Business Statistics Quiz: Cis For Means And Proportions
20 questions · exam conditions
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Cis For Means And ProportionsQuestion 1 of 20

A pharmaceutical company tests a new drug and finds that 78 out of 200 patients show improvement. They construct a 95% confidence interval for the proportion of patients who improve: (0.325, 0.455). The company's CEO announces: "We are 95% certain that the drug helps between 32.5% and 45.5% of patients, so we should proceed with FDA approval since over 30% benefit." What statistical issue should be raised with this interpretation?

The confidence level is too low; FDA approval requires 99% confidence intervals.
The sample size of 200 is too small to make reliable inferences about drug effectiveness.
The interpretation correctly describes the interval, but 'improvement' may not equal 'drug effectiveness' without a control group.
The CEO incorrectly calculated the sample proportion; 78 out of 200 is 39%, not within the stated interval.
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Business Statistics Quiz

Business Statistics Quiz: Cis For Means And Proportions

Practice Cis For Means And Proportions in Business Statistics 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 Cis For Means And Proportions, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Statistics.

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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 pharmaceutical company tests a new drug and finds that 78 out of 200 patients show improvement. They construct a 95% confidence interval for the proportion of patients who improve: (0.325, 0.455). The company's CEO announces: "We are 95% certain that the drug helps between 32.5% and 45.5% of patients, so we should proceed with FDA approval since over 30% benefit." What statistical issue should be raised with this interpretation?

  1. The confidence level is too low; FDA approval requires 99% confidence intervals.
  2. The sample size of 200 is too small to make reliable inferences about drug effectiveness.
  3. The interpretation correctly describes the interval, but 'improvement' may not equal 'drug effectiveness' without a control group. (correct answer)
  4. The CEO incorrectly calculated the sample proportion; 78 out of 200 is 39%, not within the stated interval.
Explanation: Choice C is correct. While the CEO's interpretation of the confidence interval is statistically accurate, there's a crucial distinction between 'improvement' and 'drug effectiveness.' Without a control group, we don't know how many patients would have improved without the drug. The confidence interval estimates the proportion who improved, but this could include natural recovery, placebo effects, or other factors unrelated to drug efficacy. Choice A is wrong because there's no universal FDA requirement for 99% intervals. Choice B is incorrect because 200 is generally adequate for proportion estimation. Choice D is wrong because 78/200 = 0.39, which falls within the stated interval (0.325, 0.455).

Question 2

An environmental agency measures air pollution levels and constructs a 95% confidence interval for the mean daily pollution index: (45.2, 52.8). A city councilman states: "This interval is too wide to be useful for policy decisions. If we increased our sample size by a factor of 4, the new interval would be (47.0, 51.0), which is much more precise." Assuming the councilman's calculation is correct, what can be concluded about his reasoning?

  1. The reasoning is sound; larger samples always produce more useful confidence intervals for policy decisions.
  2. The conclusion is premature because environmental data often violates normality assumptions required for confidence intervals.
  3. The reasoning is flawed because increasing sample size by a factor of 4 should reduce the margin of error by a factor of 4, not 2.
  4. The calculation demonstrates correct understanding of how sample size affects interval width, but 'usefulness' depends on the specific policy context. (correct answer)
Explanation: When you encounter confidence interval questions involving sample size changes, focus on the mathematical relationship between sample size and margin of error, but also consider the broader context of statistical interpretation. Let's examine the councilman's calculation. The original interval (45.2, 52.8) has a margin of error of 3.8, while his proposed interval (47.0, 51.0) has a margin of error of 2.0. When sample size increases by a factor of 4, the margin of error should decrease by a factor of 4=2\sqrt{4} = 2. Indeed, 3.8 ÷ 2 = 1.9 ≈ 2.0, so his calculation is mathematically correct. However, the key insight is distinguishing between statistical precision and practical usefulness. The councilman correctly understands how sample size affects interval width, but his claim about "usefulness for policy decisions" requires context-specific judgment that statistics alone cannot provide. Answer A is wrong because it makes an absolute claim about usefulness without considering policy context. Answer B incorrectly focuses on normality assumptions, which isn't the central issue here—the calculation appears valid regardless. Answer C is wrong because it misunderstands the relationship: increasing sample size by 4 should reduce margin of error by 4=2\sqrt{4} = 2, not by 4 itself. Answer D correctly identifies that while the statistical calculation is sound, determining what constitutes "useful precision" depends on the specific policy decisions being made—some policies might require even tighter intervals, while others might find the original interval adequate. Remember: statistical precision and practical usefulness are related but distinct concepts. Always consider the real-world application when evaluating confidence interval "adequacy."

