AP Statistics Unit 5: Sampling Distributions Practice Exam

Assessment for Unit 5: Sampling Distributions

Estimated Time: 60 minutes

Exam Details

Questions: 31 total (31 multiple choice)
Time: 60 minutes (recommended)
Topics Covered: Unit 5: Sampling Distributions
Format: Multiple Choice questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (31 Questions)

Select the one best answer for each question.

1. A marketing consultant for a large chain of coffee shops wants to estimate the proportion of all customers who purchase a pastry with their coffee. The consultant observes a random sample of 150 customers at various locations and finds that 48 of them purchased a pastry. Which of the following correctly identifies the population parameter and the sample statistic?

(A)The parameter is the 150 customers observed, and the statistic is the 48 customers who bought a pastry.(B)The parameter is the proportion of all customers who purchase a pastry, and the statistic is 0.32.(C)The parameter is 0.32, and the statistic is the proportion of all customers who purchase a pastry.(D)The parameter is the total number of customers who visit the chain, and the statistic is the 150 customers observed.

2. A high school principal wants to determine the mean time students spend on homework per night. She selects a random sample of 50 students and calculates the mean time to be 2.5 hours. A second administrator selects a different random sample of 50 students from the same high school and calculates the mean time to be 2.8 hours. Which of the following best explains why the two means are different?

(A)The sample sizes are too small to produce consistent results.(B)At least one of the administrators must have made a calculation error or selected a biased sample.(C)The difference is due to sampling variability, which is the natural variation in statistics from sample to sample.(D)The population parameter has changed between the time the first sample and the second sample were taken.

3. A computer simulation is designed to mimic the process of taking random samples of size n = 30 from a large population where the true proportion of successes is p = 0.60. The simulation generates 500 such samples and calculates the sample proportion for each. A dotplot is created to display these 500 values. Which of the following best describes what a single dot on this plot represents?

(A)One individual from the population who is considered a 'success'.(B)The true population proportion, p = 0.60.(C)The sample proportion, p-hat, obtained from one sample of size 30.(D)The average of the 500 sample proportions.

4. Researchers are investigating the weight of a specific species of turtle. The mean weight of all turtles of this species is known to be 34.5 pounds. A researcher collects a random sample of 20 turtles and finds the mean weight of the sample to be 31.2 pounds. Which of the following symbols correctly correspond to the values 34.5 and 31.2, respectively?

(A) $\mu $= 34.5 and $\bar{x} $= 31.2 (B) $\bar{x} $= 34.5 and $\mu $= 31.2 (C) p = 34.5 and $\hat{p} $= 31.2 (D) $\mu $= 31.2 and $\bar{x} $= 34.5

5. A factory produces bolts with a target diameter of 10 mm. A quality control engineer takes a random sample of 10 bolts each hour to measure their diameter. Over the course of a week, the engineer plots the distribution of the sample means collected each hour. Which of the following best describes the values plotted on this graph?

(A)The diameters of individual bolts produced by the factory.(B)The parameters of the population of all bolts produced.(C)The statistics calculated from multiple samples, showing how the sample mean varies from sample to sample.(D)The ideal target diameter of 10 mm repeated for every sample taken.

6. [Skill: 3.A | Topic: 5.2] The lifespan of a certain brand of car battery is normally distributed with a mean of 48 months and a standard deviation of 4 months. What is the probability that a randomly selected battery will last between 42 and 52 months?

(A)0.1587(B)0.7745(C)0.8413(D)0.9332

7. [Skill: 3.A | Topic: 5.2] A standardized test has scores that are normally distributed with a mean of 500 and a standard deviation of 100. A university admits students who score in the top 10% of the distribution. Which of the following is closest to the minimum score required for admission?

