- A city transportation department wants to estimate the average daily commute time for its residents. They randomly select 500 residents from the city's voter registration list and ask for their commute time. Why might the resulting estimate be biased?
Answer: The voter registration list may exclude certain residents, like non-citizens or those who are not registered to vote.
The sampling frame (the list from which the sample is drawn) is the voter registration list. This list is not a complete list of all residents. It excludes non-citizens, residents who are not eligible to vote (e.g., due to age or felony convictions), and eligible residents who have not registered. These groups may have different commuting patterns, leading to a biased sample that is not representative of all city residents. This is a form of undercoverage bias. While A and C are potential issues, B describes a systematic flaw in the sampling frame itself.
- A state's Department of Education wants to estimate the average number of hours high school students spend on homework per week. They randomly select 500 students from a list of all enrolled high school students in the state. The sample average is found to be 8.5 hours. Which of the following is the most valid inference from this study?
Answer: It is reasonable to estimate that the average weekly homework time for all high school students in the state is close to 8.5 hours.
A random sample allows for generalization to the population from which it was drawn. However, a sample statistic (8.5 hours) is an estimate of the population parameter, not an exact value. Therefore, it's reasonable to infer that the population average is 'close to' 8.5 hours. Choice A is too certain. Choice C makes an incorrect assumption about the range of individual student data. Choice D is incorrect because sampling variability means a different sample would likely yield a different average.
- A researcher randomly surveyed 100 students from University A and 100 students from University B about their satisfaction with campus dining. The proportion of satisfied students was higher at University A than at University B. Which statement is the most valid statistical inference?
Answer: This study provides evidence that student satisfaction with dining is likely higher at University A than at University B.
Since the samples are random, the results can be used to make an inference about the populations of the two universities. However, the conclusion must be stated with a degree of uncertainty. Choice B uses appropriate language ('provides evidence,' 'likely higher'). Choice A is too certain ('definitively better'). Choice C makes a prediction about an individual, which is not appropriate. Choice D suggests a cause (tuition costs) that is not supported by the data provided.
- A study found that in a random sample of adults from a certain city, those who owned pets had lower average blood pressure than those who did not. Which conclusion can be properly drawn from this data?
Answer: There is an association between pet ownership and lower blood pressure among adults in this city.
This was an observational study based on a sample, not a controlled experiment. It can show an association or correlation between two variables, but it cannot prove causation. Therefore, concluding that owning a pet causes lower blood pressure (A and D) is not justified. There could be other factors (confounding variables) at play. Choice C is an overgeneralization; the study is about averages, not every individual.
- A quality-control inspector at a factory selects a simple random sample of 200 light bulbs from a single production run of 50,000 bulbs. In the sample, 9 bulbs are defective. Based on the table below, which statement represents the most defensible inference about the entire production run?
Answer: Approximately 4.5% of the bulbs in the production run are defective, and this estimate applies only to this production run.
A random sample supports an inference to the population from which it was drawn — in this case, the single production run of 50,000 bulbs. The point estimate is 9/200 = 4.5%, but it is an estimate, not an exact count, so (A) is too strong. (C) overgeneralizes beyond the population sampled (only one run was sampled). (D) is a common misconception: inferential validity depends on random sampling, not on the sample being a large percentage of the population.
- Four independent researchers each drew a simple random sample from the same population of 10,000 voters and computed the sample proportion supporting a ballot measure. Their results are shown in the table. Assuming all samples were properly drawn, which researcher's estimate should be considered the most reliable for inferring the true population proportion?
Answer: Researcher 3
With simple random samples, the reliability of a proportion estimate increases as the sample size increases (the standard error decreases as 1/√n). Researcher 3 used the largest sample (n = 1,200), so the estimate is the most precise. The value of the sample proportion itself does not affect reliability; only sample size does. Choices A, B, and D correspond to smaller samples and therefore larger expected sampling error.
- A researcher surveyed a simple random sample of 400 adults in City X about weekly reading hours. The results are summarized in the table shown. The population of City X is 120,000 adults. Which of the following is the best inference the researcher can make?
Answer: An estimated 30% of adults in City X read 0–2 hours per week, with some sampling variability expected.
