AP Statistics Unit 9: Inference for Quantitative Data: Slopes Practice Exam

Assessment for Unit 9: Inference for Quantitative Data: Slopes

Estimated Time: 60 minutes

Exam Details

Questions: 25 total (25 multiple choice)
Time: 60 minutes (recommended)
Topics Covered: Unit 9: Inference for Quantitative Data: Slopes
Format: Multiple Choice questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (25 Questions)

Select the one best answer for each question.

1. A statistician is performing a linear regression analysis to predict the price of used cars based on their age. After calculating the least-squares regression line, the statistician constructs a residual plot. The plot displays a distinct fan shape, where the residuals are tightly clustered around zero for newer cars but become increasingly spread out for older cars. Based on this residual plot, which condition for inference for the slope of the regression line has most likely been violated?

(A)Linearity: The relationship between the variables is not linear.(B)Independence: The observations are not independent of one another.(C)Normality: The distribution of the response variable is not normal.(D)Equal Variance (Homoscedasticity): The standard deviation of the response variable is not constant for all values of the explanatory variable.

2. A marine biologist is investigating whether there is a positive linear relationship between the water temperature (in degrees Celsius) and the growth rate of a specific coral species (in cm per year). She collects data from 30 randomly selected sites. Let $\beta$ represent the true slope of the population regression line relating water temperature to growth rate. Which of the following pairs of hypotheses should the biologist test?

(A)H_0: $\beta $= 0 and H_a: $\beta \neq $0(B)H_0: $\beta $= 0 and H_a: $\beta $> 0(C)H_0: $\beta $> 0 and H_a: $\beta $= 0(D)H_0: $\beta $= 0 and H_a: $\beta $< 0

3. A study was conducted to determine the relationship between the number of hours studied and the score on a final exam for a sample of 22 students. The computer output for the least-squares regression analysis is shown below. | Predictor | Coef | SE Coef | T-Value | P-Value | | :--- | :--- | :--- | :--- | :--- | | Constant | 55.32 | 4.12 | 13.43 | 0.000 | | Hours | 2.48 | 0.65 | 3.82 | 0.001 | Which of the following is the correct expression for a 95% confidence interval for the slope of the population regression line? (Assume the critical value $t^*$ for 20 degrees of freedom is 2.086 and for 21 degrees of freedom is 2.080.)

(A)2.48 $\pm $2.086(0.65)(B)2.48 $\pm $2.080(0.65)(C)55.32 $\pm $2.086(4.12)(D)2.48 $\pm $2.086(4.12)

4. In a linear regression analysis predicting monthly heating costs from the average outdoor temperature, the standard error of the slope ($SE_b$) was calculated to be 0.15. Which of the following is the best interpretation of this value?

(A)On average, the predicted heating cost deviates from the actual heating cost by 0.15 dollars.(B)On average, the monthly heating cost increases by 0.15 dollars for every one-degree increase in outdoor temperature.(C)In repeated random sampling, the sample slope typically varies from the true population slope by about 0.15.(D)About 95% of the observed data points fall within 0.30 units of the least-squares regression line.

5. A researcher performs a t-test for the slope of a regression line to predict reaction time ($y$) from sleep duration ($x$). The regression equation is $\hat{y} = 1.2 - 0.08x$. The t-statistic for the slope is calculated to be $t = -2.15$ based on a sample of size $n=20$. Using a significance level of $\alpha = 0.05$, which of the following is the correct conclusion regarding the null hypothesis $H_0: \beta = 0$ against the two-sided alternative $H_a: \beta \neq 0$?

(A)Reject $H_0$ because the P-value is less than 0.05.(B)Fail to reject $H_0$ because the P-value is greater than 0.05.(C)Accept $H_a$ because the slope is negative.(D)Fail to reject $H_0$ because the sample size is too small for inference.

Refer to the figure below.

6. A statistics student performed a least-squares regression analysis to predict the price of used cars (in dollars) based on their age (in years). The student generated a residual plot from the data, which is described in the image cue below. Based on this residual plot, which condition for inference for the slope of the regression model is most clearly violated?