(A)600(B)628(C)664(D)696

8. [Skill: 3.A | Topic: 5.2] The weights of apples in a harvest are normally distributed with a mean of 150 grams and a standard deviation of 15 grams. Which of the following correctly compares the probability of selecting an apple weighing more than 165 grams, $P(X > 165)$, to the probability of selecting an apple weighing at least 165 grams, $P(X \ge 165)$?

(A)P(X > 165) < P(X ≥ 165) because the second probability includes the weight of exactly 165 grams.(B)P(X > 165) = P(X ≥ 165) because the distribution is continuous and the area at a single point is zero.(C)P(X > 165) > P(X ≥ 165) because the strict inequality excludes the lower bound.(D)The probabilities cannot be compared without knowing the sample size.

9. [Skill: 3.A | Topic: 5.2] Two different manufacturing machines produce bolts. The diameter of bolts from Machine A follows a normal distribution $N(10, 0.2)$ mm. The diameter of bolts from Machine B follows a normal distribution $N(20, 0.5)$ mm. A bolt is selected from each machine. Bolt A has a diameter of 10.5 mm and Bolt B has a diameter of 21.0 mm. Which bolt is relatively larger compared to its own distribution?

(A)Bolt A, because it is 0.5 mm above its mean while Bolt B is 1.0 mm above its mean.(B)Bolt A, because its z-score is 2.5, which is greater than Bolt B's z-score of 2.0.(C)Bolt B, because its z-score is 2.0, which is greater than Bolt A's z-score of 1.0.(D)Bolt B, because a deviation of 1.0 mm is greater than a deviation of 0.5 mm.

10. [Skill: 4.B | Topic: 5.2] A biologist collects data on the length of a rare species of beetle. The data indicates that the distribution of lengths is strongly skewed to the right with a mean of 25 mm and a standard deviation of 3 mm. The biologist wishes to calculate the probability that a randomly selected beetle is longer than 30 mm. Why is it inappropriate to use the normal distribution function (e.g., normalcdf) for this calculation?

(A)The sample size is not given, so the Central Limit Theorem cannot be applied.(B)The normal distribution is only appropriate for symmetric, bell-shaped distributions, not skewed ones.(C)The standard deviation is too small relative to the mean.(D)The variable is discrete, not continuous, so a normal approximation is invalid.

11. A statistician is studying the distribution of the weights of a specific species of turtle. The population of turtle weights is known to be strongly skewed to the right. To estimate the sampling distribution of the sample mean weight, the statistician decides to use a simulation. Which of the following best describes the correct procedure to simulate the sampling distribution of the sample mean for samples of size n = 40?

(A)Take one random sample of 40 turtles from the population, calculate the mean weight, and repeat this process 1,000 times. Plot the distribution of the 1,000 sample means.(B)Take one random sample of 40 turtles from the population and plot the distribution of the individual weights of these 40 turtles.(C)Take one random sample of 1,000 turtles from the population, calculate the mean weight, and use this single value to estimate the center of the distribution.(D)Take 1,000 random samples of size n = 5 from the population, calculate the mean for each, and plot the distribution of these means.

12. The distribution of the amount of time customers spend in a grocery store is bimodal and strongly skewed to the right with a mean of 25 minutes and a standard deviation of 12 minutes. If random samples of size n = 50 are repeatedly taken from this population, which of the following best describes the shape, center, and spread of the sampling distribution of the sample mean?

(A)Shape: Bimodal and skewed right; Mean: 25; Standard Deviation: 12(B)Shape: Approximately normal; Mean: 25; Standard Deviation: 12(C)Shape: Approximately normal; Mean: 25; Standard Deviation: 1.697(D)Shape: Bimodal and skewed right; Mean: 25; Standard Deviation: 1.697

13. Which of the following statements correctly describes the Central Limit Theorem (CLT)?

(A)The CLT states that as the sample size increases, the distribution of the sample data will approach a normal distribution.(B)The CLT states that if the population distribution is normal, the sampling distribution of the sample mean will be normal for any sample size.(C)The CLT states that for a sufficiently large sample size, the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population distribution.(D)The CLT states that the mean of the sampling distribution is always equal to the mean of the population, regardless of sample size.

