Praxis Math Quiz: Draw Statistical Inferences
20 questions · exam conditions
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Draw Statistical InferencesQuestion 1 of 20

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?

Exactly 80,000 residents.
Approximately 80,000 residents.
At least 80,000 residents.
Significantly more than 80,000 residents.
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Praxis Math Quiz

Praxis Math Quiz: Draw Statistical Inferences

Practice Draw Statistical Inferences in Praxis Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Draw Statistical Inferences, giving you a quick way to practice the rules, question types, and explanations that matter most for Praxis Math.

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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.

All questions

Question 1

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?

  1. Exactly 80,000 residents.
  2. Approximately 80,000 residents. (correct answer)
  3. At least 80,000 residents.
  4. Significantly more than 80,000 residents.
Explanation: 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.

Question 2

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?

  1. The sample of 2,000 was not large enough to begin with.
  2. The 400 respondents may not be representative of the original 2,000 employees selected. (correct answer)
  3. The percentage of satisfied employees (75%) is too high to be believable.
  4. A random sample cannot be used to measure a subjective concept like job satisfaction.
Explanation: 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.

Question 3

A study conducted in January on a random sample of residents in a northern city showed that 90% were concerned about snow removal services. A local official inferred that the vast majority of residents are concerned about snow removal year-round. What is a potential flaw in this inference?

  1. The inference is flawed because people's opinions can change over time and may be influenced by the season. (correct answer)
  2. The sample was not random, so no inference can be made about the city's residents.
  3. A sample percentage of 90% is too high, suggesting the data collection was biased.
  4. The official should have surveyed residents in a southern city to compare the results.
Explanation: The survey was conducted at a specific point in time (January) when snow is a current issue. Generalizing these results to the entire year is problematic. Residents' concern about snow removal is likely much higher in the middle of winter than it would be in July. The inference fails to account for this temporal or seasonal effect. The stem states the sample was random (B). A high percentage does not automatically mean bias (C). Comparing to another city is irrelevant to the flaw in the inference about this specific city (D).

Question 4

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?

  1. The average high school student in the state spends exactly 8.5 hours on homework per week.
  2. 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. (correct answer)
  3. All high school students in the state spend between 8 and 9 hours on homework per week.
  4. If another random sample of 500 students were taken, the average would also be 8.5 hours.
Explanation: 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.

Question 5

Two different polling organizations conducted random surveys to estimate the proportion of voters who favor Candidate A. Poll X surveyed 400 randomly selected voters. Poll Y surveyed 1,600 randomly selected voters. Both polls used the same methodology. Which statement best describes the expected difference between the two polls' results?

  1. The estimate from Poll Y is more likely to be closer to the true proportion of all voters. (correct answer)
  2. The estimate from Poll X is more likely to be accurate because the smaller sample is easier to manage.
  3. Both polls will produce the exact same estimate because they used the same methodology.
  4. The results of both polls are equally reliable, as both used random sampling.
Explanation: Assuming both are random samples, a larger sample size generally leads to less sampling variability and a more precise estimate of the population parameter. Therefore, the estimate from Poll Y (with 1,600 voters) is more likely to be closer to the actual population proportion than the estimate from Poll X (with 400 voters). Choice B is incorrect. Choice C is false due to sampling variability. Choice D is incorrect because, while both are random, the larger sample size provides greater reliability.

Question 6

A survey of a random sample of 1,200 homeowners in a state found that 65% had a vegetable garden. The survey report states that the margin of error is ±3%. Which of the following is a correct interpretation of this result?