Question 3

A statistician constructs a 99% confidence interval for the mean weight of packages shipped by a company and obtains (2.3 kg, 2.7 kg). The company's quality control manager asks: "If we collected 100 more samples of the same size and constructed 100 more 99% confidence intervals, how many would contain the true mean weight?" What is the best response?

  1. Exactly 99 intervals would contain the true mean weight.
  2. It depends on whether the current interval (2.3 kg, 2.7 kg) actually contains the true mean weight.
  3. Between 95 and 100 intervals would contain the true mean weight.
  4. We expect about 99 intervals to contain the true mean weight, but the actual number could vary. (correct answer)
Explanation: When you encounter questions about confidence intervals and repeated sampling, focus on understanding what confidence level actually means in the long run versus what happens in any single instance. A 99% confidence interval means that if you repeated the sampling process many times and constructed a confidence interval each time, about 99% of those intervals would contain the true population mean. This is a long-run frequency interpretation—it describes the behavior of the method over many repetitions, not a guarantee for any specific set of intervals. Answer D correctly captures this concept. While we expect approximately 99 out of 100 intervals to contain the true mean, the actual number will vary due to random sampling variation. You might get 97, 98, 100, or even 101 intervals containing the true mean in any particular set of 100 intervals. Answer A is wrong because it suggests certainty—exactly 99 intervals would always contain the true mean. Random sampling doesn't work with such precision. Answer B incorrectly suggests that the performance of future intervals depends on whether the current interval contains the true mean. Each interval's success or failure is independent of others. Answer C is too restrictive in its range. While 95-100 is plausible, the actual number could reasonably fall outside this range due to sampling variability. Remember: confidence level describes the long-term success rate of the method, not a precise prediction for any finite number of intervals. Always think "approximately" or "about" when interpreting confidence levels in repeated sampling scenarios.

Question 4

A university administrator constructs 90% confidence intervals for the mean GPA in four different colleges within the university. The results are: Engineering (2.95, 3.15), Business (3.05, 3.25), Liberal Arts (3.10, 3.35), and Sciences (3.20, 3.40). A faculty member concludes: "Since we used 90% confidence intervals, we can be 90% confident that all four true college mean GPAs fall within their respective intervals simultaneously." What is wrong with this reasoning?

  1. Individual 90% confidence intervals cannot be combined; the overall confidence level for all four intervals is much lower than 90%. (correct answer)
  2. The sample sizes for the four colleges are likely different, making the confidence intervals incomparable.
  3. 90% confidence intervals are inappropriate for comparing multiple groups; ANOVA should be used instead.
  4. The confidence intervals overlap too much to draw meaningful conclusions about differences between colleges.
Explanation: Choice A is correct. When multiple confidence intervals are constructed independently, the overall confidence that ALL intervals simultaneously contain their respective true parameters is much lower than the individual confidence level. If each interval has 90% confidence, the probability that all four contain their true parameters is approximately (0.90)^4 = 65.6%, assuming independence. Choice B is irrelevant to the simultaneous confidence issue. Choice C is wrong because confidence intervals are appropriate for estimation; the issue is about simultaneous confidence, not methodology. Choice D addresses interval overlap but misses the main statistical error about simultaneous confidence levels.

Question 5

A retail chain wants to estimate the proportion of customers who use mobile payment methods. They survey customers at 5 locations and find the following 95% confidence intervals: Store A (0.23, 0.41), Store B (0.31, 0.49), Store C (0.28, 0.46), Store D (0.35, 0.53), Store E (0.40, 0.58). The marketing director concludes: "Since all intervals overlap with the range 0.35-0.41, we can be confident that the true proportion for each store is between 35% and 41%." What is the flaw in this reasoning?