14. Consider the following three scenarios involving the sampling distribution of the sample mean: I. The population is skewed right. The sample size is n = 10. II. The population is approximately normal. The sample size is n = 10. III. The population is skewed left. The sample size is n = 40. In which of the scenarios can the sampling distribution of the sample mean be safely modeled as approximately normal?

(A)I only(B)II only(C)II and III only(D)I, II, and III

15. [Skill: 4.A | Topic: 5.4] Which of the following statements best explains why the sample mean, $\bar{x}$, is considered an unbiased estimator of the population mean, $\mu$?

(A)The sample mean $\bar{x}$ will always be equal to the population mean $\mu$ for any given sample.(B)The mean of the sampling distribution of the sample mean is equal to the population mean $\mu$.(C)As the sample size increases, the sample mean $\bar{x}$ gets closer to the population mean $\mu$.(D)The standard deviation of the sampling distribution of $\bar{x}$ decreases as the sample size increases.

16. [Skill: 3.B | Topic: 5.4] A store manager wants to estimate the maximum number of customers in the store at any one time (the population maximum). The true population maximum is known to be 50 customers. To test different estimators, the manager takes 1,000 random samples of size $n=10$ from the records and calculates three statistics for each sample: the Sample Mean multiplied by 2, the Sample Maximum, and the Sample Median multiplied by 2. The means of these sampling distributions are shown in the table below. | Statistic | Mean of the 1,000 Estimates | | :--- | :--- | | 2 $\times$ Sample Mean | 49.8 | | Sample Maximum | 45.2 | | 2 $\times$ Sample Median | 54.1 | Based on the simulation results, which statistic appears to be an unbiased estimator for the population maximum?

(A)2 $\times$ Sample Mean only(B)Sample Maximum only(C)2 $\times$ Sample Median only(D)Both the Sample Maximum and 2 $\times$ Sample Median

17. [Skill: 3.C | Topic: 5.4] A researcher is estimating the population proportion $p$ using the sample proportion $\hat{p}$. Initially, the researcher uses a sample size of $n = 50$. If the researcher increases the sample size to $n = 200$, how does this affect the bias and the variability of the estimator $\hat{p}$?

(A)The bias decreases and the variability decreases.(B)The bias remains the same and the variability decreases.(C)The bias decreases and the variability remains the same.(D)The bias remains the same and the variability increases.

18. A national survey found that 64% of households in a country subscribe to a streaming service. A random sample of 400 households is selected from this population. Let $\hat{p}$ represent the proportion of households in the sample that subscribe to a streaming service. What are the mean and standard deviation of the sampling distribution of $\hat{p}$?

(A)Mean = 0.64; Standard Deviation = 0.024(B)Mean = 0.64; Standard Deviation = 0.048(C)Mean = 256; Standard Deviation = 9.6(D)Mean = 256; Standard Deviation = 0.024

19. A manufacturing process produces electronic components where the probability of a defect is $p = 0.08$. A quality control manager wants to estimate the proportion of defective components by taking a random sample of size $n$. Which of the following is the smallest sample size $n$ that will satisfy the Large Counts condition, allowing the sampling distribution of the sample proportion to be approximated by a normal distribution?

(A)n = 10(B)n = 30(C)n = 100(D)n = 125

20. A botanist estimates that 75% of the seeds of a specific plant species will germinate. The botanist plants 200 randomly selected seeds. Which of the following represents the probability that the sample proportion of germinated seeds, $\hat{p}$, is less than 0.70?

