- A hospital administrator wants to test the claim that 15% of patients miss their follow-up appointment. A random sample of 250 patients shows that 52 missed their follow-up appointment. Which inference procedure is most appropriate to test the administrator's claim about the population proportion?
Answer: One-sample z test for a population proportion
This question involves testing a specific claim about a population proportion (that 15% of patients miss appointments). The one-sample z test for a population proportion is designed to test hypotheses about a single population proportion against a claimed value. The one-sample z interval (option B) would estimate the proportion, not test a claim. The two-sample z test (option C) compares two groups. The chi-square tests (options D and E) serve different purposes with categorical data. When testing a claim about a specific value for a population proportion, use the one-sample z test. - A school district wants to know whether the proportion of students who prefer taking tests on a computer differs between middle school and high school students. A random sample of 120 middle school students and 150 high school students is selected. Each student is asked whether they prefer computer-based tests (Yes/No). Which inference procedure is most appropriate to compare the two groups' preferences?
Answer: Two-proportion z test
This question tests your ability to select the appropriate procedure for comparing proportions between two independent groups. We have two distinct populations (middle school and high school students) and want to compare the proportion who prefer computer-based tests in each group. The two-proportion z test is designed specifically for this scenario - comparing proportions between two independent samples. The chi-square test for association would also work here but is less direct, while the one-sample proportion test only handles a single group. The matched-pairs test requires paired data, which we don't have since these are different students. When comparing proportions between two independent groups with a binary response (Yes/No), the two-proportion z test provides the most straightforward and powerful approach. - A university compares the proportion of students who pass a certification exam (Pass/Fail) across four different majors (Business, Biology, Computer Science, Education). A random sample from each major is taken and results are recorded in a 4×2 table of counts. The university wants to test whether pass rate is the same for all majors. Which inference procedure is most appropriate?
Answer: Chi-square test for association (independence) in a two-way table
In AP Statistics, inference for categorical data with multiple groups and binary outcomes uses tests for equal proportions across groups. The university's 4x2 table of pass/fail by four majors tests if pass rates are the same, fitting the chi-square test for association (independence) to check for differences. This procedure evaluates if major and pass status are related. Distractors like goodness of fit are for one variable; one-sample z-interval estimates single proportions; two-sample z-test is for two groups; and matched-pairs t-test is quantitative paired. Mini-lesson: For multi-group proportion comparisons, use chi-square homogeneity (same as association), ensuring expected counts ≥5. Proper selection avoids single-group or pairwise methods for broader comparisons.
- A university wants to determine whether the proportion of students who live on campus is the same across four class years (First-year, Sophomore, Junior, Senior). A random sample of 80 students from each class year is selected, and each student is recorded as living on campus (Yes/No). Which inference procedure is most appropriate?
Answer: Chi-square test for association (independence) in a 4×2 table
This question involves examining whether the proportion of students living on campus is the same across four class years. We can view this as testing for association between two categorical variables: class year (4 categories) and residence status (2 categories: on/off campus). The chi-square test for association in a 4×2 table tests whether these variables are independent - that is, whether the proportion living on campus is the same across all class years. The goodness of fit test would require specific expected proportions for each class, which we don't have. The two-proportion z test can only compare two groups at a time, not four simultaneously. When testing whether proportions are equal across more than two groups, the chi-square test for association provides a single, comprehensive test of the null hypothesis that all proportions are equal. - A cafeteria manager believes that students choose lunch options in the following proportions: 50% hot meal, 30% salad, 20% sandwich. On one day, a random sample of 200 students is observed: 86 choose hot meal, 70 choose salad, and 44 choose sandwich. Which inference procedure is most appropriate to evaluate whether the observed distribution matches the manager's stated proportions?
Answer: Chi-square test for goodness of fit
This problem involves testing whether observed frequencies for lunch choices match the manager's stated distribution (50% hot meal, 30% salad, 20% sandwich). The chi-square test for goodness of fit is specifically designed to test if observed data fits a hypothesized distribution for a single categorical variable. The chi-square test for independence (option A) requires two variables. The z-procedures (options B and D) work with proportions or binary outcomes. The t-test (option E) is for quantitative data. When testing if observed frequencies match expected proportions for one categorical variable, use chi-square goodness of fit.
- A company wants to test whether a new training program changes the proportion of employees who pass a certification exam on the first attempt. Last year (before the program), 90 of 150 randomly selected employees passed on the first attempt. This year (after the program), 110 of 160 randomly selected employees passed on the first attempt. Which inference procedure is most appropriate to test for a change?