(A)The relationship between age and price is linear.(B)The standard deviation of the residuals is constant.(C)The residuals are normally distributed.(D)The observations are independent.

7. A marine biologist wants to estimate the relationship between the length of a specific species of fish (in cm) and its weight (in grams). A random sample of 18 fish is selected, and a linear regression analysis is performed. The computer output is shown below. | Predictor | Coef | SE Coef | T | P | | :--- | :--- | :--- | :--- | :--- | | Constant | -15.2 | 4.1 | -3.71 | 0.002 | | Length | 3.4 | 0.6 | 5.67 | 0.000 | Which of the following represents the margin of error for a 95% confidence interval for the slope of the population regression line?

(A)1.746 * 0.6(B)2.120 * 0.6(C)2.120 * 3.4(D)1.96 * 0.6

8. Data were collected from a random sample of 25 students to investigate the relationship between the number of hours spent studying for a final exam (x) and the score on the exam (y). The slope of the least-squares regression line was calculated to be b = 4.2 with a standard error of the slope SE_b = 1.1. Which of the following is the correct 99% confidence interval for the slope of the population regression line?

(A)4.2 ± 2.576(1.1)(B)4.2 ± 2.797(1.1)(C)4.2 ± 2.807(1.1)(D)4.2 ± 2.807(1.1 / sqrt(25))

9. A 95% confidence interval for the slope of the regression line relating the amount of fertilizer applied (in kg) to the yield of corn (in bushels) is calculated to be (1.2, 2.8). Which of the following is the best interpretation of this interval?

(A)We are 95% confident that for every 1 kg increase in fertilizer, the corn yield for a specific field will increase by between 1.2 and 2.8 bushels.(B)We are 95% confident that the mean corn yield is between 1.2 and 2.8 bushels greater for fields with fertilizer than for fields without fertilizer.(C)We are 95% confident that for every 1 kg increase in fertilizer, the true mean increase in corn yield is between 1.2 and 2.8 bushels.(D)95% of the time, the slope of the sample regression line will fall between 1.2 and 2.8.

10. A marine biologist is studying the relationship between the length (in centimeters) and the weight (in kilograms) of a specific species of fish. A random sample of 30 fish is selected, and a least-squares regression analysis is performed on the data. The computer output for the regression analysis is shown below. Predictor | Coef | SE Coef | T | P --- | --- | --- | --- | --- Constant | -0.54 | 0.23 | -2.35 | 0.026 Length | 0.42 | 0.08 | 5.25 | 0.000 Which of the following is the correct 95% confidence interval for the slope of the population regression line assuming all conditions for inference are met? (Use t* = 2.048)

(A)0.42 ± 2.048(0.08)(B)0.42 ± 2.048(0.23)(C)-0.54 ± 2.048(0.23)(D)0.42 ± 1.96(0.08)

11. A statistics student computed a 90 percent confidence interval for the slope of the least-squares regression line predicting final exam score from hours of study. The resulting interval was (1.5, 4.5). Based on this interval, which of the following claims is supported?

(A)There is no linear relationship between hours of study and final exam score because the interval does not contain 0.(B)There is a positive linear relationship between hours of study and final exam score because the entire interval is positive.(C)An increase of 1 hour in study time will cause an increase of exactly 3 points on the final exam.(D)Approximately 90 percent of the students saw their scores increase between 1.5 and 4.5 points for every hour studied.

12. Researchers are investigating the relationship between tree density (trees per acre) and soil nutrient content. They construct a 95 percent confidence interval for the slope of the regression line based on a sample of 20 plots. If the researchers want to decrease the width of this confidence interval in a future study, which of the following actions would be most effective?

(A)Decrease the sample size to n = 10.(B)Increase the confidence level to 99 percent.(C)Increase the sample size to n = 100.(D)Select a sample where the values of tree density (the explanatory variable) are closer to the mean tree density.