Random samples yield estimates of population parameters, accompanied by sampling variability. The sample proportion 120/400 = 30% is an estimate, so (C) is correct. (A) and (B) treat sample statistics as exact population values. (D) misstates the purpose of inferential statistics — well-designed random samples of this size can support inferences to much larger populations.
- A quality control inspector at a factory randomly selects 200 smartphones and finds that 6 have a defective screen. Based on this sample, which of the following is the most appropriate conclusion?
Answer: The sample data provides evidence to estimate that around 3% of the factory's smartphones are defective.
The sample proportion of defects is 6/200 = 0.03, or 3%. This is a point estimate for the true population proportion. An inference should be worded with some uncertainty. Choice C correctly uses this estimate and appropriate language ('evidence to estimate that around 3%'). Choice A is too certain. Choice B provides a range but states it with certainty, which is not possible from a sample. Choice D ignores sampling variability.
- A survey question asks, "Given the documented negative effects of sugary drinks on public health, do you support a small tax on these beverages to fund children's health initiatives?" Why might the data collected from this question not be reliable for inferring public opinion?
Answer: The question contains leading language that is likely to influence respondents to answer in favor of the tax.
This is an example of a leading question. By priming the respondent with phrases like 'documented negative effects' and 'fund children's health initiatives,' the question frames the tax in a positive light and encourages a 'yes' answer. This wording introduces bias and means the results will likely not reflect the true, uninfluenced opinion of the population. The other options are less significant flaws compared to the biased wording of the question.
- A researcher sends a survey to 2,000 randomly selected employees at a large company. Only 400 employees complete and return the survey. The researcher finds that 75% of the respondents are satisfied with their job. Why is it difficult to make a valid inference about the job satisfaction of all employees?
Answer: The 400 respondents may not be representative of the original 2,000 employees selected.
This situation describes a potential nonresponse bias. The low response rate (400 out of 2,000, or 20%) is a major concern. The employees who chose to respond might be systematically different from those who did not. For example, employees who are very satisfied or very dissatisfied might be more likely to respond, skewing the results. Therefore, the 400 respondents may not be representative of the entire workforce, or even of the 2,000 who were initially sampled.
- A random sample of 200 moviegoers at a local theater was asked to rate a new movie on a scale of 1 to 10. The average rating was 7.8. Which of the following is the most important factor in determining if this result can be generalized to all moviegoers in the country?
Answer: The sample was taken at a single local theater, not a representative selection of theaters nationwide.
For an inference to be generalized to a large population (all moviegoers in the country), the sample must be representative of that population. A sample taken from a single theater in one location is unlikely to be representative of the entire country's moviegoers. Tastes and demographics can vary significantly by region. Therefore, the limited scope of the sampling location is the most critical barrier to generalization. The sample size (C) is less important if the sampling method is flawed.
- A random sample of 30 households in a suburb reveals a mean annual income of $95,000. However, the sample includes one household with an income of $1,500,000. How does this one high-income household likely affect the inference about the suburb's mean household income?
Answer: It makes the sample mean a likely overestimation of the true mean income of all households in the suburb.
The mean is sensitive to extreme values (outliers). In this case, the very high income of one household will pull the sample mean upwards. Therefore, the sample mean of $95,000 is likely higher than the actual mean of the entire population of households in the suburb. The outlier causes the sample mean to be a potentially biased estimator of the population mean. It does not invalidate the study (D), but it does suggest that the median might have been a better measure of center.
- A researcher is studying the effectiveness of a new reading program for third graders. A random sample of 100 third graders from a large urban school district is selected. After a semester in the program, the sample shows a statistically significant improvement in reading scores. To which population can these results be most reliably generalized?
Answer: All third graders in the large urban school district from which the sample was drawn.
Inferences from a sample are most reliably generalized to the population from which the sample was randomly drawn. In this case, the sample was drawn from third graders in a specific large urban school district. Generalizing to all students in the district (B) or all third graders in the country (A) is too broad. Stating the results cannot be generalized at all (C) is incorrect because a random sample was used.
- A survey of a random sample of 500 city residents found that 40% use public transportation at least once a week. If the city's population is 200,000, what is the most reasonable estimate for the total number of residents who use public transportation at least once a week?
Answer: Approximately 80,000 residents.