Answer: Two-proportion z test comparing last year vs. this year
This scenario involves comparing proportions from two different time periods to test for a change after implementing a training program. We have two independent samples: employees from last year (before training) and employees from this year (after training). The two-proportion z test is appropriate for comparing these two independent proportions to determine if there's a significant difference. The one-sample test would only use this year's data and couldn't assess change. The matched-pairs test would require the same employees tested before and after, which isn't the case here. The chi-square goodness of fit test checks against a specific distribution, not for changes between groups. When comparing proportions from two different time periods with independent samples, the two-proportion z test effectively tests whether the intervention caused a significant change. - A university wants to estimate the proportion of all undergraduates who have taken at least one online course. A simple random sample of 200 undergraduates finds that 78 have taken at least one online course. Which inference procedure is most appropriate to estimate the population proportion?
Answer: One-sample z interval for a population proportion
This problem asks for estimating a single population proportion (the proportion of all undergraduates who have taken online courses) using sample data. The one-sample z interval for a population proportion is the appropriate procedure for constructing a confidence interval to estimate this parameter. The one-sample z test (option A) would test a claim, not estimate. The chi-square tests (options C and E) are for different scenarios involving categorical data. The t-interval (option D) is for estimating a population mean with quantitative data. When estimating a single population proportion from sample data, use the one-sample z interval. - A hospital wants to estimate the proportion of patients who would recommend the emergency department to a friend. A simple random sample of 250 recent patients is contacted; 172 say "Yes" and 78 say "No." Which inference procedure is most appropriate to estimate the true recommendation proportion with a confidence interval?
Answer: One-sample z interval for a population proportion
Selecting appropriate inference procedures in AP Statistics for categorical data includes estimating single proportions with confidence intervals. The hospital has one random sample of 250 patients with Yes/No responses on recommendations, aiming to estimate the true proportion with a confidence interval, making the one-sample z-interval for a population proportion the best choice. This method uses the sample proportion and standard error to construct an interval, assuming large sample conditions are met. Distractors like the two-sample z-interval are for comparing two groups, not one; chi-square tests handle multiple categories or associations, not single-proportion estimation; and the one-sample t-interval is for means. The one-sample z-test is for hypothesis testing, not interval estimation. Mini-lesson: Distinguish between tests (for claims) and intervals (for estimation), and for proportions, use z-procedures when np and n(1-p) ≥10. Aligning the procedure to the goal ensures accurate inference without confusing sample types or data scales. - A biologist observes 400 offspring from a plant cross and records flower color: Red, Pink, or White. A genetics model predicts a 1:2:1 ratio (25% Red, 50% Pink, 25% White). The observed counts are Red 92, Pink 198, White 110. The biologist wants to test whether the observed distribution differs from the predicted ratio. Which inference procedure is most appropriate?
Answer: Chi-square test for goodness of fit
Selecting inference procedures in AP Statistics for categorical data includes testing observed counts against a predicted ratio for one variable. The biologist has 400 offspring in three color categories, testing against a 1:2:1 ratio, making the chi-square goodness-of-fit test appropriate to compare observed to expected frequencies. This assesses if the data fit the genetic model. Distractors like chi-square association require two variables; two-sample z-test compares groups; one-sample z-interval estimates a single proportion; and matched-pairs sign test is for paired data. Mini-lesson: For one sample with multiple categories and expected proportions, use goodness of fit with expected counts ≥5. This alignment distinguishes from association or proportion comparison methods.
- A public health official wants to estimate the difference in the proportion of residents who have received a flu shot between two counties. In County A, a random sample of 250 residents finds 160 vaccinated. In County B, a random sample of 220 residents finds 120 vaccinated. Which inference procedure is most appropriate to estimate the difference in proportions?
Answer: Two-proportion z interval for pA−pB
This problem asks for estimation rather than hypothesis testing, as indicated by the word 'estimate' in the question. We have two independent samples (County A and County B) and want to estimate the difference in vaccination proportions between them. The two-proportion z interval provides a confidence interval for the difference p₁ - p₂, which directly answers the question. The two-proportion z test would test a hypothesis about the difference but wouldn't provide an estimate. The one-sample proportion interval only handles a single proportion, while the chi-square test for association tests for relationships but doesn't estimate differences. When the goal is to estimate (not test) the difference between two proportions from independent samples, use the two-proportion z interval to construct a confidence interval for the difference. - A marketing team claims that customers choose among three package designs (A, B, C) in equal proportions. In a random sample of 300 customers, 120 chose A, 96 chose B, and 84 chose C. Which inference procedure is most appropriate to assess whether the observed choices are consistent with the company's claim?