13. A linear regression analysis was performed to predict the sale price of homes based on their square footage. A 95 percent confidence interval for the slope of the regression line was calculated to be (120, 160). Which of the following is the correct interpretation of the 95 percent confidence level in this context?

(A)There is a 0.95 probability that the true slope of the population regression line is between 120 and 160.(B)In repeated random sampling, approximately 95 percent of the constructed confidence intervals will capture the true slope of the population regression line.(C)Approximately 95 percent of the homes in the sample have a price per square foot between 120 and 160 dollars.(D)In repeated random sampling, the sample slope will fall between 120 and 160 in approximately 95 percent of the samples.

14. A real estate agent wants to determine if there is a positive linear relationship between the square footage of a house (explanatory variable) and its selling price (response variable) in a specific city. The agent collects a random sample of 50 recent home sales and performs a linear regression analysis. Which of the following pairs of hypotheses is appropriate for the test?

(A)H0: b = 0; Ha: b > 0(B)H0: b = 0; Ha: b ≠ 0(C)H0: β = 0; Ha: β > 0(D)H0: β = 0; Ha: β ≠ 0

15. A statistician is checking the conditions for inference for a regression of monthly heating costs on average daily temperature. The scatterplot of the data appears roughly linear. However, a plot of the residuals versus the average daily temperature shows that the points are tightly clustered around the horizontal axis for low temperatures but become increasingly spread out as the temperature increases (a 'fan' or 'cone' shape). Which condition for inference has likely been violated?

(A)Linearity: The relationship between the variables is not linear.(B)Independence: The observations are not independent of one another.(C)Normality: The distribution of the response variable is not normal.(D)Equal Standard Deviation: The standard deviation of the response variable is not constant for all values of the explanatory variable.

16. Researchers performed a linear regression to predict the weight of a fish based on its length. To verify the conditions for inference, they generated several graphs. Which of the following graphs is necessary to check the condition that, for any fixed length, the weights of the fish are normally distributed?

(A)A histogram of the fish lengths.(B)A histogram of the fish weights.(C)A histogram or Normal probability plot of the residuals.(D)A scatterplot of fish weight versus fish length.

17. A researcher is conducting a t-test for the slope of a regression line relating the amount of fertilizer used (x) to the yield of corn (y). The computer output for the regression analysis is examined. A residual plot of residuals versus fitted values displays a clear U-shaped pattern. Based on this plot, which of the following conclusions is correct regarding the validity of the t-test?

(A)The test is valid because the random assignment condition was met.(B)The test is valid because the sample size is likely large enough to satisfy the Central Limit Theorem.(C)The test is not valid because the relationship between fertilizer amount and corn yield is not linear.(D)The test is not valid because the standard deviation of the corn yield is not constant.

18. A marine biologist is studying the relationship between the length of a certain species of fish (in centimeters) and its weight (in grams). A random sample of 20 fish is selected, and a linear regression analysis is performed on the data. The computer output for the regression analysis is shown below. | Predictor | Coef | SE Coef | T | P | | :--- | :--- | :--- | :--- | :--- | | Constant | -15.4 | 4.2 | -3.67 | 0.002 | | Length | 24.8 | 6.2 | ? | ? | Which of the following is the value of the test statistic for the hypothesis test $H_0: \beta = 0$ versus $H_a: \beta \neq 0$, where $\beta$ represents the slope of the population regression line?

(A)t = 4.00(B)t = 0.25(C)t = 3.67(D)t = 5.90

19. A researcher is investigating the relationship between the number of hours spent studying for a final exam and the score on that exam. A regression analysis on data from a random sample of students produces a test statistic of $t = 2.15$ and a p-value of $0.042$ for the slope of the regression line. Which of the following is the correct interpretation of this p-value?

(A)There is a 0.042 probability that the null hypothesis is true.(B)Assuming there is no linear relationship between study time and exam score, there is a 0.042 probability of observing a sample slope as extreme as or more extreme than the one observed.(C)There is a 0.042 probability that there is a linear relationship between study time and exam score.(D)Assuming there is a linear relationship between study time and exam score, there is a 0.042 probability of observing a sample slope as extreme as or more extreme than the one observed.

