Ap Stats Unit 7 Progress Check Mcq Part C

Let's dive into a detailed explanation of the AP Statistics Unit 7 Progress Check MCQ Part C. This comprehensive guide will dissect each question, providing clear explanations, solutions, and relevant statistical context. We'll explore the concepts of sampling distributions, confidence intervals, hypothesis testing, and error analysis crucial to mastering Unit 7 and excelling in AP Statistics.

AP Statistics Unit 7 Progress Check MCQ Part C: A Deep Dive

The AP Statistics Unit 7 Progress Check MCQ Part C typically focuses on applying the concepts of sampling distributions, confidence intervals, and hypothesis testing in various scenarios. Let's address the kind of questions you might encounter. Because the actual questions of the College Board Progress Checks are not public, we will craft representative questions to provide a thorough understanding of the underlying concepts.

Question 1: Understanding Sampling Distributions

Question: A large university wants to estimate the proportion of students who support a new campus policy. They take a random sample of 400 students and find that 220 support the policy. Which of the following statements about the sampling distribution of the sample proportion is correct?

(A) The sampling distribution is approximately normal with a mean of 0.55 and a standard deviation that can be calculated using the formula √(p(1-p)/n), where p = 0.55 and n = 400. (B) The sampling distribution is approximately normal with a mean of 0.55, but the standard deviation cannot be determined without knowing the population size. (C) The sampling distribution is approximately normal only if the population size is at least 4000. (D) The sampling distribution is not approximately normal because the sample size is too small. (E) The sampling distribution is approximately normal with a mean of 0.55 and a standard deviation of √(0.55(0.45)/400). The 10% condition is met if the population size is at least 4000.

Solution and Explanation:

• Core Concept: This question tests your understanding of the sampling distribution of a sample proportion. Key ideas include:

  • Shape: Under certain conditions (large enough sample size), the sampling distribution of the sample proportion is approximately normal.
  • Center: The mean of the sampling distribution is equal to the population proportion p (or, in this case, our best estimate of p, which is the sample proportion).
  • Spread: The standard deviation of the sampling distribution is given by the formula √(p(1-p)/ n), where n is the sample size. The standard deviation is also known as the standard error of the sample proportion.
  • 10% Condition: For the standard deviation formula to be valid, the sample size should be no more than 10% of the population size. This ensures independence of observations.

• Step-by-Step Analysis:

  1. Calculate the sample proportion: p̂ = 220/400 = 0.55
  2. Check for approximate normality: The rule of thumb is that n p ≥ 10 and n(1-p) ≥ 10. Here, 400 * 0.55 = 220 and 400 * 0.45 = 180, both of which are greater than 10. So, the sampling distribution is approximately normal.
  3. Calculate the standard deviation: σ*_p̂* = √(0.55(0.45)/400) ≈ 0.0249
  4. 10% Condition: To ensure independence, the population size (number of students at the university) should be at least 10 times the sample size, i.e., at least 4000.

• Correct Answer: (E) is the most accurate. It correctly identifies the shape, center, spread, and the condition required for using the standard deviation formula.

• Why other options are wrong:

  • (A) While it gets the mean and general formula right, it doesn't explicitly mention the 10% condition, which is important.
  • (B) The standard deviation can be determined with the given information.
  • (C) The normality condition depends on n p and n(1-p) being large enough, not just the population size.
  • (D) The sample size is large enough for the sampling distribution to be approximately normal, given p̂ = 0.55.

Question 2: Constructing a Confidence Interval

Question: A researcher wants to estimate the average height of trees in a forest. He randomly selects 50 trees and measures their heights. The sample mean height is 62 feet, and the sample standard deviation is 12 feet. Construct a 95% confidence interval for the population mean height.

(A) 62 ± 1.96 * (12/√50) (B) 62 ± 2.009 * (12/√50) (C) 62 ± 1.96 * (12/50) (D) 62 ± 1.645 * (12/√50) (E) 62 ± 2.576 * (12/√50)

Solution and Explanation:

• Core Concept: This question tests your ability to construct a confidence interval for a population mean when the population standard deviation is unknown. This requires using a t-distribution.

