AP Statistics Unit 2: Exploring Two-Variable Data Practice Exam

Assessment for Unit 2: Exploring Two-Variable Data

Estimated Time: 60 minutes

Exam Details

Questions: 34 total (34 multiple choice)
Time: 60 minutes (recommended)
Topics Covered: Unit 2: Exploring Two-Variable Data
Format: Multiple Choice questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (34 Questions)

Select the one best answer for each question.

1. A real estate agent wants to determine if the size of a house (in square feet) can be used to predict its selling price (in dollars). In this context, which of the following correctly identifies the explanatory variable and the response variable?

(A)Explanatory variable: Selling price; Response variable: Size of house(B)Explanatory variable: Size of house; Response variable: Selling price(C)Explanatory variable: The real estate agent; Response variable: The house(D)Explanatory variable: The specific location; Response variable: Selling price

2. A researcher is studying a population of 500 high school students to investigate if there is an association between the amount of time spent on social media (in minutes per day) and the student's grade point average (GPA). Which of the following best describes the individuals and the variables in this study?

(A)The individuals are the 500 students; the variables are social media time and GPA.(B)The individuals are social media time and GPA; the variable is the 500 students.(C)The individual is the researcher; the variables are the 500 students.(D)The individuals are the high schools; the variables are the average social media time and average GPA.

3. A scatterplot is constructed to show the relationship between the age of a used car (in years) and its resale value (in dollars). The points on the plot show a trend where as the age increases, the resale value generally decreases. Which of the following questions is this graph best suited to answer?

(A)What is the exact resale value of a 5-year-old car?(B)Does an increase in the age of a car cause the resale value to decrease?(C)Is there an association between the age of a car and its resale value?(D)Are the resale values of used cars normally distributed?

4. A scientist plots the height of 20 randomly selected oak trees against the last digit of the phone number of the person who planted them. The resulting plot shows a cloud of points with no discernible pattern or direction. Which of the following is the most appropriate conclusion based on this data?

(A)There is a strong nonlinear relationship between tree height and phone numbers.(B)The apparent lack of pattern suggests that any association between the two variables is likely due to random chance or that no association exists.(C)The sample size is too small to determine if the variables are quantitative or categorical.(D)Because the data is numerical, there must be a positive correlation that is currently hidden.

5. A survey was conducted to investigate the relationship between age group and preferred method of watching movies. The data are summarized in the two-way table below. | | Streaming | Theater | Cable TV | Total | |---|---|---|---|---| | **18–29** | 120 | 40 | 10 | 170 | | **30–50** | 80 | 50 | 70 | 200 | | **51+** | 30 | 40 | 60 | 130 | | **Total** | 230 | 130 | 140 | 500 | Which of the following calculates the conditional relative frequency of participants who prefer Streaming, given that they are in the 18–29 age group?

(A)120 / 500(B)120 / 230(C)120 / 170(D)170 / 500

6. A local transit authority collected data on two variables: 'Time of Day' (Morning Rush, Afternoon, Evening Rush) and 'Train Line' (Line A, Line B). A segmented bar chart was created with 'Time of Day' on the x-axis and relative frequency on the y-axis. The segments within each bar represent the proportion of riders using Line A and Line B. If the segmented bar chart shows that the segment for Line A has a relative frequency of 0.60 for Morning Rush, 0.60 for Afternoon, and 0.60 for Evening Rush, which of the following is the most valid conclusion?

(A)There is a strong association between Time of Day and Train Line because the proportions are high.(B)There is no association between Time of Day and Train Line because the conditional relative frequencies of Line usage are the same across all times of day.(C)Line A carries more passengers than Line B during the Morning Rush.(D)The number of passengers is identical for all three times of day.

7. In a study of 200 pets, 50 are cats, and 150 are dogs. Of the cats, 10 are strictly indoor animals. Of the dogs, 30 are strictly indoor animals. The value $\frac{10}{200} = 0.05$ represents which of the following?

(A)The marginal relative frequency of indoor animals.(B)The conditional relative frequency of being an indoor animal given the pet is a cat.(C)The joint relative frequency of pets that are both cats and strictly indoor animals.(D)The conditional relative frequency of being a cat given the pet is a strictly indoor animal.