  1. The director is incorrectly assuming that overlapping confidence intervals indicate similar population parameters.
  2. The director should average all the intervals to get a single estimate rather than looking for overlap regions.
  3. The director is confusing the intersection of intervals with the confidence level for individual parameters. (correct answer)
  4. Store locations are not independent, so confidence intervals cannot be compared across different stores.
Explanation: Choice C is correct. The director is incorrectly interpreting the intersection of confidence intervals (0.35-0.41) as a region where we can be confident about each individual store's true proportion. However, each confidence interval provides information about its own store's parameter, and the intersection doesn't inherit the 95% confidence level. The true proportion for any individual store could be anywhere within its own interval, not necessarily in the intersection. Choice A is partially related but doesn't capture the specific error about confidence levels. Choice B suggests an inappropriate averaging method. Choice D incorrectly assumes store locations can't be compared independently.

Question 6

Two researchers independently construct confidence intervals for the same population mean using different samples. Researcher A obtains a 90% CI of (23.1, 26.9) and Researcher B obtains a 90% CI of (24.8, 28.2). A colleague concludes that Researcher B's interval is "more accurate" because it's entirely above 24, while A's interval includes values below 24, and "we know the true mean must be at least 24 from prior studies." What is the primary flaw in this reasoning?

  1. The colleague incorrectly assumes that wider confidence intervals are always less accurate than narrower ones.
  2. The colleague is using external information not reflected in the confidence interval procedure to judge accuracy. (correct answer)
  3. 90% confidence intervals are too low; 95% intervals should be used when comparing different researchers' results.
  4. The colleague should average the two intervals to get the most accurate estimate of the population mean.
Explanation: Choice B is correct. The colleague is imposing external knowledge ("the true mean must be at least 24") to evaluate which interval is "more accurate." However, confidence intervals are evaluated based on their construction method and the data used, not on whether they align with prior beliefs. Each interval is valid based on its own sample. Choice A is incorrect because the colleague isn't comparing interval widths. Choice C is wrong because the confidence level doesn't affect the validity of comparing results. Choice D is incorrect because averaging intervals from different samples isn't a standard statistical procedure and doesn't improve accuracy.

Question 7

A 95% confidence interval for the mean monthly rent of a one-bedroom apartment in a large city is calculated to be ($1,850, $2,050). Which of the following statements is the most accurate interpretation of this interval?

  1. There is a 95% probability that the true mean monthly rent for all one-bedroom apartments in this city is between $1,850 and $2,050.
  2. If we were to repeat the sampling process many times, approximately 95% of the confidence intervals constructed would contain the true mean monthly rent. (correct answer)
  3. Approximately 95% of the one-bedroom apartments in the city have a monthly rent between $1,850 and $2,050.
  4. If a new sample of apartments is taken, there is a 95% probability that its sample mean rent will be between $1,850 and $2,050.
Explanation: The correct interpretation of a 95% confidence interval relates to the reliability of the estimation method. It means that if the same sampling procedure were repeated many times, we would expect 95% of the resulting intervals to capture the true, unknown population parameter (in this case, the mean rent). The confidence is in the method, not in any single interval. A is incorrect because it assigns a probability to the true parameter. The true mean is a fixed value, not a random variable, so it doesn't make sense to talk about the probability of it being in a certain range. C is incorrect because a confidence interval is for a population parameter (the mean), not for the distribution of individual data points. D is incorrect because the interval is an estimate for the population mean, not a prediction for a future sample mean.

Question 8

An e-commerce company tests two different website layouts, A and B, to see which yields a higher mean customer session duration. Independent random samples are used. The 95% confidence interval for the mean session duration for layout A is (120, 150) seconds, and for layout B is (140, 180) seconds. What is the most appropriate conclusion?