• Formula for Confidence Interval: x̄ ± t* * (s / √n), where:

  • x̄ is the sample mean
  • t* is the critical t-value for the desired confidence level and degrees of freedom
  • s is the sample standard deviation
  • n is the sample size

• Step-by-Step Analysis:

  1. Identify the given values: x̄ = 62, s = 12, n = 50, Confidence Level = 95%
  2. Determine the degrees of freedom: df = n - 1 = 50 - 1 = 49
  3. Find the critical t-value: Using a t-table or calculator, find the t-value for a 95% confidence level with 49 degrees of freedom. This value is approximately 2.009. (Note: If the degrees of freedom isn't directly on the table, use the closest value). Using 1.96 (the z-score) is incorrect as we are using the sample standard deviation to estimate the population standard deviation. Therefore, we must use a t-distribution.
  4. Calculate the margin of error: Margin of Error = 2.009 * (12/√50) ≈ 3.40
  5. Construct the confidence interval: 62 ± 3.40 which is 62 ± 2.009 * (12/√50)

• Correct Answer: (B) is the correct confidence interval.

• Why other options are wrong:

  • (A) Uses the z-score (1.96) instead of the t-value, which is inappropriate when the population standard deviation is unknown.
  • (C) Incorrectly divides the sample standard deviation by n instead of √n.
  • (D) Uses an incorrect z-score (1.645, corresponding to a 90% confidence level) instead of the correct t-value.
  • (E) Uses an incorrect z-score (2.576) corresponding to a 99% confidence level.

Question 3: Hypothesis Testing - P-value Interpretation

Question: A researcher conducts a hypothesis test to determine if the average commute time for workers in a city is greater than 30 minutes. The null hypothesis is H₀: μ = 30, and the alternative hypothesis is H_a: μ > 30. The calculated p-value is 0.035. Which of the following is the correct interpretation of the p-value?

(A) There is a 3.5% chance that the null hypothesis is true. (B) There is a 3.5% chance that the alternative hypothesis is true. (C) Assuming the average commute time is 30 minutes, there is a 3.5% chance of observing a sample mean as large as, or larger than, the one obtained. (D) There is a 3.5% chance that the researcher will make a Type I error. (E) There is a 3.5% chance that the researcher will make a Type II error.

Solution and Explanation:

• Core Concept: This question focuses on the correct interpretation of a p-value in the context of hypothesis testing.

• Definition of P-value: The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming the null hypothesis is true.

• Step-by-Step Analysis:

  1. Understanding the Null Hypothesis: H₀: μ = 30 (The average commute time is 30 minutes).
  2. Understanding the Alternative Hypothesis: H_a: μ > 30 (The average commute time is greater than 30 minutes).
  3. P-value of 0.035: This means that if the average commute time is actually 30 minutes, there's a 3.5% chance of getting a sample mean commute time as high or higher than the one the researcher observed.

• Correct Answer: (C) is the correct interpretation.

• Why other options are wrong:

  • (A) The p-value is not the probability that the null hypothesis is true. Hypothesis tests don't prove or disprove hypotheses; they only provide evidence for or against them.
  • (B) The p-value is not the probability that the alternative hypothesis is true.
  • (D) The p-value is related to the risk of making a Type I error, but it's not directly the probability of making one. The significance level (alpha) is the probability of making a Type I error if the null hypothesis is true.
  • (E) The p-value has no direct relationship to the probability of making a Type II error (failing to reject a false null hypothesis).

Question 4: Type I and Type II Errors

Question: A pharmaceutical company is testing a new drug. The null hypothesis is that the drug has no effect (H₀: effect = 0), and the alternative hypothesis is that the drug has a positive effect (H_a: effect > 0). A Type I error would occur if:

(A) The company concludes the drug has no effect when it actually does have a positive effect. (B) The company concludes the drug has a positive effect when it actually has no effect. (C) The company correctly concludes the drug has no effect. (D) The company correctly concludes the drug has a positive effect. (E) The company fails to conduct the hypothesis test.

Solution and Explanation:

• Core Concept: This question tests your understanding of Type I and Type II errors in hypothesis testing.

• Definitions:

  • Type I Error (False Positive): Rejecting the null hypothesis when it is actually true.
  • Type II Error (False Negative): Failing to reject the null hypothesis when it is actually false.

• Step-by-Step Analysis:

  1. Null Hypothesis: H₀: effect = 0 (Drug has no effect)
  2. Alternative Hypothesis: H_a: effect > 0 (Drug has a positive effect)
  3. Type I Error: Concluding the drug has a positive effect (rejecting H₀) when, in reality, the drug has no effect (H₀ is true).

• Correct Answer: (B) describes a Type I error.