8. A survey asked 200 high school students about their preferred mode of transportation to school. The results, categorized by grade level (Underclassman vs. Upperclassman), are shown in the table below. | | Bus | Car | Walk | Total | |---|---|---|---|---| | **Underclassman** | 50 | 20 | 30 | 100 | | **Upperclassman** | 30 | 50 | 20 | 100 | | **Total** | 80 | 70 | 50 | 200 | Which of the following represents the conditional relative frequency of students who prefer to walk, given that they are Underclassmen?

(A)0.15(B)0.25(C)0.30(D)0.60

9. A local cafe collected data on 200 drink orders, categorizing them by drink type (Coffee or Tea) and customer type (Student or Non-Student). The data is summarized in the contingency table below. | | Coffee | Tea | Total | |---|---|---|---| | **Student** | 45 | 15 | 60 | | **Non-Student** | 85 | 55 | 140 | | **Total** | 130 | 70 | 200 | What is the marginal relative frequency of customers who ordered Coffee?

(A)0.45(B)0.65(C)0.75(D)0.85

10. A study investigated the relationship between handedness (Left-handed vs. Right-handed) and eye color (Blue vs. Brown). The data collected is displayed in the table below. | | Blue Eyes | Brown Eyes | Total | |---|---|---|---| | **Left-Handed** | 10 | 40 | 50 | | **Right-Handed** | 30 | 120 | 150 | | **Total** | 40 | 160 | 200 | Based on the table, which of the following statements is true regarding the association between handedness and eye color?

(A)There is an association because the number of right-handed people with brown eyes (120) is much larger than the number of left-handed people with brown eyes (40).(B)There is an association because the conditional relative frequency of having blue eyes is different for left-handed people compared to right-handed people.(C)There is no association because the conditional relative frequency of having brown eyes is the same for both left-handed and right-handed people.(D)There is no association because the number of people in the study who are right-handed is three times the number of people who are left-handed.

11. Market researchers surveyed 200 adults to determine if there is an association between age group (Under 30 vs. 30 and Over) and preference for a new product (Like vs. Dislike). The results are as follows: - **Under 30:** 60 Like, 40 Dislike (Total 100) - **30 and Over:** 40 Like, 60 Dislike (Total 100) Which of the following best supports the conclusion that there is an association between age group and product preference?

(A)The number of people who Like the product is equal to the number of people who Dislike the product in the total sample.(B)The conditional relative frequency of liking the product is 0.60 for the 'Under 30' group and 0.40 for the '30 and Over' group.(C)The joint relative frequency of being 'Under 30' and liking the product is 0.30.(D)The marginal relative frequency of being 'Under 30' is the same as the marginal relative frequency of being '30 and Over'.

12. [Skill: 2.A | Topic: 2.4] A high school guidance counselor collects data on the grade point average (GPA) and the number of hours spent playing video games per week for a random sample of 50 students. The counselor constructs a scatterplot with hours spent playing video games on the horizontal axis and GPA on the vertical axis. The graph shows a pattern where students who play more hours of video games tend to have lower GPAs. The points are moderately scattered around a straight line that slopes downward. Which of the following best describes the relationship between the number of hours spent playing video games and GPA shown in the scatterplot?

(A)There is a positive, linear, and moderate association.(B)There is a negative, linear, and moderate association.(C)There is a negative, non-linear, and strong association.(D)There is no apparent association between the variables.

13. [Skill: 2.A | Topic: 2.4] A researcher is studying the effect of water temperature on the heart rate of a specific species of water flea. The researcher varies the water temperature (in degrees Celsius) and measures the resulting heart rate (in beats per minute) for several fleas. When constructing a scatterplot to display the data from this experiment, which of the following describes the correct placement of the variables?

(A)Water temperature should be plotted on the x-axis because it is the response variable.(B)Water temperature should be plotted on the x-axis because it is the explanatory variable.(C)Heart rate should be plotted on the x-axis because it is the explanatory variable.(D)Heart rate should be plotted on the x-axis because it is the response variable.