  1. Layout B is definitively superior because its interval contains higher values than Layout A's interval.
  2. There is no statistically significant difference between the mean session durations for the two layouts because the intervals overlap.
  3. Layout A is superior because its interval is narrower, indicating a more precise estimate of the mean.
  4. Since the intervals overlap, there is not sufficient evidence to conclude one layout is superior; a formal hypothesis test for the difference in means is needed. (correct answer)
Explanation: When comparing two confidence intervals from independent samples, overlap suggests that the true means may not be different. However, simply observing overlap is not a formal test. The proper procedure is to conduct a hypothesis test for the difference between two means (or construct a confidence interval for the difference). Because the intervals overlap, we cannot conclude there is a significant difference based on this informal comparison. A is incorrect because it ignores the uncertainty represented by the overlap. B makes too strong a claim. While overlap suggests no difference, it does not prove it. A formal test could still find a significant difference, especially if the overlap is small. C is incorrect; precision (a narrower interval) does not imply a better outcome (a higher mean).

Question 9

A company is considering a new marketing campaign. It will be launched only if there is strong evidence that more than 30% of the target audience expresses interest. A survey of 500 people is conducted, and a 95% confidence interval for the proportion of interested individuals is calculated to be [0.28, 0.34]. Based on this interval, what is the most logical business decision?

  1. Launch the campaign, as the upper bound of the interval (34%) is above the 30% threshold.
  2. Do not launch the campaign, as the lower bound of the interval (28%) is below the 30% threshold.
  3. The result is inconclusive because the interval contains plausible values on both sides of the 30% decision threshold. (correct answer)
  4. Launch the campaign, as the center of the interval (31%) is above the 30% threshold, which is the most likely true value.
Explanation: The confidence interval represents a range of plausible values for the true population proportion. Since this range includes values both below 30% (e.g., 29%) and above 30% (e.g., 33%), there is not strong evidence to conclude that the true proportion is greater than 30%. The data is consistent with the true proportion being 29% (don't launch) as well as it being 32% (launch). Therefore, the result is inconclusive regarding the decision criterion. A and B are incorrect because they involve 'cherry-picking' one end of the interval while ignoring the other. D is incorrect because it relies only on the point estimate and ignores the sampling error represented by the margin of error. The entire interval must be considered.

Question 10

A researcher at a university calculates a 95% confidence interval for the mean GPA of all students as [3.15, 3.35]. However, the data was collected by surveying only students who were studying in the main library on a Tuesday afternoon. Which of the following is the most significant issue with interpreting this confidence interval?

  1. The confidence level of 95% is too low; a 99% interval should have been used for academic data.
  2. The interval is too wide to be useful, indicating that a much larger sample should have been taken.
  3. The sampling method is a convenience sample, introducing bias that makes the interval an unreliable estimate for the entire student population. (correct answer)
  4. The true mean GPA of all students is likely not within this interval because the sample size was probably too small to satisfy the Central Limit Theorem.
Explanation: The validity of a confidence interval depends critically on the assumption of random sampling from the target population. Surveying students in the library creates a convenience sample, which is likely biased. Students in the library may be more studious and have higher GPAs than the general student population. This selection bias means the interval, no matter how precisely calculated, is likely not a valid estimate for the mean GPA of all students. A is incorrect; the choice of confidence level is a matter of preference, not a flaw in the study's validity. B is a judgment about precision, not validity. The core issue is bias, not the width of the interval. D is incorrect; the primary issue is bias from the sampling method, not sample size. A biased sample, no matter how large, will produce a biased estimate.

Question 11

A large clinical trial for a new weight-loss drug is conducted. A 95% confidence interval for the mean difference in weight loss between the drug group and the placebo group (drug - placebo) is [0.1 kg, 0.3 kg] after three months. Which statement provides the best interpretation?

