- A teacher wants to determine whether a new vocabulary app improves quiz scores. The teacher randomly assigns 40 students to use the app for two weeks and 40 different students to use traditional worksheets for two weeks. Afterward, all students take the same vocabulary quiz, and the response variable is the quiz score (0–100). The teacher wants to test whether the mean score differs between the two groups. Which inference procedure is most appropriate?
Answer: Two-sample t test for a difference in means
This experiment compares mean quiz scores between two independent groups: students using the app versus students using worksheets. With random assignment creating two separate groups of 40 students each, and quantitative outcome data (quiz scores 0-100), a two-sample t-test is appropriate. Choice A is incorrect because students aren't paired - different students are in each group. Choice C tests proportions, but quiz scores are quantitative. Choice D tests categorical distributions. Choice E estimates a single mean rather than comparing two. Random assignment to different treatments with independent groups always calls for a two-sample test when comparing means. - A school district wants to know whether a new online homework platform changes the mean time (in minutes) students spend on math homework per night. A random sample of 60 students is selected; each student reports their homework time for one week before the platform and for one week after the platform. The district wants to test whether the mean change (after − before) is different from 0. Which inference procedure is most appropriate?
Answer: Matched-pairs t test for the mean difference
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. - A university wants to test whether the proportion of students who prefer in-person office hours differs between first-year and senior students. A random sample of 150 first-year students and an independent random sample of 140 seniors are surveyed; each student answers "prefer in-person" or "do not prefer in-person." Which inference procedure is most appropriate?
Answer: Two-proportion z test for a difference in proportions
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. - A retailer wants to estimate the difference in the proportion of customers who make a purchase when greeted at the door versus not greeted. Over two weeks, a random sample of 500 customers on days with greeters is recorded as purchase/no purchase, and an independent random sample of 500 customers on days without greeters is recorded similarly. Which inference procedure is most appropriate to construct a confidence interval for the difference in purchase proportions?
Answer: Two-proportion z interval for p1−p2
This question evaluates selecting intervals for comparing proportions in AP Statistics. The two-proportion z interval (choice A) is suitable for estimating the difference in purchase proportions from two independent samples. Common errors include choice B (two-proportion z test), which is for hypothesis testing, not intervals, and choice D (one-proportion z interval), for a single group. Choice C (matched-pairs t interval) applies to paired quantitative data, not categorical. Mini-lesson: To align procedures, note the number of groups (two independent for comparison), variable (categorical for z proportions), and type (confidence interval). This distinction ensures the method captures the comparative estimation needed. - A school district wants to know whether a new online homework platform changes students' average weekly math homework time. From 6 randomly selected middle schools, administrators randomly assign half of the students in each school to use the new platform and the other half to keep the usual system for 8 weeks. At the end, each student reports their average homework minutes per week. Which inference procedure is most appropriate to test whether the mean homework time differs between the two systems?
Answer: Two-sample t test for a difference in means (independent groups)
This question tests the skill of selecting an appropriate inference procedure in AP Statistics, specifically for comparing means from two groups. The two-sample t test for a difference in means (choice C) is most appropriate because the study involves two independent groups of students (new platform vs. usual system) with quantitative data on homework time, and the goal is to test for a difference in population means. Distractors like choice B (matched-pairs t test) are incorrect as there is no pairing between students; each is in only one group, not measured twice. Choice D (two-proportion z test) would apply to categorical data, not quantitative minutes, while choice E (chi-square) is for associations between categorical variables. A mini-lesson: Always align the procedure by identifying the response variable type (quantitative here, so t procedures), the design (independent samples, not paired), and the inference type (hypothesis test for means). This ensures the method matches the study's structure and objectives. - A sociologist wants to determine whether political party affiliation (Democrat/Republican/Independent) is associated with preferred news source (TV/online/print). She takes a random sample of 600 adults and records each person's party affiliation and preferred news source, both categorical variables. Which inference procedure is most appropriate to assess the association?
Answer: Chi-square test of independence
This question evaluates the AP Statistics skill of choosing chi-square procedures for associations between two categorical variables. The chi-square test of independence is suitable as the sociologist has a single random sample with two categorical variables (party and news source) and wants to test for association. A common distractor is the chi-square goodness-of-fit, which tests one variable against expected proportions, not relationships. The two-proportion z-test is for binary outcomes in two groups, while t-tests and intervals are for quantitative data. A mini-lesson on aligning methods: For categorical data, use chi-square; independence for two variables in one sample, goodness-of-fit for one variable vs. expectations, homogeneity for one variable across multiple populations; ensure expected counts ≥ 5 in contingency tables.
- A botanist records the height (cm) of a random sample of 25 sunflowers grown with a new fertilizer. The botanist wants to test the claim that the mean height of sunflowers grown with this fertilizer is greater than 180 cm. No population standard deviation is known. Which inference procedure is most appropriate?