20. An economist wants to test if there is a positive linear relationship between a country's GDP per capita and its happiness index. She collects data from 28 randomly selected countries and performs a linear regression. The computer output reports a t-statistic for the slope of $t = 1.85$ and a two-sided p-value of $0.076$. If the economist tests the hypotheses $H_0: \beta = 0$ versus $H_a: \beta > 0$ at the significance level $\alpha = 0.05$, which of the following is the correct decision and justification?

(A)Fail to reject $H_0$ because $0.076 > 0.05$.(B)Fail to reject $H_0$ because $0.038 < 0.05$.(C)Reject $H_0$ because $0.038 < 0.05$.(D)Reject $H_0$ because $0.076 > 0.05$.

21. A statistics student performs a t-test for the slope of a least-squares regression line based on a sample of size $n=32$. The null hypothesis is that the true population slope is zero ($H_0: \beta = 0$). Which of the following describes the sampling distribution of the test statistic $t = \frac{b}{SE_b}$ if the null hypothesis is true and the conditions for inference are met?

(A)A normal distribution with mean 0 and standard deviation 1.(B)A t-distribution with 30 degrees of freedom.(C)A t-distribution with 31 degrees of freedom.(D)A t-distribution with 32 degrees of freedom.

22. A marine biologist is studying the relationship between the length of a certain species of fish (in centimeters) and its weight (in grams). A random sample of 18 fish is selected, and a linear regression analysis is performed on the data. The computer output for the regression is shown below. Predictor | Coef | SE Coef | T-Value | P-Value --- | --- | --- | --- | --- Constant | -120.5 | 24.3 | -4.96 | 0.000 Length | 28.4 | 1.8 | 15.78 | 0.000 Which of the following is the correct 95% confidence interval for the slope of the population regression line, assuming all conditions for inference are met?

(A)28.4 ± 1.746(1.8)(B)28.4 ± 2.120(1.8)(C)28.4 ± 2.120(24.3)(D)-120.5 ± 2.120(24.3)

23. An economist wants to determine if there is a linear relationship between the unemployment rate and the inflation rate in a specific country over the last 30 years. A least-squares regression line is fitted to the data, and a hypothesis test for the slope is conducted. The hypotheses are H_0: β = 0 versus H_a: β ≠ 0. The test yields a t-statistic of t = -1.85 and a p-value of 0.075. Using a significance level of α = 0.05, which of the following is the correct conclusion?

(A)Reject H_0; there is convincing evidence of a linear relationship between unemployment rate and inflation rate.(B)Reject H_0; there is convincing evidence that the true slope is negative.(C)Fail to reject H_0; there is not convincing evidence of a linear relationship between unemployment rate and inflation rate.(D)Fail to reject H_0; this proves there is no linear relationship between unemployment rate and inflation rate.

Refer to the figure below.

24. A student collects data on the height (in inches) and vertical jump (in inches) of 25 athletes. A regression analysis is performed, and the student wishes to calculate a confidence interval for the slope of the regression line. A residual plot of the data is shown below. Based on the residual plot, which condition for inference is most clearly violated?

(A)Linearity: The relationship between the variables is not linear.(B)Independence: The observations are not independent of each other.(C)Normality: The distribution of the residuals is not approximately normal.(D)Equal Standard Deviation: The standard deviation of the residuals is not constant across all values of the explanatory variable.

25. Researchers analyzed the relationship between the amount of fertilizer applied (x) and the yield of corn (y). The computer output provided a standard error of the slope (SE_b) of 0.45. Which of the following best interprets this value?

(A)On average, the corn yield is expected to vary by 0.45 units from the predicted regression line.(B)In repeated random sampling, the estimated slope of the regression line typically varies by about 0.45 from the true population slope.(C)For every 1-unit increase in fertilizer, the corn yield increases by 0.45 units on average.(D)The standard deviation of the residuals for this regression model is 0.45.

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