  1. The result is statistically significant because the interval does not contain zero, but the effect may not be practically significant due to the small magnitude. (correct answer)
  2. The result is not statistically significant because the amount of weight loss is too small to be meaningful for most patients.
  3. The drug is proven to be highly effective, as the confidence interval clearly shows a positive and statistically significant effect on weight loss.
  4. Since the interval is very narrow, the study's results are likely unreliable due to a small sample size.
Explanation: This question tests the distinction between statistical significance and practical significance. Statistical significance is established because the interval for the difference in means, [0.1 kg, 0.3 kg], does not contain 0. This indicates a real effect. However, a mean effect of only 0.1 to 0.3 kg of extra weight loss may be too small to be considered meaningful or valuable in a real-world context, which is the concept of practical significance. B confuses the two concepts, incorrectly stating the result is not statistically significant. C overstates the conclusion. 'Highly effective' is a judgment of practical significance, which is questionable here. D incorrectly links a narrow interval to a small sample size. A narrow interval indicates high precision, which is typically the result of a large sample size.

Question 12

A food processing company claims that its bags of chips contain an average of 250 grams. A consumer advocacy group tests this claim by weighing a random sample of bags. They construct a 99% confidence interval for the mean weight, which turns out to be [247.5, 249.5] grams. What conclusion can be drawn regarding the company's claim based on a 1% significance level?

  1. The company's claim of a 250-gram average is rejected, as this value falls outside the 99% confidence interval. (correct answer)
  2. The company's claim cannot be rejected, because the difference between 249.5 grams and 250 grams is practically insignificant.
  3. The company's claim is plausible, as the interval is very close to 250 grams, indicating only minor sampling error.
  4. No conclusion can be drawn about the claim without performing a formal t-test, as confidence intervals are only for estimation.
Explanation: There is a direct duality between a two-sided hypothesis test and a confidence interval. A (1 - α) confidence interval contains all the null hypothesis values that would not be rejected by a two-sided test at the α significance level. Here, the 99% confidence interval corresponds to a 1% significance level (α = 0.01). Since the company's claimed mean of 250 grams is not contained within the [247.5, 249.5] interval, we have statistically significant evidence to reject the claim. B incorrectly substitutes practical significance for statistical significance. C incorrectly concludes that being 'close' is sufficient. In hypothesis testing, if the value is outside the interval, it is rejected. D is incorrect; the confidence interval provides all the information needed to perform the equivalent hypothesis test.

Question 13

A researcher constructs a 95% confidence interval for a population mean based on a sample of size n=100n=100. The resulting interval is [145, 155]. Assuming a Z-distribution was appropriate (i.e., using Z∗=1.96Z^*=1.96), what is the approximate value of the sample standard deviation (s)?

  1. 5.0
  2. 2.5
  3. 51.0
  4. 25.5 (correct answer)
Explanation: This requires working backward. First, find the margin of error (ME). ME = (Upper Bound - Lower Bound) / 2 = (155 - 145) / 2 = 5. The formula for ME is ME=Z∗×snME = Z^* \times \frac{s}{\sqrt{n}}. We can plug in the known values: 5=1.96×s1005 = 1.96 \times \frac{s}{\sqrt{100}}. This simplifies to 5=1.96×s105 = 1.96 \times \frac{s}{10}. Solving for s: 50=1.96×s50 = 1.96 \times s, so s=501.96≈25.51s = \frac{50}{1.96} \approx 25.51. A is incorrect; this is the margin of error. B is incorrect; this results from forgetting to multiply by n=10\sqrt{n}=10. C is incorrect; this results from multiplying by 1.96 instead of dividing, or a similar arithmetic error.

Question 14

An operations manager at a logistics company wants to estimate the mean delivery time for a specific route. A 95% confidence interval, based on a sample of deliveries, is calculated to be [3.2 hours, 4.8 hours]. The company has a service-level agreement (SLA) that promises customers delivery within 4.0 hours. What does the confidence interval suggest about the company's performance relative to its SLA?