14. [Skill: 4.B | Topic: 2.4] A real estate agent analyzes the relationship between the square footage of houses ($x$) and their selling prices ($y$) in a large subdivision. The scatterplot of the data reveals a strong, positive, linear association. Which of the following is the best interpretation of this association?

(A)Increasing the square footage of a house causes the selling price to increase.(B)Houses with larger square footage tend to have higher selling prices.(C)Houses with larger square footage tend to have lower selling prices.(D)There is no relationship between the square footage of a house and its selling price.

15. [Skill: 2.A | Topic: 2.4] An environmental scientist collects data on the amount of rainfall (in inches) and the depth of a local river (in feet) over several months. The scatterplot of the data shows that as rainfall increases, the river depth increases rapidly at first, but then the rate of increase slows down as the river nears its capacity, creating a curved pattern. There are no points that deviate significantly from this pattern. Which of the following best describes the form and unusual features of the scatterplot?

(A)The form is linear, and there are no unusual features.(B)The form is linear, and there are clusters present.(C)The form is non-linear, and there are no unusual features.(D)The form is non-linear, and there is an outlier present.

16. A researcher collects data on two quantitative variables, $x$ and $y$, and creates a scatterplot. The graph shows a distinct curved pattern in the data points. Despite this curvature, the researcher calculates the correlation coefficient to be $r = 0.88$. Which of the following best describes the relationship between $x$ and $y$?

(A)There is a strong, positive linear relationship between $x$ and $y$ because $r$ is close to 1.(B)The calculation of $r$ must be incorrect because the correlation coefficient for curved data should be close to 0.(C)There is a strong positive association, but a linear model may not be appropriate for describing the relationship.(D)Changes in $x$ cause changes in $y$ because the correlation is statistically significant.

17. A marine biologist is studying the relationship between the length of a specific species of shark (measured in centimeters) and its weight (measured in kilograms). The correlation coefficient for the data collected is calculated to be $r = 0.92$. If the biologist were to convert the length measurements to meters and the weight measurements to pounds, what would be the new value of the correlation coefficient?

(A)The new correlation would be lower than 0.92 because meters are larger units than centimeters.(B)The new correlation would be 0.92.(C)The new correlation would be significantly higher than 0.92 due to the conversion to pounds.(D)The new correlation cannot be determined without the original raw data.

18. A high school guidance counselor analyzes data regarding the number of college applications submitted by students ($x$) and the total amount of scholarship money awarded ($y$) in dollars. The correlation coefficient is calculated to be $r = 0.65$. Which of the following is the correct interpretation of this value?

(A)For every additional college application submitted, the scholarship money awarded increases by 0.65 dollars.(B)Submitting more college applications causes students to be awarded more scholarship money.(C)There is a moderate, positive linear relationship between the number of college applications submitted and the amount of scholarship money awarded.(D)65% of the variation in scholarship money awarded can be explained by the linear relationship with the number of applications.

19. Consider the four scatterplots described below: 1. **Plot A:** Points are scattered randomly with no discernible pattern. 2. **Plot B:** Points fall exactly on a straight line sloping downward from left to right. 3. **Plot C:** Points show a loose upward trend from left to right with significant scatter. 4. **Plot D:** Points are tightly clustered around a line sloping upward from left to right. Which of the plots corresponds to a correlation coefficient of approximately $r = -1.0$?

(A)Plot A(B)Plot B(C)Plot C(D)Plot D

20. A marine biologist is studying the relationship between the length of a certain species of fish (in centimeters) and its weight (in grams). A least-squares regression analysis was performed on the data, and the computer output is shown below. Predictor | Coef | SE Coef | T | P --- | --- | --- | --- | --- Constant | -105.4 | 22.1 | -4.77 | 0.000 Length | 28.3 | 1.2 | 23.58 | 0.000 Based on the computer output, what is the predicted weight, in grams, of a fish that is 15 centimeters long?

