AP Statistics Quiz: Selecting Implementing And Communicating Inference Procedures

What this quiz covers

This quiz focuses on Selecting Implementing And Communicating Inference Procedures, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. One-sample $t$ test for a mean (correct answer)
2. One-proportion $z$ test
3. Two-sample $t$ test for a difference in means
4. Chi-square goodness-of-fit test
5. Two-proportion $z$ interval

Explanation: This question tests selecting procedures for testing a population mean in AP Statistics. The one-sample t test (choice A) is ideal for quantitative nitrate concentrations from one sample, testing if the mean exceeds 10 mg/L. Incorrect options include choice B (one-proportion z test) for categorical data, and choice C (two-sample t test) needing two groups. Choice D (chi-square goodness-of-fit) assesses categorical distributions, not means. Mini-lesson: Match procedures by checking samples (one here), variable (quantitative for t), and inference (hypothesis test). This systematic alignment avoids mismatches with multi-group or categorical methods.

Question 2

1. Two-proportion $z$ test for a difference in proportions
2. One-sample $t$ interval for a mean
3. One-proportion $z$ interval for a proportion (correct answer)
4. Chi-square test of independence
5. Matched-pairs $t$ test for a mean difference

Explanation: This problem asks for estimating a single population proportion (the proportion of all households that support the recycling fee) based on one random sample of 500 households. Since we need a confidence interval for a single proportion from categorical data (support/do not support), the one-proportion z-interval is the correct procedure. The two-proportion z-test (option A) would require two groups to compare, which we don't have. The t-procedures (options B and E) are for quantitative data, not proportions. The chi-square test (option D) is for testing relationships between variables, not estimating a single proportion. When estimating a population proportion from one sample, always use the one-proportion z-interval.

Question 3

1. Two-proportion $z$ test for a difference in proportions (correct answer)
2. Chi-square goodness-of-fit test
3. Two-sample $t$ test for a difference in means
4. One-proportion $z$ interval for a proportion
5. Matched-pairs $t$ interval for a mean difference

Explanation: This experiment compares side-effect rates (proportions) between two independent groups in a randomized experiment. Since we're testing whether the proportion experiencing side effects differs between Medication A and Medication B, the two-proportion z-test is the appropriate procedure. The chi-square goodness-of-fit test (option B) is for testing a single categorical variable against expected frequencies, not comparing two groups. The t-tests (options C and E) are for quantitative data, not proportions. The one-proportion z-interval (option D) would only analyze one group, not compare two. When comparing proportions between two independent groups, the two-proportion z-test directly addresses whether the proportions differ significantly.

Question 4

1. Matched-pairs $t$ test for a mean difference (correct answer)
2. Two-sample $t$ interval for a difference in means
3. One-sample $z$ test for a proportion
4. Chi-square goodness-of-fit test
5. One-sample $t$ interval for a mean

Explanation: This scenario involves measuring the same clients twice (before and after the stretching routine), creating paired data where each client serves as their own control. The matched-pairs t-test for a mean difference is the appropriate procedure because we're analyzing the mean of the differences between paired measurements (after minus before for each client). The two-sample t-interval (option B) would be incorrect because it assumes independent groups, not paired measurements. The one-sample procedures (options C and E) don't account for the paired nature of the data. When the same subjects are measured under two conditions, the matched-pairs design controls for individual variation and requires a matched-pairs t-test to analyze the mean difference.

Question 5

1. Chi-square goodness-of-fit test for one categorical variable (correct answer)
2. Chi-square test of independence for two categorical variables
3. Two-proportion $z$ test for a difference in proportions
4. One-sample $t$ test for a population mean
5. Matched-pairs $t$ test for a mean difference

Explanation: This scenario tests whether observed frequencies match expected frequencies for a single categorical variable with four categories. The chi-square goodness-of-fit test compares observed counts to expected counts under a specified distribution (here, equal representation would mean 125 voters per party). This test determines if the sample data provides evidence against the null hypothesis of equal distribution. Choice B tests independence between two variables, but we only have one variable (party). Choices C and E involve comparing groups or paired data. Choice D tests means, but we have categorical data. The goodness-of-fit test is specifically designed for testing distributional claims about one categorical variable.