Answer: Chi-square test for goodness of fit
This scenario involves testing whether observed frequencies match expected frequencies for a single categorical variable with three categories (package designs A, B, C). The chi-square test for goodness of fit is designed exactly for this purpose - testing if observed data fits a hypothesized distribution (equal proportions: 33.3% each). The chi-square test for independence (option B) requires two categorical variables, not one. The z-tests (options C and E) are for proportions with binary outcomes, not multiple categories. The t-test (option D) is for quantitative data. When testing if a single categorical variable follows a specific distribution, use chi-square goodness of fit.
- A company compares customer satisfaction (Satisfied/Not Satisfied) between three store locations (North, Central, South). A random sample of customers from each location is surveyed and the results are summarized in a 3×2 table of counts. The company wants to know whether satisfaction depends on location. Which inference procedure is most appropriate?
Answer: Chi-square test for association (independence) in a two-way table
In AP Statistics, inference for categorical data requires selecting procedures based on study design, here a 3x2 table comparing satisfaction across three locations to test dependence. The chi-square test for association (independence) is appropriate as it examines if satisfaction and location are related in the two-way table of counts. This method computes expected frequencies under independence and tests for significant differences. Distractors like the two-sample z-interval are for two groups only, not three; goodness of fit is for one variable; the one-sample z-test is single-proportion; and matched-pairs t-test is quantitative paired data. Mini-lesson: For more than two groups with categorical outcomes, use chi-square for homogeneity (equivalent to association here), checking expected counts ≥5. This ensures the method matches the multi-group comparison without reducing to pairwise tests.
- A marketing firm claims that 40% of consumers prefer Brand A over Brand B. To check the claim, a simple random sample of 150 consumers is surveyed; 48 prefer Brand A and 102 prefer Brand B. The firm wants to test whether the true preference proportion differs from 0.40. Which inference procedure is most appropriate?
Answer: One-sample z test for a population proportion
This question tests the AP Statistics skill of picking inference methods for categorical data, specifically testing a claim about a single population proportion. The marketing firm surveys one sample of 150 consumers with preferences for Brand A or B, testing if the proportion preferring A differs from 0.40, which fits the one-sample z-test for a population proportion. This procedure calculates a z-statistic to assess the null hypothesis against the sample data. Distractors include the one-sample z-interval, which estimates rather than tests; the two-sample z-test compares two groups; and chi-square tests are for distributions or associations, not a single binary proportion test. Chi-square association would require two variables. Mini-lesson: For hypothesis tests on one proportion, verify random sampling and large sample conditions, distinguishing from estimation or multi-group comparisons. Proper alignment avoids applying multi-category or interval methods to simple proportion tests. - A researcher studies whether preferred study environment is related to class year. A random sample of students is classified by class year (Freshman, Sophomore, Junior, Senior) and preferred environment (Quiet, Background Music, Group Setting). The data are summarized in a 4×3 two-way table of counts. The researcher wants to know if class year and preferred environment are associated in the population. Which inference procedure is most appropriate?
Answer: Chi-square test for association (independence) in a two-way table
This AP Statistics skill focuses on choosing procedures for categorical data, particularly when examining relationships between two categorical variables. The researcher has a 4x3 table of counts for class year and preferred study environment, wanting to test for association in the population, which calls for the chi-square test for association (independence) to determine if the variables are related. This procedure is suitable as it analyzes the two-way table to see if observed frequencies differ significantly from expected under independence. Common distractors are the chi-square goodness of fit, which is for one variable against expected proportions, not two variables; the two-sample z-test compares proportions between two groups, not multiple categories; and t-tests are for means, not counts. The matched-pairs t-interval is for paired quantitative data. Mini-lesson: For two-way tables, use chi-square association if testing independence or homogeneity of proportions across groups, checking conditions like random sampling and expected counts ≥5. This alignment prevents misapplying single-sample or quantitative methods to categorical associations.
- A school district wants to know whether the proportion of students who prefer block scheduling differs between two grade levels. A random sample of 120 ninth-graders and 150 twelfth-graders was surveyed. Among ninth-graders, 54 preferred block scheduling; among twelfth-graders, 93 preferred block scheduling. Which inference procedure is most appropriate to answer the research question?
Answer: Two-sample z test for a difference in proportions
This question asks about comparing proportions between two independent groups (ninth-graders and twelfth-graders), making the two-sample z test for a difference in proportions the appropriate choice. We have two distinct populations with binary outcomes (prefer block scheduling or not), and we want to test if the population proportions differ. The chi-square test for goodness of fit (option C) tests against a specific distribution, not comparing two groups. The t-tests (options D and E) are for quantitative data, not categorical. When comparing proportions between two independent samples, always use the two-sample z test. - Researchers want to determine whether there is an association between smoking status (smoker, nonsmoker) and exercise frequency (0–1 days/week, 2–4 days/week, 5–7 days/week) among adults in a city. They take a random sample of 420 adults and record each person's smoking status and exercise category. Which inference procedure is most appropriate?