Answer: One-sample t test for a population mean
This problem involves testing a claim about a single population mean when the population standard deviation is unknown. With a sample of 25 sunflowers and no known population standard deviation, we must use a one-sample t-test. The t-distribution accounts for the additional uncertainty from estimating the population standard deviation with the sample standard deviation. Choice A is wrong because it assumes a known population standard deviation. Choice C compares two groups, but we only have one. Choice D tests proportions, not means. Choice E tests homogeneity across multiple categorical groups, which doesn't apply to this quantitative, single-sample scenario. - A city wants to estimate the proportion of households that support a proposed recycling fee. A simple random sample of 500 households is contacted, and each household responds either "support" or "do not support." The parameter of interest is the population proportion who support the fee. Which inference procedure is most appropriate?
Answer: One-proportion z interval for a proportion
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. - A school district wants to know whether a new online tutoring program improves Algebra I exam scores. Two Algebra I classes were randomly selected; one class was randomly assigned to use the tutoring program for 6 weeks and the other continued with regular homework. At the end, each student took the same standardized exam and the response variable is the numerical exam score. Which inference procedure is most appropriate to determine whether the tutoring program changes the mean exam score?
Answer: Two-sample t test for a difference in means (independent groups)
This question assesses the AP Statistics skill of selecting and implementing appropriate inference procedures for comparing means from two independent groups. The two-sample t-test for a difference in means is most appropriate because the study involves two independent random samples of classes, one with tutoring and one without, and the response variable is quantitative exam scores, aiming to test if the tutoring changes the mean score. Distractors like the matched-pairs t-test might tempt if one misinterprets the classes as paired, but here the groups are independent without pairing. The one-sample t-interval is incorrect as it applies to a single population, while chi-square homogeneity and two-proportion z-test are for categorical data, not quantitative scores. A mini-lesson on aligning methods: First, determine if the variable is quantitative (use t-procedures) or categorical (use z or chi-square); then check for one sample, independent samples, or paired data; finally, decide between a test for significance or a confidence interval for estimation. - A political scientist wants to determine whether party affiliation (Democrat/Republican/Independent) is associated with preferred news source (TV/online/print). She surveys a random sample of 900 registered voters and records each voter's party affiliation and preferred news source. Which inference procedure is most appropriate to assess whether the two categorical variables are associated?
Answer: Chi-square test of independence (association) for a two-way table
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.
- A coach wants to know whether a new strength program increases athletes' vertical jump height. The coach measures each of 18 athletes' vertical jump (in cm) before the program and again after 6 weeks on the program, then computes each athlete's change (after − before). Which inference procedure is most appropriate to test whether the program increases the mean vertical jump height?
Answer: Matched-pairs t test for a mean difference
This question examines choosing inference procedures for paired quantitative data in AP Statistics. The matched-pairs t test (choice B) is appropriate as it analyzes before-and-after differences in jump height for the same athletes, testing if the mean difference is positive. Distractors like choice A (two-sample t test) treat groups as independent, ignoring the pairing, while choice C (one-proportion z test) is for categorical data, not heights. Choice D (chi-square) assesses categorical associations, not means. Mini-lesson: Align methods by noting the design (paired for repeated measures on the same subjects), variable type (quantitative), and goal (test for mean difference). Recognizing pairing prevents incorrectly using independent-sample methods and improves accuracy. - A fitness coach claims that clients who follow a new stretching routine can increase their sit-and-reach flexibility. A random sample of 32 clients measures sit-and-reach distance (cm) before starting the routine and again after 4 weeks on the routine. The research question is whether the routine increases the mean sit-and-reach distance. Which inference procedure is most appropriate?
Answer: Matched-pairs t test for a mean difference
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. - A public health official wants to test whether the proportion of households with a working smoke detector differs between two counties. She takes an independent random sample of households from County 1 and County 2 and records whether each household has a working detector (yes/no). Which inference procedure is most appropriate to test for a difference between the two population proportions?
Answer: Two-proportion z test
This question tests the AP Statistics skill of selecting tests for comparing proportions from two independent populations. The two-proportion z-test is suitable as there are independent random samples from two counties with binary detector data, testing if proportions differ. Distractors like the two-proportion z-interval are for estimation, while chi-square goodness-of-fit is for one sample's distribution. Two-sample t-test and matched-pairs t-test are for quantitative data, not binary. A mini-lesson on aligning methods: For binary data from two groups, use two-proportion z-procedures; opt for test to assess significance of difference or interval for estimation; verify large samples with successes and failures ≥ 10 in each group for normal approximation. - A manufacturer claims its light bulbs last an average of 1200 hours. A consumer group tests a random sample of 25 bulbs and records each bulb's lifetime in hours (quantitative). The group wants to determine whether the true mean lifetime is less than 1200 hours. Which inference procedure is most appropriate?