• Why other options are wrong:

  • (A) Describes a Type II error (failing to reject a false null hypothesis).
  • (C) and (D) Describe correct decisions.
  • (E) Failing to conduct the test isn't an error within the context of the hypothesis test itself.

Question 5: Impact of Sample Size on Confidence Intervals

Question: A researcher wants to estimate the proportion of voters who support a particular candidate. She constructs a 95% confidence interval based on a random sample of voters. How will the width of the confidence interval change if the sample size is increased, assuming the sample proportion remains the same?

(A) The width of the confidence interval will increase. (B) The width of the confidence interval will decrease. (C) The width of the confidence interval will remain the same. (D) The width of the confidence interval will double. (E) The effect on the width cannot be determined without knowing the specific sample proportion.

Solution and Explanation:

• Core Concept: This question tests your understanding of the relationship between sample size and the width of a confidence interval.

• Relationship: As the sample size increases, the standard error (and thus the margin of error) decreases. A smaller margin of error results in a narrower confidence interval.

• Formula Reminder: The margin of error for a proportion is related to √(p(1-p)/n). Therefore, as n increases, the margin of error decreases.

• Correct Answer: (B) is correct. Increasing the sample size will decrease the width of the confidence interval.

• Why other options are wrong:

  • (A) Incorrect; increasing sample size always decreases the width.
  • (C) Incorrect; the width will change.
  • (D) Incorrect; the width doesn't simply double.
  • (E) Incorrect; the effect can be determined.

Question 6: Conditions for Inference about a Mean

Question: A researcher wants to perform a t-test to determine if the mean blood pressure of a population is different from 120 mmHg. He collects a sample of 30 individuals. Which of the following conditions must be met to ensure the validity of the t-test?

(A) The population standard deviation must be known. (B) The population must be normally distributed. (C) The sample size must be greater than 30. (D) Either the population is normally distributed, or the sample size is large enough (n ≥ 30) to invoke the Central Limit Theorem. (E) The data must be collected using a stratified random sample.

Solution and Explanation:

• Core Concept: This question tests your understanding of the conditions required for performing inference about a population mean using a t-test.

• Conditions for t-test:

  • Randomness: The data must be collected from a random sample or a randomized experiment.
  • Independence: Observations must be independent. If sampling without replacement, the sample size should be no more than 10% of the population size.
  • Normality: Either the population is normally distributed, or the sample size is large enough (typically n ≥ 30) to invoke the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normal.

• Correct Answer: (D) accurately describes the normality condition.

• Why other options are wrong:

  • (A) The population standard deviation is not required for a t-test. A t-test is used when the population standard deviation is unknown and estimated by the sample standard deviation.
  • (B) While a normally distributed population satisfies the normality condition, it's not strictly necessary if the sample size is large enough.
  • (C) While a sample size of 30 is a common guideline for the Central Limit Theorem, it's not a strict requirement; the population could be approximately normal even with a smaller sample size.
  • (E) Stratified random sampling is a method for collecting data, but it's not a requirement for a t-test itself. A simple random sample is sufficient.

Question 7: Choosing the Correct Hypothesis Test

Question: A researcher wants to determine if there is a relationship between two categorical variables: gender (male/female) and favorite color (red/blue/green). Which of the following hypothesis tests is most appropriate?

(A) One-sample t-test (B) Two-sample t-test (C) Paired t-test (D) Chi-square test for independence (E) Linear regression t-test

Solution and Explanation:

• Core Concept: This question tests your ability to choose the correct hypothesis test based on the type of data and the research question.

• Different Hypothesis Tests:

  • t-tests: Used to compare means of one or two groups.
  • Chi-square test for independence: Used to determine if there is a relationship between two categorical variables.
  • Linear regression t-test: Used to test the significance of the slope in a linear regression model (relationship between two numerical variables).

• Step-by-Step Analysis:

  1. Identify the data types: Both gender and favorite color are categorical variables.
  2. Identify the research question: The researcher wants to know if there's a relationship between these two categorical variables.

• Correct Answer: (D) The chi-square test for independence is the appropriate test for determining if there is a relationship between two categorical variables.

• Why other options are wrong:

  • (A), (B), (C) t-tests are used for comparing means, not for analyzing categorical relationships.
  • (E) Linear regression is for numerical variables.

Question 8: Power of a Test

Question: A hypothesis test is conducted with a significance level of α = 0.05. If the true population parameter is such that the null hypothesis is false, which of the following would increase the power of the test?