(A)319.1 grams(B)424.5 grams(C)529.9 grams(D)1475.6 grams

21. Data were collected on the depth of a dive (in meters) and the water temperature (in degrees Celsius) for dives ranging from 5 meters to 40 meters deep. The least-squares regression line for the data is given by the equation: $\hat{y} = 28 - 0.5x$ where $\hat{y}$ is the predicted water temperature and $x$ is the depth of the dive. Which of the following best explains why it would be inappropriate to use this model to predict the water temperature at a depth of 90 meters?

(A)The correlation coefficient would likely be close to 0 at that depth.(B)A depth of 90 meters is an outlier and would drastically change the slope of the regression line.(C)Predicting for a depth of 90 meters involves extrapolation, and the linear relationship may not hold outside the observed data range.(D)The predicted temperature would be negative, which is impossible for water in a liquid state.

22. A linear regression model is developed to predict the battery life of a laptop (in hours) based on the screen brightness setting (measured as a percentage from 0 to 100). The regression equation is: $\widehat{\text{Battery}} = 12.5 - 0.08(\text{Brightness})$ What is the predicted battery life for a laptop with the screen brightness set to 75%?

(A)6.0 hours(B)6.5 hours(C)9.4 hours(D)18.5 hours

23. A statistics student collects data on the number of hours spent studying for a final exam ($x$) and the score received on the exam ($y$). The least-squares regression line is calculated to be $\hat{y} = 55 + 4x$. One specific student in the sample studied for 6 hours and received a score of 82. What is the predicted exam score for this student according to the model?

(A)3(B)27(C)79(D)82

24. A statistics student is analyzing the relationship between the number of hours studied (x) and the score received on a final exam (y). The least-squares regression line for the data is given by the equation $\hat{y} = 45 + 3.5x$. One student in the study studied for 6 hours and received a score of 62. What is the value of the residual for this student, and how is it interpreted?

(A)Residual = -4; the student scored 4 points lower than the model predicted.(B)Residual = 4; the student scored 4 points higher than the model predicted.(C)Residual = -4; the student studied 4 hours less than the model predicted.(D)Residual = 66; the model predicted the student would score a 66.

Refer to the figure below.

25. A researcher collects data on the age of a car (in years) and its resale value (in dollars). After fitting a least-squares regression line to the data, the researcher constructs the residual plot shown below. Based on the residual plot, is a linear model appropriate for describing the relationship between the age of the car and its resale value?

(A)Yes, because the residuals are relatively small compared to the predicted values.(B)Yes, because the sum of the residuals is approximately zero.(C)No, because there is a distinct curved pattern in the residual plot.(D)No, because the residuals are not all equal to zero.

26. The residual plot for a linear regression model relating the height of a tree (x) to the diameter of its trunk (y) is shown. One specific point on the residual plot is located at $(15, 2.3)$. Which of the following is the correct interpretation of this point?

(A)For a tree with a height of 15 feet, the predicted trunk diameter is 2.3 inches.(B)For a tree with a height of 15 feet, the actual trunk diameter is 2.3 inches more than the model predicted.(C)For a tree with a height of 15 feet, the actual trunk diameter is 2.3 inches less than the model predicted.(D)For a tree with a predicted trunk diameter of 15 inches, the residual is 2.3 inches.

27. A biologist is studying the growth of bacteria over time. She fits a least-squares regression line to her data and calculates the residuals. For a specific data point, the residual is negative. Which of the following statements must be true regarding this data point?

(A)The data point lies above the least-squares regression line.(B)The predicted value for this data point is greater than the observed (actual) value.(C)The correlation between time and growth is negative.(D)The linear model is not appropriate for this data set.

28. A statistics student is analyzing the relationship between the number of hours spent studying for a final exam ($x$) and the score received on that exam ($y$) for a class of 30 students. The summary statistics for the data are as follows: Mean number of hours studied: $\bar{x} = 12$ Standard deviation of hours studied: $s_x = 3$ Mean exam score: $\bar{y} = 78$ Standard deviation of exam scores: $s_y = 9$ Correlation coefficient: $r = 0.8$ Based on these summary statistics, which of the following is the equation of the least-squares regression line for predicting the exam score from the number of hours studied?