  1. The company is successfully meeting the SLA, because the lower bound of 3.2 hours is well below the 4.0-hour target.
  2. The company is not meeting the SLA, because the sample mean delivery time must be 4.0 hours, which is exactly the limit.
  3. The company is failing to meet the SLA, as the upper bound of 4.8 hours proves that deliveries are often late.
  4. The company cannot be confident it is meeting the SLA, as the interval contains plausible values both below and above the 4.0-hour target. (correct answer)
Explanation: The confidence interval represents a range of plausible values for the true mean delivery time. Since the interval [3.2, 4.8] contains values that meet the SLA (e.g., 3.8 hours) and values that fail to meet the SLA (e.g., 4.5 hours), there is not enough statistical evidence to confidently conclude that the mean delivery time is at or below 4.0 hours. The result is inconclusive with respect to the SLA. A is incorrect as it only considers the best-case scenario within the interval. B correctly identifies the sample mean as 4.0 hours but incorrectly ignores the uncertainty; the true mean could be different. C is incorrect as it only considers the worst-case scenario. The true mean could still be below 4.0 hours.

Question 15

Two independent research firms estimate the market share for a particular product. Firm A uses a sample of 400 consumers and calculates a 95% confidence interval of [0.22, 0.28]. Firm B uses a sample of 1600 consumers and also calculates a 95% confidence interval, which is [0.24, 0.26]. Both intervals estimate the same parameter. Which statement is the most likely explanation for the difference between the two intervals?

  1. Firm A's result is more reliable because its interval is wider, capturing a larger range of possible values.
  2. Firm B's interval is narrower because it was constructed using a larger sample size, leading to a more precise estimate. (correct answer)
  3. The two firms must have sampled from different populations, as it is impossible for two CIs for the same parameter to be different.
  4. Firm B must have used a lower confidence level than Firm A, which is why its interval is narrower.
Explanation: The width of a confidence interval is inversely related to the square root of the sample size. A larger sample size leads to a smaller margin of error and thus a narrower, more precise confidence interval, all else being equal. Firm B used a sample four times larger than Firm A's, which should result in an interval approximately half as wide, which is consistent with the results. A is incorrect; a wider interval implies less precision, not more reliability. C is incorrect; due to sampling variability, it is expected that two different samples will produce two different confidence intervals. D is incorrect as the problem states both firms used a 95% confidence level.

Question 16

A human resources department compares the mean scores on a job satisfaction survey between two divisions, A and B. A 95% confidence interval for the difference in mean scores (μA−μB\mu_A - \mu_B) is calculated to be [-3.5, 1.5]. What is the proper interpretation of this result?

  1. Division A has a significantly lower mean satisfaction score than Division B, because the majority of the interval range is negative.
  2. There is no statistically significant difference in mean job satisfaction scores between the two divisions, because the interval contains the value of zero. (correct answer)
  3. Division B has a significantly lower mean satisfaction score than Division A, because the interval extends into positive territory.
  4. The true difference in mean scores is likely -1.0, the center of the interval, indicating Division A is less satisfied on average.
Explanation: When interpreting a confidence interval for the difference between two means (or proportions), the key value to look for is zero. If the interval contains zero, it means that a zero difference between the two population means is a plausible value. Therefore, we cannot conclude that a statistically significant difference exists at the corresponding significance level (in this case, α = 0.05). A is incorrect; the location of the majority of the range is irrelevant. The inclusion of zero is the deciding factor. C is a misinterpretation of the interval's meaning. D focuses only on the point estimate (-1.0) and ignores the margin of error. The interval indicates the true difference could plausibly be zero or even positive (up to 1.5).

Question 17

A financial analyst presents the following statement to a management committee: 'Based on historical data from the past five years, we constructed a 95% confidence interval for the mean annual return of Investment Portfolio X. The interval is [6.2%, 9.8%]. Therefore, we can be 95% confident that next year's return for Portfolio X will fall between 6.2% and 9.8%.'

Which of the following best describes the fundamental flaw in the analyst's statistical interpretation?

  1. The historical data from the past five years may not be representative of future performance, invalidating the conclusion.
  2. The analyst should have used a t-distribution instead of a Z-distribution because the population standard deviation is unknown.
  3. The conclusion is too strong; it should state there is a 95% probability, not 95% confidence, that the return will be in the interval.
  4. The confidence interval estimates the long-run average annual return, not the specific return for a single future year, which is subject to much greater variability. (correct answer)
Explanation: This is another example of confusing a confidence interval for a mean with a prediction interval for a single observation. The interval [6.2%, 9.8%] is an estimate for the average annual return over the long run. The actual return in any single year is a random variable with much more variance than the sample mean. The analyst is incorrectly applying an interval for a population average to predict a single future outcome. A is a valid practical concern about using historical data, but it is a modeling assumption issue, not a flaw in the statistical interpretation of what the interval represents. B is a minor computational detail, not the fundamental error in interpretation. C is incorrect because it suggests swapping 'confidence' for 'probability,' which is the classic fallacy of CI interpretation.