The best estimate from the sample is that 40% of the population has the characteristic. We can apply this percentage to the total population: 40% of 200,000 is 0.40 * 200,000 = 80,000. However, since this is based on a sample, it is an estimate, not an exact number. Therefore, 'approximately 80,000' is the most appropriate phrasing. 'Exactly' is too certain. 'At least' and 'significantly more than' are not supported by the point estimate from the sample.
- A researcher analyzed data from a random sample of cities and found a positive correlation between the number of ice cream shops and the number of crimes. The researcher concluded that the presence of ice cream shops leads to an increase in crime. Why is this conclusion likely flawed?
Answer: A confounding variable, such as population size or warmer weather, is likely responsible for the association.
This is a classic example of confusing correlation with causation. A third, unobserved variable (a confounding variable) is likely influencing both variables. Larger cities tend to have both more ice cream shops and more crime. Similarly, crime rates and ice cream sales both tend to rise in warmer weather. The ice cream shops are not causing the crime; they are both associated with another factor. This is an observational study, so causal claims cannot be made.
- A city park ranger wants to estimate the proportion of visitors who use the park's east entrance. The ranger considers four sampling plans, described in the table. Which plan is most likely to yield a random sample that supports a valid inference about all park visitors on a typical day?
Answer: Plan D
Plan D randomizes both the day and the times, spreading the sample across conditions in a way approximating a random sample of visitors on a typical day. Plan A (only Saturday morning at the east entrance) introduces location and time bias guaranteed to overestimate east-entrance use. Plan B samples only at one entrance, excluding other visitors. Plan C (volunteers) is a voluntary-response sample, not random. Only (D) approximates proper random sampling of the target population.
- A researcher takes a simple random sample of 50 households from a neighborhood of 2,000 households and records monthly water use (in gallons). Summary statistics from the sample are shown. Which of the following inferences is MOST appropriate for the neighborhood population?
Answer: The mean monthly water use for all 2,000 households is likely close to 3,420 gallons, but the true mean could differ due to sampling variability.
The sample mean is a point estimate of the population mean; the correct interpretation is that the population mean is likely near the estimate but subject to sampling variability — statement (B). (A) treats the estimate as exact. (C) misunderstands standard deviation — it describes spread of individuals, not quality of the sample mean as an estimate. (D) incorrectly assumes sample range bounds the population range.
- A telephone survey was conducted to estimate support for a political candidate. Calls were made to randomly selected landline telephone numbers. The results showed 58% support for the candidate. Which group of people was most likely underrepresented in this survey, potentially biasing the results?
Answer: Younger people and lower-income individuals, who are less likely to have landline phones.
This survey method suffers from undercoverage bias. In recent years, many people, particularly younger and lower-income individuals, have given up landlines in favor of only using cell phones. By sampling only from landline numbers, the survey is likely to miss a significant portion of these demographics, making the sample unrepresentative of the entire adult population. The other options describe potential nonresponse biases but B describes the most systematic and well-known flaw of landline-only polling.
- A high school has 1,600 students. A researcher obtains a simple random sample of 100 students and records whether each owns a bicycle. The results are shown in the bar graph. Which statement is the most appropriate inference for the population of 1,600 students?
Answer: Approximately 65% of students in the school own a bicycle; the true percentage is likely close to, but not exactly, 65%.
The sample proportion 65/100 = 65% is a point estimate for the school's population with expected sampling variability — this is the proper inference. (A) treats the estimate as exact. (C) generalizes beyond the sampled population to other schools, which is not supported. (D) is incorrect; a random sample of 100 is generally sufficient to estimate a proportion within a reasonable margin of error.
- A news website posts a poll asking readers, "Do you support the proposed city-wide ban on plastic bags?" Out of 5,000 responses, 85% voted in favor of the ban. The website concludes that the vast majority of city residents support the ban. Why is this conclusion likely not valid?
Answer: The sample is not random and is likely biased because it consists of self-selected volunteers with strong opinions.
This is a voluntary response sample. People who choose to participate in such polls often have strong feelings about the issue, making them unrepresentative of the general population. This self-selection bias makes the inference invalid. Choice A is incorrect; 5,000 is a very large sample size. The problem is the sampling method, not the size. Choice B is an opinion and not a statistical reason. Choice D is a possibility, but the most significant and certain flaw is the sampling method.