Question 6

1. Chi-square test of independence for a two-way table (correct answer)
2. Chi-square goodness-of-fit test for one categorical variable
3. Two-proportion $z$ interval for a difference in proportions
4. Two-sample $t$ test for a difference in means
5. One-proportion $z$ test for a single proportion

Explanation: This problem involves testing for association between two categorical variables: sleep category (3 levels) and stress category (2 levels). The chi-square test of independence examines whether the distribution of one variable depends on the value of the other variable. With 300 students classified into a 3×2 contingency table, this test determines if sleep and stress are independent or associated. Choice B tests a single variable's distribution, not association between two. Choice C estimates a difference in proportions, but we have more than two categories for sleep. Choices D and E involve quantitative data or single proportions. The independence test is the standard procedure for examining relationships in two-way tables.

Question 7

1. One-proportion $z$ test for a population proportion (correct answer)
2. Two-proportion $z$ test for a difference in proportions
3. One-sample $t$ test for a population mean
4. Chi-square test of independence for a two-way table
5. One-proportion $z$ interval for a population proportion

Explanation: This problem tests a claim about a single population proportion. With 200 customers providing binary responses (prefer new design: yes/no) and a claimed proportion of 0.60, a one-proportion z-test is appropriate. This test determines whether the sample provides evidence that the true proportion differs from 0.60. Choice B compares two proportions, but we have one. Choice C tests means, not proportions. Choice D tests independence between two variables, but we have one. Choice E constructs an interval rather than testing a specific claim. When testing a hypothesis about a single proportion, use the one-proportion z-test.

Question 8

1. Chi-square test of independence for a two-way table
2. One-proportion $z$ test for a proportion
3. Chi-square goodness-of-fit test (correct answer)
4. Two-sample $t$ interval for a difference in means
5. Matched-pairs $t$ interval for a mean difference

Explanation: This scenario tests whether the observed distribution of nests across four categories (North, South, East, West) differs from a specific expected distribution (equal use, meaning 25% in each area). The chi-square goodness-of-fit test is designed for exactly this purpose: comparing observed frequencies in one categorical variable to expected frequencies based on a hypothesized distribution. The test of independence (option A) requires two categorical variables, but we only have one (nesting area). The proportion tests (options B and E) are for binary outcomes, not multiple categories. The t-procedures (options D and E) are for quantitative data. When testing whether a single categorical variable follows a specific distribution, use the chi-square goodness-of-fit test.

Question 9

1. Two-sample $t$ test for a difference in means
2. Chi-square test of independence
3. One-proportion $z$ test
4. Test for the slope in simple linear regression (using a $t$ test for $\beta_1$) (correct answer)
5. Matched-pairs $t$ test for a mean difference

Explanation: This problem involves testing for a linear relationship between two quantitative variables (years of education and annual income), specifically whether the slope is positive. The t-test for the slope in simple linear regression (testing whether β₁ > 0) is the appropriate procedure for determining if there's a significant positive linear relationship. The two-sample t-test (option A) compares means between two groups, not relationships between variables. The chi-square test (option B) is for categorical variables. The one-proportion z-test (option C) is for proportions, not relationships. The matched-pairs test (option E) is for paired differences, not regression. When testing whether a linear relationship exists between two quantitative variables, use the t-test for the regression slope.

Question 10

1. Chi-square test of independence (association) for a two-way table (correct answer)
2. Chi-square goodness-of-fit test for one categorical variable
3. Two-proportion $z$ test for a difference in proportions
4. Two-sample $t$ test for a difference in means
5. One-proportion $z$ interval for $p$

Explanation: This question assesses selecting inference procedures for associations between two categorical variables in AP Statistics. The chi-square test of independence (choice A) is ideal to test if party affiliation and news source are associated, using counts in a two-way table from one sample. Distractors such as choice B (chi-square goodness-of-fit) apply to one variable against an expected distribution, not two variables, while choice C (two-proportion z test) compares proportions from two groups, not multiple categories. Choice D (two-sample t test) is for quantitative means, not categorical data. Mini-lesson: Align methods by classifying variables (both categorical here, so chi-square), checking for one vs. two variables (two for independence), and specifying the inference (test for association). This systematic check helps avoid common errors in procedure selection.