Question 18

A polling organization reports that 42% of voters support a ballot measure, with a margin of error of ±3 percentage points at 95% confidence. A news anchor states: "This means we can be 95% sure that if the election were held today, between 39% and 45% of voters would vote for the measure." What is wrong with this interpretation?

  1. The confidence interval applies to the population proportion, not to future voting behavior in an election. (correct answer)
  2. The margin of error should be applied to both sides, giving a range of 36% to 48%.
  3. The interpretation incorrectly uses "95% sure" instead of "95% confident."
  4. Nothing is wrong; this is a correct interpretation of the confidence interval for a proportion.
Explanation: Choice A is correct. The confidence interval estimates the current population proportion who support the measure, not future voting behavior. Voting behavior can differ from stated support due to turnout patterns, last-minute changes of mind, or differences between the polled population and actual voters. Choice B is wrong because ±3 percentage points correctly gives 39% to 45%. Choice C is incorrect because "95% sure" and "95% confident" convey the same meaning in this context. Choice D is wrong because the interpretation conflates current opinion with future voting behavior.

Question 19

A technology company surveys employee satisfaction and reports: "Based on our sample of 150 employees, we are 95% confident that between 62% and 78% of our workforce is satisfied with their job. This means there's only a 5% chance we're wrong about employee satisfaction levels." A statistician reviewing this statement would most likely object to which aspect?

  1. The sample size of 150 is too small to justify 95% confidence for a proportion.
  2. The phrase 'only a 5% chance we're wrong' misrepresents what the confidence level means. (correct answer)
  3. The interval is too wide; employee satisfaction should be measured with 99% confidence intervals.
  4. The company should report the exact sample proportion rather than a confidence interval for internal surveys.
Explanation: Choice B is correct. The phrase 'only a 5% chance we're wrong' incorrectly suggests there's a 5% probability that this specific interval doesn't contain the true proportion. However, the 5% refers to the long-run error rate of the confidence interval procedure - if we repeated this process many times, about 5% of the intervals wouldn't contain the true parameter. For this specific interval, we don't know the probability it's 'wrong.' Choice A is incorrect because 150 is adequate for proportion estimation. Choice C is wrong because there's no statistical requirement for 99% intervals in this context. Choice D is incorrect because confidence intervals provide more useful information than point estimates alone.

Question 20

A financial analyst calculates a 95% confidence interval for the mean price-to-earnings (P/E) ratio of stocks in the technology sector. The resulting interval is [28.5, 31.5]. A colleague interprets this as '95% of technology stocks have a P/E ratio between 28.5 and 31.5.' Why is this interpretation incorrect?

  1. The interpretation is incorrect because a 99% confidence level is required for financial data, making the 95% interval invalid.
  2. The interpretation is only incorrect if the distribution of P/E ratios is skewed; it is correct for a normal distribution.
  3. The confidence interval provides a plausible range for the population mean P/E ratio, not for the P/E ratios of individual stocks. (correct answer)
  4. The interpretation is incorrect because it should refer to the sample mean P/E ratio, not the P/E ratios of individual stocks.
Explanation: This is a classic misconception. A confidence interval for a mean estimates a single value: the population mean (average). It does not describe the distribution of the individual data points in the population. The range containing 95% of individual stock P/E ratios would be much wider and is described by different statistical measures (like standard deviation or a prediction interval), not a confidence interval for the mean. A is incorrect; the choice of confidence level is a user decision, not a rule. B is incorrect; the interpretation is always wrong, regardless of the population distribution. D is incorrect; the interval is an estimate for the population mean, not the sample mean (which is the center of the interval).