- A forestry agency is monitoring two forests, one with a new pest and one without. They want to estimate the difference in the proportion of damaged trees between the two forests. They take a random sample of 200 trees from the infested forest and a separate random sample of 250 trees from the healthy forest. Which procedure should they use?
Answer: A two-sample z-interval for the difference in the proportions of damaged trees.
The correct procedure is a two-sample z-interval for the difference in proportions. The goal is to 'estimate the difference' (which calls for an interval) in proportions (damaged/not damaged) between two independent populations (the two forests). Distractor A is a one-sample procedure and would not provide the comparison needed. Distractor B is a hypothesis test, but the goal is estimation. Distractor C is incorrect because it refers to means, not proportions; the data for each tree is categorical.
- A national study reported that the average number of books checked out per library visitor is 2.4. The director of a local library system wants to determine if visitors to her libraries differ from this national average. She collects data from a random sample of 250 visitor records from her system. What is the most appropriate statistical procedure?
Answer: A one-sample t-test for a mean to compare her library's average to the national value of 2.4.
The correct procedure is a one-sample t-test for a mean. The director is comparing a single sample from her library system to a pre-existing, known population value (the national average of 2.4). The goal is to test if her library's mean 'differs,' which implies a hypothesis test. Distractor A is incorrect because the national average is a fixed parameter for the test, not a second sample with its own mean and standard deviation. Distractor B is incorrect because the goal is to test against the national average, not just to estimate the local average. Distractor D is incorrect because the data (number of books) is quantitative, not categorical.
- An educational psychologist is studying the effect of background music on test performance. Fifty students are randomly assigned to two groups: 25 take a test in a quiet room, and 25 take the same test in a room with classical music playing. The psychologist wants to determine if the presence of music has a statistically significant effect on the mean test scores. Which is the proper procedure?
Answer: A two-sample t-test for the difference in means, to compare the average scores of the two independent groups.
The correct procedure is a two-sample t-test for the difference in means. The study design involves two independent groups created by random assignment. The goal is to determine if there is a 'statistically significant effect,' which is a question answered by a hypothesis test comparing the means of the two groups. Distractor A is for estimation, not for testing significance. Distractor B is incorrect because students were randomly assigned to separate groups, making them independent, not paired. A paired design would involve each student taking tests under both conditions. Distractor C is for categorical data (e.g., pass/fail), but the test scores are quantitative.
- A biologist believes the average length of a certain fish species in a polluted river is less than the established average of 34 cm in unpolluted rivers. She collects a random sample of 25 fish from the polluted river. The population standard deviation is unknown. Which procedure should she use to test her belief?
Answer: A one-sample t-test for a mean, to compare the sample mean to the hypothesized value of 34 cm.
The correct procedure is a one-sample t-test for a mean. The biologist is testing a hypothesis about a single population mean against a known value (34 cm). Since the population standard deviation is unknown, a t-procedure is appropriate. Distractor A is incorrect because a z-test requires the population standard deviation to be known. Distractor C is incorrect because data was collected from only one sample (the polluted river), not two. Distractor D is incorrect because the goal is to test the biologist's belief (a hypothesis test), not just to estimate the average length.
- A medical researcher is comparing recovery times for two surgical procedures. Data are collected from two independent random samples of patients. The researcher wants to answer the question: 'By how many days, on average, does recovery time differ between the two procedures?' Which of the following procedures directly addresses this question?
Answer: A two-sample t-interval for the difference in mean recovery times.
The correct procedure is a two-sample t-interval. The question asks 'By how many days...?', which implies estimation of the magnitude of the difference, not just a test for significance. A confidence interval provides a range of plausible values for this difference. Distractor A, a hypothesis test, would answer 'Is there a significant difference?', but not 'how large is the difference?'. Distractor C is a one-sample procedure and answers a different question. Distractor D is a useful exploratory tool but is not an inferential procedure that provides a numerical answer to the question.
- A city's recycling program manager claims a new awareness campaign was effective. The historical participation rate was 60%. After the campaign, a survey of 500 households found 321 participating. To test if the rate is now significantly higher than 60%, which procedure is appropriate?