Answer: One-sample t test for a mean
This question evaluates the AP Statistics skill of choosing hypothesis tests for a single mean against a specific claim. The one-sample t-test for a mean is appropriate because the group has a single random sample of quantitative bulb lifetimes and tests if the mean is less than 1200 hours. Common distractors include the one-sample t-interval for estimation, not testing, and the two-sample t-test for comparing groups. One-proportion z-test and chi-square goodness-of-fit are for categorical data. A mini-lesson on aligning methods: For one quantitative sample testing against a value, use one-sample t-test; specify alternative (one-sided here); check randomness, independence, and normality; interpret p-value against significance level. - A biologist expects a 1:2:1 ratio of phenotypes (Type 1, Type 2, Type 3) in the offspring of a certain cross. She observes 160 offspring and records the phenotype category for each. Which inference procedure is most appropriate to test whether the observed counts differ from the expected 1:2:1 distribution?
Answer: Chi-square goodness-of-fit test
This question assesses procedures for testing observed counts against an expected distribution in AP Statistics. The chi-square goodness-of-fit test (choice A) is best to compare observed phenotype counts to the 1:2:1 ratio for one categorical variable. Distractors such as choice B (chi-square independence) require two variables, not one, while choice D (one-sample t test) is for quantitative means, not counts. Choice C (one-proportion z test) handles single proportions, not multi-category distributions. Mini-lesson: Align methods by identifying categories (one variable with expected proportions for goodness-of-fit), data type (categorical counts), and goal (test vs. expected). This prevents confusing it with association tests and refines selection skills.
- A public health analyst wants to estimate the mean amount of sugar (grams, quantitative) in a particular brand of cereal. A simple random sample of 18 boxes is collected and the sugar content per serving is measured for each box. Which inference procedure is most appropriate to create a confidence interval for the population mean sugar content?
Answer: One-sample t interval for a mean
This question requires selecting the appropriate procedure for estimating a single population mean. We have one sample of 18 boxes from one brand of cereal, measuring a quantitative variable (sugar content in grams). Since we want to create a confidence interval for the population mean sugar content and the population standard deviation is unknown, a one-sample t interval is the correct choice. The sample size is relatively small (n = 18), and we don't know the population standard deviation, so we cannot use a z interval. This is not a comparison problem (ruling out two-sample procedures) and the response is quantitative (ruling out proportion procedures). The one-sample t interval will provide an estimate of the true mean sugar content with a margin of error. - A marketing firm wants to estimate the mean amount of money (in dollars) spent per week on streaming services by adults in a certain city. The firm takes a simple random sample of 80 adults and records each person's weekly spending. The firm wants a 90% confidence interval for the population mean spending. Which inference procedure is most appropriate?
Answer: One-sample t interval for a population mean
This problem requires constructing a confidence interval for a single population mean. With 80 adults providing quantitative data (dollars spent) and no known population standard deviation, a one-sample t-interval is appropriate. The t-distribution accounts for estimating the population standard deviation from the sample. Choice B constructs intervals for proportions, not means. Choice C compares two groups, but we have one. Choice D tests independence of categorical variables. Choice E assumes a known population standard deviation, which isn't given. For estimating a population mean with unknown σ, always use the t-interval regardless of sample size. - A city council wants to know whether support for a proposed recycling ordinance is distributed equally among four political parties (A, B, C, D) in the city. A random sample of 500 registered voters is taken, and each voter is classified by party. The observed counts are compared to what would be expected if the parties were equally represented in the sample. Which inference procedure is most appropriate?
Answer: Chi-square goodness-of-fit test for one categorical variable
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.
- A public health researcher wants to estimate the difference in the proportion of adults who received a flu shot this year between Urban County and Rural County. Independent random samples are taken: 400 adults from Urban County and 350 adults from Rural County, and each adult is classified as vaccinated or not. The researcher wants a 95% confidence interval for pUrban−pRural. Which inference procedure is most appropriate?
Answer: Two-proportion z interval for pUrban−pRural
This scenario requires estimating the difference between two population proportions using independent samples. The researcher has two distinct groups (Urban County and Rural County) with categorical data (vaccinated or not), and wants a confidence interval for the difference p_Urban - p_Rural. A two-proportion z-interval is the appropriate procedure when comparing proportions from two independent samples. Choice B is incorrect because we have categorical data (vaccinated/not), not quantitative measurements. Choice C tests a single proportion, not a difference. Choice D assumes paired data, but these are independent samples. Choice E tests the distribution of a single categorical variable, not comparing two groups. - A manufacturer wants to test whether the mean fill volume of a drink bottle is 500 mL. A simple random sample of 36 bottles from today's production line is selected, and each bottle's fill volume (mL, quantitative) is measured. The population standard deviation is not known. Which inference procedure is most appropriate to test H0:μ=500 vs. Ha:μ=500?
Answer: One-sample t test for a mean
This question involves hypothesis testing for a single population mean when the population standard deviation is unknown. We have one sample of 36 bottles, measuring a quantitative variable (fill volume in mL), and want to test whether the mean equals 500 mL. Since the population standard deviation is not known, we must use a one-sample t test rather than a z test. The null hypothesis is μ = 500 and the alternative is μ ≠ 500 (two-sided). Even though the sample size is reasonably large (n = 36), without knowing σ, we cannot use a z test. This is a single sample problem (not comparing groups) with a quantitative response (not proportions), making the one-sample t test the only appropriate choice for testing whether the production process is properly calibrated.