(A)$P\left(Z < \frac{0.70 - 0.75}{\sqrt{\frac{0.70(0.30)}{200}}}\right)$(B)$P\left(Z < \frac{0.70 - 0.75}{\sqrt{\frac{0.75(0.25)}{200}}}\right)$(C)$P\left(Z < \frac{0.75 - 0.70}{\sqrt{\frac{0.75(0.25)}{200}}}\right)$(D)$P\left(Z < \frac{0.70 - 0.75}{\frac{0.75(0.25)}{\sqrt{200}}}\right)$

21. Suppose that 40% of all adults in a city favor a new zoning law. Two separate research groups plan to take random samples from this population to estimate the proportion of support. Group A will take a random sample of size $n = 100$, and Group B will take a random sample of size $n = 400$. Which of the following best describes the difference between the sampling distributions of the sample proportion $\hat{p}$ for the two groups?

(A)The sampling distribution for Group B will have a mean that is 4 times larger than the mean for Group A.(B)The sampling distribution for Group B will have a standard deviation that is half the standard deviation of Group A.(C)The sampling distribution for Group B will have a standard deviation that is one-fourth the standard deviation of Group A.(D)The sampling distributions for both groups will have the same shape, center, and spread.

22. Suppose that 50% of all adults in City A prefer coffee over tea, and 40% of all adults in City B prefer coffee over tea. Independent random samples of 100 adults from City A and 100 adults from City B are selected. Let $\hat{p}_A$ and $\hat{p}_B$ represent the sample proportions of adults who prefer coffee in City A and City B, respectively. Which of the following are the mean and standard deviation of the sampling distribution of the difference in sample proportions $\hat{p}_A - \hat{p}_B$?

(A)Mean = 0.10; Standard Deviation = 0.01(B)Mean = 0.10; Standard Deviation = 0.07(C)Mean = 0.10; Standard Deviation = 0.099(D)Mean = 0.90; Standard Deviation = 0.07

23. A manufacturing plant has two machines, A and B. Historically, Machine A produces defects at a rate of $p_A = 0.05$, and Machine B produces defects at a rate of $p_B = 0.08$. A quality control manager takes independent random samples of 200 items from Machine A and 200 items from Machine B. Which of the following expressions represents the probability that the sample proportion of defects from Machine A will be greater than the sample proportion of defects from Machine B?

(A)$P\left(Z > \frac{0 - (0.05 - 0.08)}{\sqrt{\frac{0.05(0.95)}{200} + \frac{0.08(0.92)}{200}}}\right)$(B)$P\left(Z > \frac{(0.05 - 0.08) - 0}{\sqrt{\frac{0.05(0.95)}{200} + \frac{0.08(0.92)}{200}}}\right)$(C)$P\left(Z > \frac{0 - (0.05 - 0.08)}{\sqrt{\frac{0.065(0.935)}{400}}}\right)$(D)$P\left(Z < \frac{0 - (0.05 - 0.08)}{\sqrt{\frac{0.05(0.95)}{200} - \frac{0.08(0.92)}{200}}}\right)$

24. Which of the following best describes the interpretation of the standard deviation of the sampling distribution of the difference in sample proportions, $\sigma_{\hat{p}_1 - \hat{p}_2}$?

(A)It is the difference between the standard deviations of the two populations.(B)It is the probability that the difference in sample proportions equals the difference in population proportions.(C)It represents the typical distance that the difference in sample proportions varies from the true difference in population proportions in repeated independent random samples.(D)It represents the typical distance that the true difference in population proportions varies from the difference in sample proportions in repeated independent random samples.

25. [Skill: 3C | Topic: 5.7] A population of home prices in a large city is strongly skewed to the right with a mean of $350,000 and a standard deviation of $50,000. A real estate researcher selects a random sample of 5 homes and a random sample of 50 homes from this city and calculates the mean price for each sample. Which of the following best describes the shape of the sampling distribution of the sample mean for these two sample sizes?

(A)For n = 5, the shape is approximately normal; for n = 50, the shape is approximately normal.(B)For n = 5, the shape is skewed to the right; for n = 50, the shape is skewed to the right.(C)For n = 5, the shape is skewed to the right; for n = 50, the shape is approximately normal.(D)For n = 5, the shape is approximately normal; for n = 50, the shape is skewed to the right.

