression line?

• ŷ = 19.31 + 0.891x

• ŷ = 2.8621 + 0.2715x

• ŷ = 0.891 + 19.31x

• ŷ = 0.2715 + 2.8621x

• ŷ = 0.2715 + 0.891x

A random sample of households is taken. For each household, the number of hours spent watching television and the power consumption (in kWh) during a day are recorded. The table below shows computer output from a linear regression analysis on the data. Which of the following is a correct interpretation of r²?

• Number of hours of television explains 30% of the variability in power consumption.

• 30% of the increase in number of hours of television is explained by power consumption.

• 30% of the data will lie on the least-squares regression line.

• 30% of the residuals will be less than 4.185.

• All of the above are correct interpretations.

Using the least-squares regression line, ˆy = −25.5+1.5x , what is the residual for the data point at (28,19)?

• -2.5

• 2.5

• 4.33

• 16.5

• 19

Scientists examined the activity level of 7 fish at different temperatures. Fish activity was rated on a scale of 0 (no activity) to 100 (maximal activity). The temperature was measured in degrees Celsius. A computer regression printout and a residual plot are provided. Notice that the horizontal axis on the residual plot is labeled “Fitted value,” which means the same thing as “predicted value.” What is the correlation between temperature and fish activity?

• –0.95

• 0.91

• 0.45

• 0.95

• –0.91

Scientists examined the activity level of 7 fish at different temperatures. Fish activity was rated on a scale of 0 (no activity) to 100 (maximal activity). The temperature was measured in degrees Celsius. A computer regression printout and a residual plot are provided. Notice that the horizontal axis on the residual plot is labeled “Fitted value,” which means the same thing as “predicted value.”Which of the following gives a correct interpretation of 𝑠s in this setting?

• The typical distance of the temperature readings from their mean is about 4.785°C.

• The typical distance of the activity level readings from their mean is about 4.785 units.

• At a temperature of 0°C, this model predicts an activity level of 4.785 units.

• For every 1°C increase in temperature, fish activity is predicted to increase by 4.785 units.

• The typical distance of the activity level ratings from the least-squares line is about 4.785 units.

Answer explanation

This is another way of saying the typical error between the actual values and the predicted values using the linear model.

Which of the following statements is not true of the correlation 𝑟 between the lengths (in inches) and weights (in pounds) of a sample of brook trout?

• 𝑟 is measured in inches

• 𝑟 would not change if we measured the weights of the trout in kilograms instead of pounds.

• 𝑟 would not change if we measured the lengths of the trout in centimeters instead of inches.

• 𝑟 must be a value between -1 and 1.

• If longer trout tend to also be heavier, then

  𝑟 > 0.

Answer explanation

Correlation does not have units attached to it.

When we standardize the values of a variable, the distribution of standardized values has mean 0 and standard deviation 1. Suppose we measure two variables 𝑋 and 𝑌 on each of several subjects. We standardize both variables and then compute the least-squares regression line. Suppose the slope of the least-squares regression line is –0.44.

• the correlation will also be –0.44.

• the intercept will be 1.0.

• The correlation will be 1/–0.44.

• the intercept will also be –0.44.

• the correlation will be 1.0.

Scientists examined the activity level of 7 fish at different temperatures. Fish activity was rated on a scale of 0 (no activity) to 100 (maximal activity). The temperature was measured in degrees Celsius. A computer regression printout and a residual plot are provided. Notice that the horizontal axis on the residual plot is labeled “Fitted value,” which means the same thing as “predicted value.” What was the actual activity level rating for the fish at a temperature of 20°C?

• 66

• 81

• 87

• 84

• 3

Answer explanation

The line predicts that a fish would have activity level 𝑦̂ =148.62−3.2167(20) = 84.286 Looking at the residual plot, the fish with a predicted activity level of about 84 has a residual of approximately +3.