  1. It is certain that between 62% and 68% of all homeowners in the state have a vegetable garden.
  2. It is plausible that the true percentage of homeowners in the state with a vegetable garden is between 62% and 68%. (correct answer)
  3. If another survey were conducted, the result would fall somewhere between 62% and 68%.
  4. Exactly 65% of the 1,200 homeowners surveyed have a vegetable garden, and 3% do not.
Explanation: A margin of error creates a confidence interval, which is a range of plausible values for the true population parameter. The interval is 65% ± 3%, or 62% to 68%. However, this is a probabilistic statement, not a certainty. Choice B uses the correct language ('plausible'). Choice A is incorrect because we can never be 'certain'. Choice C is a misinterpretation; another sample would have its own result and its own margin of error. Choice D misinterprets the meaning of the 3% margin of error.

Question 7

To test a new fertilizer, a farmer divides a large field into 20 plots. He randomly assigns 10 plots to receive the new fertilizer and 10 plots to receive the standard fertilizer. He finds that the plots with the new fertilizer yielded, on average, 15% more corn. What is the most appropriate inference?

  1. The new fertilizer will cause a 15% increase in yield for any crop, on any farm.
  2. There is evidence that the new fertilizer causes a higher yield for this type of corn on this specific farm. (correct answer)
  3. There is an association, but not a causal link, between the new fertilizer and higher yield.
  4. The results are invalid because the study was only conducted on one farm.
Explanation: This is a randomized controlled experiment. The random assignment of treatments (fertilizers) to plots allows for a causal inference to be made. However, the experiment was conducted on one specific farm with one type of corn. Therefore, the inference of causation is limited to that population and context. Choice B correctly identifies the causal link and its limited scope. Choice A generalizes too broadly. Choice C is incorrect because random assignment allows for a causal conclusion. Choice D is too strong; the results are valid for the context in which they were produced.

Question 8

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?

  1. The sample size is too small for a large city, so the estimate will be unreliable.
  2. The voter registration list may exclude certain residents, like non-citizens or those who are not registered to vote. (correct answer)
  3. People are likely to exaggerate their commute times, leading to an artificially high average.
  4. The estimate is not biased because the sample was selected randomly from the list.
Explanation: 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.

Question 9

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?

  1. It has no effect on the inference because the sample was selected randomly.
  2. It makes the sample mean a likely overestimation of the true mean income of all households in the suburb. (correct answer)
  3. It makes the sample mean a likely underestimation of the true mean income of all households in the suburb.
  4. It invalidates the entire study, and no inference can be made about the population.
Explanation: 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.

Question 10

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?

  1. The data is from a sample, so no conclusions can be made about the relationship.
  2. A confounding variable, such as population size or warmer weather, is likely responsible for the association. (correct answer)
  3. The researcher should have used a larger sample of cities to verify the positive correlation.
  4. The correlation is negative, not positive, as ice cream is associated with positive events.
Explanation: 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.

Question 11

To estimate the average number of books read per year by adults in a town, a researcher randomly surveys 100 people at the town's public library. The average from the sample is 24 books. The researcher concludes that the average adult in the town reads about 24 books per year. Why might this inference be inaccurate?

  1. The sample size of 100 is not large enough to make any valid estimates for the entire town.
  2. The sample, although random, was drawn from a location that might overrepresent people who read frequently. (correct answer)
  3. The researcher should have calculated the median instead of the average to draw a valid conclusion.
  4. The data is invalid because some people might not remember exactly how many books they read.
Explanation: The sampling frame is biased. By surveying people at the library, the researcher is likely to get a sample that includes more avid readers than the general population of the town. This is a form of undercoverage bias, where parts of the population (in this case, non-library-goers) are not represented. Therefore, the sample average of 24 is likely an overestimate for the entire town. While A, C, and D mention potential issues, the most significant flaw affecting the inference is the biased sampling location.

Question 12

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?

  1. The sample size of 5,000 is too small to make a conclusion about the entire city's population.
  2. The poll results are invalid because the percentage in favor is unusually high for a political issue.
  3. The sample is not random and is likely biased because it consists of self-selected volunteers with strong opinions. (correct answer)
  4. The question is poorly worded and may have confused the respondents about the nature of the ban.
Explanation: 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.