26. [Skill: 3B | Topic: 5.7] The distribution of the weights of a certain variety of orange is approximately normal with a mean of 145 grams and a standard deviation of 12 grams. A crate contains 16 randomly selected oranges. Assuming the oranges in the crate can be treated as a random sample from the population, what are the mean and standard deviation of the sampling distribution of the sample mean weight for the oranges in the crate?

(A)Mean = 145 grams; Standard Deviation = 0.75 grams(B)Mean = 145 grams; Standard Deviation = 3.0 grams(C)Mean = 145 grams; Standard Deviation = 12.0 grams(D)Mean = 9.06 grams; Standard Deviation = 3.0 grams

27. [Skill: 3C | Topic: 5.7] A factory produces ceramic tiles with a mean thickness of 8 mm and a standard deviation of 0.4 mm. The distribution of tile thickness is not normally distributed. A quality control manager takes a random sample of 64 tiles to verify the production process. What is the probability that the mean thickness of the sample is greater than 8.1 mm?

(A)P(z > 0.25)(B)P(z > 2.0)(C)P(z < 2.0)(D)P(z > 16.0)

28. [Skill: 4B | Topic: 5.7] An environmental scientist is measuring the concentration of a pollutant in a river. The standard deviation of the measurements from the population is known to be 5.0 ppm. The scientist wants to estimate the mean concentration with a sampling distribution standard deviation (standard error) of exactly 1.0 ppm. Which of the following describes the required sample size?

(A)The sample size must be 5.(B)The sample size must be 10.(C)The sample size must be 25.(D)The sample size must be 50.

29. [Skill: 3.B | Topic: 5.8] A botanist is studying the growth of two different species of plants, Species A and Species B. Based on previous research, the population mean height for Species A is 15.0 cm with a standard deviation of 4.0 cm, and the population mean height for Species B is 12.0 cm with a standard deviation of 3.0 cm. The botanist selects independent random samples of 64 plants of Species A and 36 plants of Species B. What are the mean and standard deviation of the sampling distribution of the difference in sample means (Species A minus Species B)?

(A)Mean = 3.0 cm; Standard Deviation = 0.707 cm(B)Mean = 3.0 cm; Standard Deviation = 0.500 cm(C)Mean = 3.0 cm; Standard Deviation = 1.000 cm(D)Mean = 3.0 cm; Standard Deviation = 5.000 cm

30. [Skill: 3.C | Topic: 5.8] A school district is comparing the reading scores of students in two different programs, Program I and Program II. The distribution of scores for all students in Program I is strongly skewed to the left, while the distribution of scores for all students in Program II is approximately normal. Independent random samples of size $n_1$ from Program I and size $n_2$ from Program II are taken. Which of the following conditions is sufficient to guarantee that the sampling distribution of the difference in sample means ($\bar{x}_1 - \bar{x}_2$) is approximately normal?

(A)$n_1 \ge 30$ and $n_2 \ge 30$(B)$n_1 \ge 30$, regardless of the size of $n_2$(C)$n_1 + n_2 \ge 30$(D)$n_2 \ge 30$, regardless of the size of $n_1$

31. [Skill: 4.B | Topic: 5.8] A researcher calculates the parameters for the sampling distribution of the difference between two sample means, $\bar{x}_1 - \bar{x}_2$. The calculation yields a standard deviation of 1.5 units. Which of the following is the best interpretation of this value?

(A)The difference between the means of the two populations is 1.5 units.(B)The difference between the means of any two specific samples selected from these populations will be 1.5 units.(C)In repeated random sampling, the difference in sample means ($\bar{x}_1 - \bar{x}_2$) typically varies from the true difference in population means ($\mu_1 - \mu_2$) by about 1.5 units.(D)The sum of the standard deviations of the two populations is 1.5 units.

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