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# Two-Variable Data Analysis Free Response Practice Problems for AP Statistics

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By McGraw-Hill Professional
Updated on Feb 5, 2011

Review the following concepts if necessary:

### Problems

1. Given a two-variable dataset such that = 14.5, = 20, sx = 4, sy = 11, r = .80, find the least-squares regression line of y on x.
2. The data below give the first and second exam scores of 10 students in a calculus class.
1. Draw a scatterplot of these data.
2. To what extent do the scores on the two tests seem related?
3. The following is a residual plot of a linear regression. A line would not be a good fit for these data. Why not? Is the regression equation likely to underestimate or overestimate the y-value of the point in the graph marked with the square?
4. The regional champion in 10 and under 100 m backstroke has had the following winning times (in seconds) over the past 8 years:
5. How many years until you expect the winning time to be one minute or less? What's wrong with this estimate?

6. Measurements are made of the number of cockroaches present, on average, every 3 days, beginning on the second day, after apartments in one part of town are vacated. The data are as follows:
7. How many cockroaches would you expect to be present after 9 days?

8. A study found a strongly positive relationship between number of miles walked per week and overall health. A local news commentator, after reporting on the results of the study, advised everyone to walk more during the coming year because walking more results in better health. Comment on the reporter's advice.
9. Carla, a young sociologist, is excitedly reporting on the results of her first professional study. The finding she is reporting is that 72% of the variation in math grades for girls can be explained by the girls' socioeconomic status. What does this mean, and is it indicative of a strong linear relationship between math grades and socioeconomic status for girls?
10. Which of the following statements are true of a least-squares regression equation?
1. It is the unique line that minimizes the sum of the residuals.
2. The average residual is 0.
3. It minimizes the sum of the squared residuals.
4. The slope of the regression line is a constant multiple of the correlation coefficient.
5. The slope of the regression line tells you how much the response variable will change for each unit change in the explanatory variable.
11. Consider the following dataset:
12. Given that the LSRL for these data is = 26.211 – 0.25x, what is the value of the residual for x = 73? Is the point (73,7.9) above or below the regression line?

13. Suppose the correlation between two variables is r = –0.75. What is true of the correlation coefficient and the slope of the regression line if
1. each of the y values is multiplied by –1?
2. the x and y variables are reversed?
3. the x and y variables are each multiplied by –1?
14. Suppose the regression equation for predicting success on a dexterity task (y) from number of training sessions (x) is = 45 + 2.7x and that = 3.33.
15. What percentage of the variation in y is not explained by the regression on x?

16. Consider the following scatterplot. The highlighted point is both an outlier and an influential point. Describe what will happen to the correlation and the slope of the regression line if that point is removed.
17. The computer printout below gives the regression output for predicting crime rate (in crimes per 1000 population) from the number of casino employees (in 1000s).
18. Based on the output,

1. give the equation of the LSRL for predicting crime rate from number.
2. give the value of r, the correlation coefficient.
3. give the predicted crime rate for 20,000 casino employees.
19. A study was conducted in a mid-size U.S. city to investigate the relationship between the number of homes built in a year and the mean percentage appreciation for that year. The data for a 5-year period are as follows:
1. Obtain the LSRL for predicting appreciation from number of new homes built in a year.
2. The following year, 85 new homes are built. What is the predicted appreciation?
3. How strong is the linear relationship between number of new homes built and percentage appreciation? Explain.
4. Suppose you didn't know the number of new homes built in a given year. How would you predict appreciation?
20. A set of bivariate data has r2 = 0.81.
1. x and y are both standardized, and a regression line is fitted to the standardized data. What is the slope of the regression line for the standardized data?
2. Describe the scatterplot of the original data.
21. Estimate r, the correlation coefficient, for each of the following graphs:
22. The least-squares regression equation for the given data is = 3 + x. Calculate the sum of the squared residuals for the LSRL.
23. Many schools require teachers to have evaluations done by students. A study investigated the extent to which student evaluations are related to grades. Teacher evaluations and grades are both given on a scale of 100. The results for Prof. Socrates (y) for 10 of his students are given below together with the average for each student (x).
1. Do you think student grades and the evaluations students give their teachers are related? Explain.
2. What evaluation score do you think a student who averaged 80 would give Prof. Socrates?
24. Which of the following statements are true?
1. The correlation coefficient, r, and the slope of the regression line, b, always have the same sign.
2. The correlation coefficient is the same no matter which variable is considered to be the explanatory variable and which is considered to be the response variable.
3. The correlation coefficient is resistant to outliers.
4. x and y are measured in inches, and r is computed. Now, x and y are converted to feet, and a new r is computed. The two computed values of r depend on the units of measurement and will be different.
5. The idea of a correlation between height and gender is not meaningful because gender is not numerical.
25. A study of right-handed people found that the regression equation for predicting lefthand strength (measured in kg) from right-hand strength is left-hand strength = 7.1 + 0.35 (right-hand strength).
1. What is the predicted left-hand strength for a right-handed person whose righthand strength is 12 kg?
2. Interpret the intercept and the slope of the regression line in the context of the problem.

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