Question 11

1. One-proportion $z$ confidence interval (correct answer)
2. One-proportion $z$ test
3. Two-proportion $z$ confidence interval
4. One-sample $t$ confidence interval for a mean
5. Chi-square goodness-of-fit test

Explanation: This question evaluates the AP Statistics skill of choosing the correct inference procedure for estimating a single population proportion. The one-proportion z confidence interval is suitable because the researcher has a single random sample with binary yes/no responses and wants to estimate the proportion supporting the park with a 95% interval. A common distractor is the one-proportion z-test, which is for hypothesis testing rather than interval estimation, while the two-proportion z-interval applies to comparing two groups, not present here. The one-sample t-interval is for means of quantitative data, and chi-square goodness-of-fit is for testing distributions across multiple categories. A mini-lesson on aligning methods: Identify the parameter (proportion vs. mean), the number of populations (one or two), and the goal (test a claim or estimate with an interval); for proportions from large samples, use z-procedures, ensuring conditions like np ≥ 10 and n(1-p) ≥ 10 are met.

Question 12

1. Two-sample $z$ test for a difference in means
2. Matched-pairs $t$ test for the mean difference (correct answer)
3. One-sample $t$ interval for a population mean
4. Chi-square test of independence for two categorical variables
5. Two-proportion $z$ test for a difference in proportions

Explanation: This question tests your ability to recognize when paired data requires a matched-pairs t-test. The key insight is that the same 60 students are measured twice - once before and once after using the platform - creating dependent samples. When you have the same subjects measured under two different conditions, you must use a matched-pairs t-test to analyze the mean difference. The two-sample tests (choices A and E) are incorrect because they assume independent samples. Choice C is wrong because we're not estimating a single population mean but rather testing whether the mean difference equals zero. Choice D is inappropriate because we're dealing with quantitative data (time in minutes), not categorical data.

Question 13

1. One-sample $t$ interval for a mean
2. Two-sample $t$ test for a difference in means (correct answer)
3. Matched-pairs $t$ test for a mean difference
4. Two-proportion $z$ test for a difference in proportions
5. Chi-square test of independence for a two-way table

Explanation: This problem requires selecting an appropriate inference procedure for comparing mean weekly homework time between two independent groups (new platform vs. old platform). Since we have two independent random samples of students (40 in each group) and we're comparing means of a quantitative variable (homework time in minutes), the two-sample t-test for a difference in means is the correct choice. The matched-pairs test (option C) would be incorrect because the same students aren't using both platforms. The two-proportion z-test (option D) and chi-square test (option E) are for categorical data, not quantitative means. When comparing means between two independent groups, always use the two-sample t-test rather than procedures designed for paired data or proportions.

Question 14

1. Two-sample $t$ test for a difference in means (correct answer)
2. Matched-pairs $t$ test for a mean difference
3. Two-proportion $z$ test for a difference in proportions
4. One-sample $t$ interval for a population mean
5. Chi-square test of homogeneity for several populations

Explanation: This scenario compares means from two independent groups (Brand X and Brand Y batteries). With 30 batteries randomly selected from each brand and tested separately, we have independent samples of quantitative data (battery life in hours). The two-sample t-test compares the population means of the two brands. Choice B is incorrect because the batteries aren't paired - each battery is tested only once. Choice C tests proportions, not means. Choice D estimates a single mean, not comparing two. Choice E tests homogeneity of categorical distributions across populations. When comparing means from two independent groups, the two-sample t-test is the appropriate choice.

Question 15

1. Chi-square test of homogeneity for two or more populations (correct answer)
2. Chi-square goodness-of-fit test for one population
3. Two-proportion $z$ test for a difference in proportions
4. Two-sample $t$ test for a difference in means
5. Matched-pairs $t$ interval for a mean difference

Explanation: This scenario compares the distribution of a categorical variable (commuting method with 4 categories) across two populations (City 1 and City 2). The chi-square test of homogeneity determines whether the distribution of commuting preferences is the same in both cities. This test is appropriate when comparing categorical distributions across two or more independent groups. Choice B tests a single population's distribution against a specified model. Choice C only works for binary outcomes, not four categories. Choices D and E involve quantitative data or paired samples. The homogeneity test specifically addresses whether multiple populations share the same categorical distribution.