Answer: A one-sample z-test for a proportion.
The correct procedure is a one-sample z-test for a proportion. The goal is to test a hypothesis (is the rate > 60%?) for a single population based on one sample. The data is categorical (participating/not participating). The historical rate of 60% provides the null hypothesis value. Distractor A is incorrect because the data are categorical, not quantitative. Distractor B is incorrect because there is only one sample; a second sample from a control group that did not see the campaign would be needed for a two-sample test. Distractor D is used for estimation, but the goal is to test for a significant increase.
- A school district wants to compare two different online learning platforms, A and B. A group of 80 volunteer teachers is randomly assigned, with 40 using Platform A and 40 using Platform B for a semester. At the end of the semester, an effectiveness score is calculated for each teacher. The district wants to determine if there is a statistically significant difference in the mean effectiveness scores between the two platforms. Which procedure is most appropriate?
Answer: A two-sample t-test for the difference in mean scores.
The correct procedure is a two-sample t-test for the difference in means. The design involves two independent groups of teachers created by random assignment. The goal is to test for a 'statistically significant difference' between the mean scores of these two groups. Distractor A is incorrect because the teachers in the two groups are independent, not paired. A paired design might involve each teacher using both platforms. Distractor C is incorrect because effectiveness scores are quantitative, not categorical, making a chi-square test inappropriate. Distractor D is incorrect because the primary goal is to test for significance, not to estimate the magnitude of the difference.
- A researcher conducts a two-sided, two-sample t-test and obtains a p-value of 0.03. If the researcher had instead constructed a confidence interval for the difference in means using the same data, which of the following would be true?
Answer: A 95% confidence interval would not contain the value 0.
There is a direct link between a two-sided hypothesis test and a confidence interval. If the p-value is less than the significance level α, the null hypothesis is rejected. A p-value of 0.03 is less than α = 0.05. Rejecting the null hypothesis H₀: μ₁ - μ₂ = 0 is equivalent to the corresponding confidence interval not containing 0. Therefore, a 95% confidence interval (corresponding to α = 0.05) would not contain 0. Distractor A is incorrect for this reason. Distractor C is incorrect because a p-value of 0.03 is greater than α = 0.01, so we would fail to reject the null hypothesis at the 1% level, meaning a 99% confidence interval would contain 0. Distractor D is incorrect because the midpoint of the interval is the sample difference, which is non-zero if the p-value is not 1.
- A political campaign wants to know if a recent TV advertisement was effective in a particular state. They polled a random sample of 500 voters before the ad aired and a different random sample of 550 voters after the ad aired. They want to determine if there is convincing statistical evidence that the proportion of voters who support their candidate has increased. Which procedure should be used?
Answer: A two-sample z-test for a difference in proportions.
The correct procedure is a two-sample z-test for a difference in proportions. The goal is to test a hypothesis ('has the proportion... increased?'). There are two independent samples of voters taken at different times. The variable of interest is categorical (support/do not support), so a procedure for proportions is required. Distractor B is incorrect because there are two samples being compared; the 'before' poll result is a sample statistic with its own variability, not a fixed hypothesized value. Distractor C is incorrect because the data is categorical (proportions), not quantitative (means), and the samples are independent, not paired. Distractor D is incorrect because an interval estimates the size of the difference, while the goal is to test for evidence of an increase.
- A city's transportation department wants to determine if the average commute time for its residents has changed from the 28.5 minutes recorded in a census five years ago. They collect data on commute times from a new random sample of 150 residents. Which statistical procedure is most appropriate for investigating whether there is evidence of a change?
Answer: A one-sample t-test for a mean, to determine if the sample mean is significantly different from 28.5 minutes.
The correct procedure is a one-sample t-test for a mean. The goal is to test a hypothesis ('has the average... changed?') about a single population mean against a known historical value (28.5 minutes). The data comes from one new sample. A t-test is used because the population standard deviation of commute times is unknown. Distractor B is incorrect because an interval estimates a value, while the goal here is to test for a change. Distractor C is incorrect because a two-sample test requires two independent samples (e.g., from two different cities or two different time periods), not one sample compared to a fixed number. Distractor D is incorrect because commute time is a quantitative variable, so a test for a mean is appropriate, not a proportion.
