Stats Chapter

An AP®® Statistics student designs an experiment to see whether today’s high school students are becoming too calculator-dependent. She prepares two quizzes, both of which contain 40 questions that are best done using paper-and-pencil methods. A random sample of 30 students participates in the experiment. Each student takes both quizzes—one with a calculator and one without—in a random order. To analyze the data, the student constructs a scatterplot that displays a linear association between the number of correct answers with and without a calculator for the 30 students. A least-squares regression yields the equation𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑜𝑟ˆ=−1.2+0.865(𝑃𝑒𝑛𝑐𝑖𝑙)Calculator^=−1.2+0.865(Pencil) 𝑟=0.79.

• If the student had used Calculator as the explanatory variable, the correlation would remain the same.

• If the student had used Calculator as the explanatory variable, the slope of the least-squares line would remain the same.

• The standard deviation of the number of correct answers on the paper-and-pencil quizzes was smaller than the standard deviation on the calculator quizzes.

• 1 and 3 only

• 1,2, and 3

Answer explanation

The slope of the least-squares line depends on which variable is the explanatory variable and which is the response.

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.  

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.

The scatterplot shows the relationship between the number of people per television set and the number of people per physician for 40 countries, along with the least-squares regression line. In Ethiopia, there were 503 people per TV and 36,660 people per doctor. Which of the following is correct?

• Ethiopia has more people per doctor than expected, based on how many people it has per TV.

• Increasing the number of TVs in a country will attract more doctors.

• The correlation is greater than 1.

• The slope of the least-squares regression line is less than 1.

• The point for Ethiopia is decreasing the slope of the least-squares regression line.

Answer explanation

Ethiopia's residual is very large, indicating that there are many more people per doctor than is predicted by the least-squares regression line.

The first scatterplot shows the lean body mass and metabolic rate for a sample of 5 adults. For each person, the lean body mass is the subject’s total weight in kilograms less any weight due to fat. The metabolic rate is the number of calories burned in a 24-hour period. Because a person with no lean body mass should burn no calories, it makes sense to model the relationship with a direct variation function in the form 𝑦=𝑘𝑥. Models were tried using different values of 𝑘(𝑘=25,𝑘=26), and the sum of squared residuals (SSR) was calculated for each value of 𝑘. Given is a second scatterplot, this one showing the relationship between SSR and 𝑘: According to the scatterplot, what is the ideal value of 𝑘 to use for predicting metabolic rate?

• 24

• 26

• 25

• 31

• 36

Answer explanation

When 𝑘=31, the sum of the squared residuals is minimized.

What equation for the LSRL would be most reasonable for this data?

The correlation (r) between high school students' height (in inches) and weight (in pounds) for a large sample of students is found to be 0.3. If the weight measurements are converted to kg, what will be the resulting correlation?

• 0.3

• 0.66

• -0.3

• 0.136

For a particular value of an explanatory variable, the residual is -0.7. Which of the following is a true statement?

• The regression line underestimates the actual response variable value at this point.

• The regression line overestimates the actual response variable value at this point.

• . A non-linear regression would provide a better fit.

Data is obtained for a large group of college students relating the time they spent writing an essay and the grade on the essay. The equation is grade = 30.18 + 6.49(time) with r = 0.724 What percentage of the variation in grades can be explained by looking at time spent on the essay?

• 52.4%

• 72.4%

• 27.6%

• 47.6%