Question 16

1. Chi-square goodness-of-fit test
2. Two-proportion $z$ test for a difference in proportions (correct answer)
3. Matched-pairs $t$ test for a mean difference
4. One-proportion $z$ interval for a proportion
5. Two-sample $t$ test for a difference in means

Explanation: This problem compares the proportion preferring in-person office hours between two independent groups (first-year students and seniors). Since we have two independent random samples and we're testing whether the proportions differ for a binary outcome (prefer/do not prefer), the two-proportion z-test is the correct procedure. The chi-square goodness-of-fit test (option A) is for testing one categorical variable against expected frequencies, not comparing two groups. The matched-pairs test (option C) requires paired data, which we don't have. The one-proportion z-interval (option D) only analyzes one group. The two-sample t-test (option E) is for comparing means of quantitative data, not proportions. When comparing proportions between two independent groups, use the two-proportion z-test.

Question 17

1. Two-sample $t$ interval for a difference in means
2. Matched-pairs $t$ interval for a mean difference (correct answer)
3. Two-proportion $z$ test for a difference in proportions
4. One-sample $t$ test for a mean
5. Chi-square goodness-of-fit test

Explanation: This study uses a crossover design where each volunteer tests both brands of earbuds, creating paired data where each person provides two measurements (one for each brand). The matched-pairs t-interval for a mean difference is appropriate because we're estimating the mean difference in battery life between brands using paired observations. The two-sample t-interval (option A) would incorrectly treat the groups as independent, ignoring that the same people tested both brands. The proportion tests (options C and D) are for categorical data, not quantitative battery life measurements. The chi-square test (option E) is also for categorical data. When the same subjects provide measurements under two conditions, the matched-pairs design controls for person-to-person variation and requires matched-pairs analysis.

Question 18

1. Chi-square test of independence for a two-way table (correct answer)
2. Chi-square goodness-of-fit test for one categorical variable
3. Two-proportion $z$ interval for a difference in proportions
4. One-sample $t$ test for a mean
5. Matched-pairs $t$ test for a mean difference

Explanation: This problem involves examining the relationship between two categorical variables: news source (3 categories) and political affiliation (3 categories) from a single sample of 300 adults. The chi-square test of independence is designed specifically to test whether two categorical variables are associated in the population. The goodness-of-fit test (option B) only analyzes one categorical variable against expected frequencies. The two-proportion z-interval (option C) can only compare two proportions, not handle multiple categories. The t-tests (options D and E) are for quantitative data, not categorical relationships. When testing for association between two categorical variables in a two-way table, the chi-square test of independence is the standard procedure.

Question 19

1. Chi-square test of independence (two-way table of two categorical variables) (correct answer)
2. Chi-square goodness-of-fit test (compare one categorical variable to a claimed distribution)
3. Two-proportion $z$ test (compare proportions for exactly two categories)
4. Two-sample $t$ test (compare mean study hours across two groups)
5. Linear regression $t$ test for slope (predict location from year numerically)

Explanation: This question requires identifying when to use a chi-square test of independence for analyzing association between two categorical variables. We have two categorical variables: study location (4 categories) and class year (4 categories), displayed in a two-way table. The chi-square test of independence tests whether there's an association between these variables - that is, whether study location preferences depend on class year. This is different from a goodness-of-fit test, which compares one variable to a theoretical distribution. Since we're examining the relationship between two categorical variables with multiple categories each, the chi-square test of independence is the only appropriate choice. The test will determine if the distribution of study locations varies significantly across class years.

Question 20

1. One-proportion $z$ interval for $p$ (correct answer)
2. Two-proportion $z$ interval for $p_1-p_2$
3. One-sample $t$ interval for a mean
4. Chi-square goodness-of-fit test
5. Matched-pairs $t$ interval for a mean difference

Explanation: This question evaluates the ability to choose the right inference procedure for estimating a single proportion in AP Statistics. The one-proportion z interval (choice A) fits best as it constructs a confidence interval for the population proportion based on yes/no categorical data from one random sample. Incorrect options include choice B (two-proportion z interval), which requires two groups for comparison, not present here, and choice C (one-sample t interval), suited for means of quantitative data, not proportions. Choice D (chi-square goodness-of-fit) tests distributions against expected values, not for interval estimation. Mini-lesson: Match procedures by determining if the variable is categorical (proportions via z) or quantitative (means via t), the number of samples (one here), and the goal (confidence interval vs. test). This approach prevents mismatching and ensures accurate inference.