- To evaluate a new fertilizer, a botanist selects 20 plots of land. Each plot is divided in half. On one randomly chosen half, the new fertilizer is applied; on the other half, a standard fertilizer is used. After harvest, the corn yield from each half-plot is measured. The botanist wants to determine if there is strong evidence that the new fertilizer increases mean yield. Which procedure is most appropriate?
Answer: A paired t-test on the differences in yield calculated for each of the 20 plots.
The correct procedure is a paired t-test. The experimental design is a matched pairs design, where each plot of land serves as a block, and the two fertilizers are compared within that block. This pairing controls for variability between plots (e.g., soil quality, sunlight). The analysis should focus on the difference in yield within each plot. Distractor A is a very common error that fails to recognize the paired nature of the data, leading to a less powerful test. Distractor B is a one-sample procedure and doesn't involve a comparison. Distractor D is incorrect because yield is a quantitative variable, and categorizing it loses information.
- A high school counselor wants to know if there is a significant difference in the average GPA of students who participate in varsity sports versus those who do not. She collects random samples of GPAs from 50 student-athletes and 50 students not involved in sports. Which procedure is the most appropriate first step in her investigation?
Answer: A two-sample t-test for the difference in mean GPAs.
The correct procedure is a two-sample t-test for the difference in means. The counselor is comparing the mean GPAs (a quantitative variable) of two independent groups (athletes and non-athletes). The phrase 'is there a significant difference' indicates that a hypothesis test is the appropriate procedure. Distractor B is incorrect because the two groups of students are independent, not paired. Distractor C changes the question from comparing means to comparing proportions. Distractor D would estimate the size of the difference, but testing for significance is the more direct answer to the question asked.
- An e-commerce company A/B tests two website layouts. A random half of visitors see Layout A, and the other half see Layout B. The company wants to estimate the magnitude of the difference in the conversion rate (proportion of visitors making a purchase) between the two layouts to inform a business decision. What is the most appropriate procedure?
Answer: A two-sample z-interval for the difference in proportions to provide a range of plausible values for the effect size.
The correct procedure is a two-sample z-interval for the difference in proportions. The company has two independent groups (Layout A visitors, Layout B visitors) and is interested in a categorical outcome (purchase/no purchase). The goal is to 'estimate the magnitude of the difference,' which calls for a confidence interval. Distractors A and B are hypothesis tests that would determine if a significant difference exists but would not quantify its size. Distractor C is a one-sample procedure and would not provide a comparison between the two layouts.
- A university reports that 45% of its students are from out-of-state. A student newspaper journalist suspects this proportion is actually lower and surveys a simple random sample of 200 students to investigate. Which procedure is most appropriate for the journalist to use to determine if their suspicion is justified?
Answer: A one-sample z-test for a proportion.
The correct procedure is a one-sample z-test for a proportion. The journalist is testing a hypothesis ('is the proportion lower?') about a single population proportion based on one sample. The university's reported 45% serves as the hypothesized value under the null hypothesis. Distractor B is incorrect because there is only one sample being compared to a fixed value, not two samples being compared to each other. Distractor C is incorrect because the data is categorical (in-state/out-of-state), not quantitative. Distractor D is incorrect because the goal is to test a suspicion (a hypothesis test), not to estimate the proportion (a confidence interval).
- A water treatment plant manager needs to ensure the mean concentration of a chemical in the water is at the target level of 5.0 ppm. Any significant deviation requires action. Each day, she collects and analyzes 10 water samples. Which procedure is best suited for her daily check to see if the process is off-target?
Answer: A one-sample t-test for a mean to determine if there is evidence the mean concentration has deviated from 5.0 ppm.
The correct procedure is a one-sample t-test for a mean. The manager's goal is to test a hypothesis: whether the true mean concentration for that day is equal to the target of 5.0 ppm or if it has deviated. This involves comparing a single sample's mean to a known target value. Distractor A (an interval) would show the range of plausible values but doesn't provide the formal decision framework of a test. Distractor C is incorrect because the comparison is against a fixed target, not another sample. Distractor D is incorrect because the measurement is quantitative (concentration), not categorical.