(A)$\hat{y} = 49.2 + 2.4x$(B)$\hat{y} = 74.8 + 0.27x$(C)$\hat{y} = 42 + 3x$(D)$\hat{y} = 2.4 + 49.2x$

29. A marine biologist collects data on the length in inches ($x$) and weight in pounds ($y$) of a specific species of fish. The least-squares regression line for the data is calculated to be $\hat{y} = -2.3 + 0.45x$. Which of the following is the best interpretation of the slope of the regression line?

(A)For every 1-inch increase in length, the predicted weight of the fish increases by 0.45 pounds.(B)For every 1-pound increase in weight, the predicted length of the fish increases by 0.45 inches.(C)A fish with a length of 0 inches is predicted to weigh -2.3 pounds.(D)For every 1-inch increase in length, the actual weight of the fish will increase by 0.45 pounds.

30. A car dealership analyzes the relationship between the mileage on a used car (in thousands of miles) and its selling price (in dollars). The regression analysis produces the following linear model: $\widehat{\text{Price}} = 24000 - 150(\text{Mileage})$ The mileage of the cars in the dataset ranges from 10,000 to 150,000 miles. Which of the following statements best interprets the y-intercept of this regression line?

(A)For every additional 1,000 miles driven, the predicted price decreases by 150 dollars.(B)The predicted selling price of a car with 0 miles is 24,000 dollars, but this prediction may not be valid because 0 is outside the range of the data.(C)The selling price of a car is predicted to be 0 dollars when the mileage is 160,000 miles.(D)The initial selling price of all cars on the lot is 24,000 dollars.

31. Researchers are investigating the relationship between the amount of dietary fiber consumed per day (in grams) and cholesterol levels (in mg/dL) in a group of adults. The correlation coefficient between the two variables is $r = -0.60$. Which of the following is the correct interpretation of the coefficient of determination, $r^2$?

(A)36% of the variation in cholesterol levels is explained by the linear relationship with dietary fiber consumption.(B)60% of the variation in cholesterol levels is explained by the linear relationship with dietary fiber consumption.(C)There is a weak negative linear relationship between dietary fiber consumption and cholesterol levels.(D)The model correctly predicts cholesterol levels 36% of the time.

32. A set of bivariate data exhibits a strong, positive linear relationship. One observation, Point A, has an x-value that is significantly larger than the x-values of all other data points. The y-value of Point A is consistent with the linear trend established by the other data points. Which of the following best describes the effect of Point A on the least-squares regression line and the correlation coefficient?

(A)Point A is a high-leverage point. Including it in the analysis makes the correlation coefficient closer to 1.(B)Point A is an outlier with a large residual. Including it in the analysis makes the correlation coefficient closer to 0.(C)Point A is an influential point that drastically changes the slope. Removing it would significantly increase the slope.(D)Point A has low leverage but a high residual. Including it in the analysis decreases the coefficient of determination.

33. A researcher collects data on two variables, x and y, and fits a least-squares regression line to the data. The residual plot for this linear model shows a distinct curved pattern. The researcher then transforms the data by taking the natural logarithm of the y-values (ln y) and plots ln(y) against x. The residual plot for the transformed data shows a random scatter of points above and below zero with no discernible pattern. Which of the following conclusions is best supported by this analysis?

(A)The linear model was appropriate for the original data because the residuals summed to zero.(B)An exponential model is appropriate for the relationship between x and y because the residual plot for the transformed data shows no pattern.(C)A power model is appropriate for the relationship between x and y because the transformation involved logarithms.(D)The transformation was unsuccessful because the correlation coefficient of the transformed data is not exactly 1 or -1.

34. Consider a scatterplot of 20 points where the x-values range from 10 to 90. A least-squares regression line is calculated. Point Q has an x-value of 50 (which is the mean of the x-values) and a y-value that is far above the regression line. Which of the following statements correctly predicts the change in the slope and the coefficient of determination (r²) if Point Q is removed from the data set?

(A)The slope will change significantly, and r² will increase.(B)The slope will change significantly, and r² will decrease.(C)The slope will not change significantly, and r² will increase.(D)The slope will not change significantly, and r² will decrease.

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