(A) Decreasing the sample size. (B) Increasing the significance level (α). (C) Decreasing the probability of a Type I error. (D) Making a two-tailed test instead of a one-tailed test. (E) None of the above.

Solution and Explanation:

• Core Concept: This question assesses understanding of the power of a hypothesis test and the factors that influence it.

• Definition of Power: The power of a test is the probability of correctly rejecting the null hypothesis when it is false. It's the probability of avoiding a Type II error.

• Factors Affecting Power:

  • Sample Size: Increasing the sample size increases power.
  • Significance Level (α): Increasing α increases power (but also increases the risk of a Type I error).
  • Effect Size: A larger effect size (the difference between the true population parameter and the value stated in the null hypothesis) increases power.
  • Variability: Decreasing the variability (standard deviation) in the population increases power.
  • One-tailed vs. Two-tailed: For a given α, a one-tailed test has more power than a two-tailed test if the true parameter is in the direction specified by the alternative hypothesis.

• Correct Answer: (B) Increasing the significance level (α) increases the power of the test.

• Why other options are wrong:

  • (A) Decreasing the sample size decreases power.
  • (C) Decreasing the probability of a Type I error (which is done by decreasing α) decreases power.
  • (D) Making a two-tailed test instead of a one-tailed test (when the true parameter is in the direction specified by the one-tailed test) decreases power.

Question 9: Impact of Non-response Bias

Question: A survey is conducted to estimate the proportion of adults who support a new public policy. A large random sample is selected, but a significant number of people refuse to participate in the survey. What type of bias is most likely to affect the results?

(A) Sampling bias (B) Non-response bias (C) Response bias (D) Undercoverage bias (E) Voluntary response bias

Solution and Explanation:

• Core Concept: This question tests your knowledge of different types of bias in sampling and surveys.

• Types of Bias:

  • Sampling Bias: Occurs when the sample is not representative of the population.
  • Non-response Bias: Occurs when a significant portion of the selected sample does not respond to the survey. This is problematic if non-respondents differ systematically from respondents on the characteristic being measured.
  • Response Bias: Occurs when respondents provide inaccurate or untruthful answers.
  • Undercoverage Bias: Occurs when some members of the population are less likely to be included in the sample.
  • Voluntary Response Bias: Occurs when individuals self-select to participate in a survey, leading to an unrepresentative sample.

• Correct Answer: (B) Non-response bias is the most likely bias when a significant number of people refuse to participate.

• Why other options are wrong:

  • (A) Sampling bias might be present if the initial sample selection was flawed, but the key issue here is the refusal to participate.
  • (C) Response bias is possible if people give inaccurate answers, but the primary problem is the lack of answers.
  • (D) Undercoverage bias could be present depending on the sampling method, but the non-response issue is more direct.
  • (E) Voluntary response bias applies when people actively choose to participate, which isn't the case here (people were selected for the sample).

Question 10: Choosing the Right Statistic

Question: A researcher wants to compare the effectiveness of two different teaching methods. He randomly assigns students to two groups: one group receives Method A, and the other receives Method B. After a semester, both groups take the same standardized test. Which statistic is most appropriate for comparing the average test scores of the two groups?

(A) A single sample mean. (B) The difference in sample proportions. (C) A single sample proportion. (D) The difference in sample means. (E) A correlation coefficient.

Solution and Explanation:

• Core Concept: This question tests your understanding of choosing the appropriate statistic to analyze data based on the study design and type of variable being measured.

• Step-by-Step Analysis:

  1. Identify the data type: The standardized test scores are numerical, continuous data.
  2. Identify the goal: The researcher wants to compare the average test scores of two groups.

• Correct Answer: (D) The difference in sample means is the appropriate statistic to compare the average test scores of two independent groups.

• Why other options are wrong:

  • (A) A single sample mean would only describe one group, not compare two.
  • (B) The difference in sample proportions is for comparing categorical data, not numerical scores.
  • (C) A single sample proportion is for describing a proportion within one group, not comparing averages.
  • (E) A correlation coefficient measures the linear relationship between two variables, not the difference in means between two groups.

By understanding the underlying statistical concepts and practicing with various examples, you can confidently tackle the AP Statistics Unit 7 Progress Check MCQ Part C. Remember to focus on the definitions of key terms, the conditions for inference, and the appropriate test or statistic for different situations. Good luck!