- A manufacturer of a new 'Long Life' battery wishes to quantify its advantage over its 'Standard' battery. Researchers collect random samples of 50 of each battery type and measure their lifespans in hours. To provide a range of plausible values for how much longer the 'Long Life' battery lasts on average, which procedure should be used?
Answer: A two-sample t-interval for the difference in means.
The correct procedure is a two-sample t-interval for the difference in means. The goal is to estimate ('provide a range of plausible values') the difference between two populations. The data comes from two independent random samples (Standard and Long Life). The variable (lifespan) is quantitative. Distractor A describes a test, but the goal is estimation. Distractor B is incorrect because the samples are independent, not paired. Distractor C is incorrect because it would only estimate the mean lifespan of one battery type, not the difference between the two.
- To compare the cost of groceries at two supermarket chains, a researcher creates a list of 40 common grocery items. They purchase all 40 items at a store from Chain A and the same 40 items at a store from Chain B. The goal is to determine if there's sufficient evidence of a difference in the average cost per item between the chains. Which procedure is most appropriate?
Answer: A paired t-test on the differences in price for each of the 40 matched items.
The correct procedure is a paired t-test. The data are paired because the same 40 items were priced at both stores. The appropriate analysis is to calculate the price difference for each item and then perform a one-sample t-test on these differences to see if the mean difference is significantly different from zero. Distractor A is a common error; it fails to account for the paired design, which controls for variability among items. Distractor C is for categorical data, not quantitative prices. Distractor D is an interval for estimation, but the goal is to test for evidence of a difference.
- A coffee shop owner believes that customers buy more pastries on rainy days than on non-rainy days. Over a period of 60 days, she records daily pastry sales and whether the day was rainy or not. She has data from 18 rainy days and 42 non-rainy days. Which statistical procedure is most suitable for testing her belief?
Answer: A two-sample t-test for the difference in mean sales on rainy versus non-rainy days.
The correct procedure is a two-sample t-test for the difference in means. The data can be separated into two independent groups: sales on rainy days and sales on non-rainy days. The owner's belief can be formulated as a hypothesis comparing the mean sales of these two groups. Distractor A is incorrect because the days are not paired; they are independent observations classified into two categories. Distractor B is incorrect because it involves only one group and doesn't make the comparison central to the owner's belief. Distractor C is plausible but less direct; the data is categorical (rainy/not rainy), not quantitative (amount of rainfall), making the two-sample test the most direct approach.
- A pharmaceutical company is developing a new drug to lower cholesterol. Researchers enroll 100 subjects with high cholesterol and measure each subject's level. After one month of treatment with the new drug, they measure each subject's cholesterol level again. To quantify the drug's effect, they want to provide a range of plausible values for the average reduction in cholesterol. What is the most appropriate procedure?
Answer: A one-sample t-interval for the mean difference, calculated from the paired 'before' and 'after' measurements for each subject.
The correct procedure is a one-sample t-interval for the mean difference. The data are paired because there are two measurements (before and after) for each subject. The analysis should be performed on the differences. The goal is to estimate ('provide a range of plausible values'), which calls for a confidence interval. Distractor A is incorrect because it treats the data as two independent samples, ignoring the paired nature of the design. Distractors B and D are incorrect because they describe hypothesis tests, whereas the goal is estimation.
- Researchers are investigating a potential association between regular exercise and resting heart rate. They recruit volunteers and classify them as 'regular exercisers' or 'non-exercisers'. Then they measure the resting heart rate of each volunteer. Which procedure is most appropriate for determining if the mean resting heart rate is different for the two groups?
Answer: A two-sample t-test for the difference in mean heart rate.
The correct procedure is a two-sample t-test for the difference in means. The study compares two independent groups ('regular exercisers' and 'non-exercisers') on a quantitative variable (resting heart rate). The goal is to test for a difference in the means of these two groups. Distractor A is incorrect because the groups are independent, not paired. Distractor C is for two categorical variables; heart rate is quantitative. Distractor D is less appropriate because exercise status is a binary categorical variable, not a quantitative one suitable for correlation.