Question 13

A high school principal wants to gauge student opinion on a new cafeteria menu. She surveys the first 50 students who enter the library on a Monday morning. She finds that 80% of them are in favor of the new menu. She infers that most students at the school like the new menu. What is the most significant flaw in this inference?

  1. The sample size of 50 is too small to represent a high school of 1,000 students.
  2. The survey was conducted on a Monday, which may not be a typical day for students.
  3. The sampling method is not random; students in the library may not be representative of all students. (correct answer)
  4. The survey question was likely biased in favor of the new menu, leading to a high approval rate.
Explanation: This is a convenience sample, not a random sample. Students who are in the library in the morning might be different from the general student population (e.g., more studious, less likely to eat breakfast in the cafeteria). This lack of random selection means the sample is likely biased, and the results cannot be reliably generalized to all students. While the sample size (A) could be larger, the primary flaw is the sampling method. B and D are possible issues, but C describes the most certain and significant flaw.

Question 14

A polling company wants to ensure its random sample of 1,000 adults is representative of the state's population in terms of age. They divide the state's population into age groups (18-29, 30-49, 50-64, 65+) and then draw random samples from each group in proportion to their size in the population. What is the primary benefit of this sampling method?

  1. It guarantees that the survey's conclusions will be 100% accurate for the entire state.
  2. It allows the company to make causal claims about the relationship between age and opinions.
  3. It reduces the required sample size needed to achieve the desired level of precision.
  4. It ensures the sample accurately reflects the age distribution of the population, improving the reliability of inferences. (correct answer)
Explanation: This method is called stratified random sampling. Its primary purpose is to ensure that key subgroups of a population are properly represented in the sample. By forcing the sample's age distribution to match the population's, it prevents the possibility that a simple random sample would, by chance, over- or under-represent an age group, thus leading to more reliable and representative results for the overall population.

Question 15

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?

  1. People who are retired, as they may not answer calls from unknown numbers.
  2. Younger people and lower-income individuals, who are less likely to have landline phones. (correct answer)
  3. Supporters of the opposing candidate, who might refuse to participate in the survey.
  4. People living in rural areas, where landline service may be less reliable.
Explanation: 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.

Question 16

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?

  1. All third graders in the country, as the sample size is sufficiently large for national generalization.
  2. All students in the large urban school district from which the sample was drawn.
  3. Only the 100 students who participated in the study, as results cannot be generalized.
  4. All third graders in the large urban school district from which the sample was drawn. (correct answer)
Explanation: 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.

Question 17

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?

  1. The question is too long for a typical survey, and respondents may not read it carefully.
  2. The question contains leading language that is likely to influence respondents to answer in favor of the tax. (correct answer)
  3. The survey should have been administered to a random sample of children, not adults.
  4. The topic of taxes is too complex to be accurately measured in a single survey question.
Explanation: 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.

Question 18

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?

  1. Owning a pet causes a reduction in blood pressure for adults in this city.
  2. There is an association between pet ownership and lower blood pressure among adults in this city. (correct answer)
  3. All adults in the city who own pets have lower blood pressure than all adults who do not.
  4. To lower their blood pressure, all adults in this city should be advised to get a pet.
Explanation: 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.

Question 19

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?

  1. The food service at University A is definitively better than the food service at University B.
  2. This study provides evidence that student satisfaction with dining is likely higher at University A than at University B. (correct answer)
  3. Any student who transfers from University B to University A will be more satisfied with the campus dining.
  4. The difference in satisfaction is caused by the higher tuition costs at University A.
Explanation: 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.

Question 20

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?

  1. The average rating of 7.8 is statistically high enough.
  2. The sample was taken at a single local theater, not a representative selection of theaters nationwide. (correct answer)
  3. The sample size of 200 is large enough for a valid national conclusion.
  4. The rating scale of 1 to 10 is a universally understood measurement.
Explanation: 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.