Answer: Chi-square test for independence
This question involves examining the relationship between two categorical variables: smoking status (2 categories) and exercise frequency (3 categories). The chi-square test for independence is specifically designed to test whether two categorical variables are associated. The chi-square goodness of fit (option B) tests a single variable against expected frequencies. The z-tests (options C and D) are for proportions, typically with binary outcomes. The matched-pairs t-test (option E) requires paired quantitative data. When investigating associations between two categorical variables in a contingency table, always use the chi-square test for independence.
- A political scientist wants to know whether party affiliation (Democrat, Republican, Independent) is associated with preferred news source (TV, Online, Print). A random sample of 500 registered voters is surveyed, and each voter is classified into one category for each variable. Which inference procedure is most appropriate?
Answer: Chi-square test for association (independence)
This scenario examines the relationship between two categorical variables: party affiliation (3 categories) and news source preference (3 categories). The goal is to determine whether these variables are associated - whether party affiliation and news preference are related or independent. The chi-square test for association (independence) is specifically designed to test whether two categorical variables are independent, regardless of the number of categories each has. The proportion tests are limited to specific proportions and can't handle the full 3×3 contingency table structure. The goodness of fit test only handles one variable against expected proportions. When testing for association between two categorical variables, especially with multiple categories each, the chi-square test for independence is the standard and most appropriate procedure.
- A marketing team wants to test whether the distribution of preferred smartphone brand among customers matches last year's distribution: 40% Brand A, 35% Brand B, and 25% Brand C. This year, a random sample of 300 customers is asked to name their preferred brand (A/B/C), and the observed counts are recorded. Which inference procedure is most appropriate?
Answer: Chi-square test for goodness of fit
This problem requires testing whether the current distribution of brand preferences matches a specific expected distribution from last year (40%, 35%, 25%). We have observed counts from this year's sample and want to test if they significantly deviate from last year's proportions. The chi-square goodness of fit test is designed precisely for this purpose - comparing observed frequencies to expected frequencies based on a hypothesized distribution. The two-proportion z test only compares two groups, while the chi-square test for association requires two variables. The one-sample proportion test only handles a single proportion, not a distribution across multiple categories. When testing whether categorical data follows a specific distribution pattern with known expected proportions, the chi-square goodness of fit test is the appropriate choice.
- A city council surveys a random sample of 500 residents and asks their preferred transportation mode for commuting: Car, Public Transit, Bike/Walk, or Other. The observed counts are Car 260, Public Transit 140, Bike/Walk 80, Other 20. The council wants to test whether the distribution of preferences matches a previously published national distribution (Car 55%, Public Transit 25%, Bike/Walk 15%, Other 5%). Which inference procedure is most appropriate?
Answer: Chi-square test for goodness of fit
In AP Statistics, selecting inference procedures for categorical data involves matching the study design to the right test, here testing if observed counts in multiple categories fit expected proportions from a known distribution. The city council has one sample of 500 residents categorized into four transportation modes, comparing to national percentages, which fits the chi-square goodness-of-fit test to assess if the sample distribution matches the hypothesized one. This is appropriate because it calculates expected counts based on the national proportions and compares them to observed counts. Distractors include the chi-square test for association, which requires two categorical variables in a two-way table, not a single variable against fixed proportions; the one-sample z-test is for a single proportion, not multiple categories; and the two-sample z-interval compares two groups, not a distribution. The matched-pairs sign test is for paired categorical data, irrelevant here. A mini-lesson: Always identify if it's one sample with multiple categories (goodness of fit) versus two variables (association), and verify conditions like expected counts ≥5 per category. This ensures the procedure aligns with the goal of testing fit to a model.
- A restaurant manager claims that customers choose among three lunch specials (Pasta, Salad, Sandwich) in equal proportions. Over 200 randomly selected lunch orders on weekdays, the counts are recorded by special. The manager wants to test whether the observed distribution differs from the claimed equal distribution. Which inference procedure is most appropriate?
Answer: Chi-square test for goodness of fit
This scenario requires testing whether observed frequencies match a specific expected distribution. The manager claims equal proportions (1/3, 1/3, 1/3) for the three lunch specials, and we need to test if the observed counts deviate significantly from this expectation. The chi-square goodness of fit test is specifically designed for this purpose - comparing observed frequencies to expected frequencies based on a hypothesized distribution. The two-proportion z test only compares two groups, while the chi-square test for association examines relationships between two variables rather than testing against a specific distribution. The t-test options are inappropriate as we're dealing with categorical data, not means. When testing whether categorical data follows a specific distribution pattern, always use the chi-square goodness of fit test.
