College Statistics Quiz: Ci For Difference Of Proportions
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Ci For Difference Of ProportionsQuestion 1 of 20

An experiment was conducted to determine if a new fertilizer (F) increases the proportion of seeds that germinate compared to an old fertilizer (O). From two independent random samples, the 95% confidence interval for pF−pOp_F - p_O was calculated to be (0.08, 0.22). The research team wants to report the 95% confidence interval for the difference pO−pFp_O - p_F. What is this interval?

(0.08, 0.22)
(-0.22, -0.08)
(-0.08, -0.22)
The interval cannot be determined without the original sample sizes and proportions.
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College Statistics Quiz

College Statistics Quiz: Ci For Difference Of Proportions

Practice Ci For Difference Of Proportions in College Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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Question 1

An experiment was conducted to determine if a new fertilizer (F) increases the proportion of seeds that germinate compared to an old fertilizer (O). From two independent random samples, the 95% confidence interval for pF−pOp_F - p_O was calculated to be (0.08, 0.22). The research team wants to report the 95% confidence interval for the difference pO−pFp_O - p_F. What is this interval?

  1. (0.08, 0.22)
  2. (-0.22, -0.08) (correct answer)
  3. (-0.08, -0.22)
  4. The interval cannot be determined without the original sample sizes and proportions.
Explanation: If a confidence interval for a difference A−BA-B is (L,U)(L, U), then the confidence interval for the difference B−AB-A is (−U,−L)(-U, -L). This is because the point estimate changes sign (from p^F−p^O\hat{p}_F - \hat{p}_O to p^O−p^F\hat{p}_O - \hat{p}_F) and the margin of error remains the same. The original interval is (p^F−p^O)±ME(\hat{p}_F - \hat{p}_O) \pm ME. The new interval is (p^O−p^F)±ME=−(p^F−p^O)±ME(\hat{p}_O - \hat{p}_F) \pm ME = -(\hat{p}_F - \hat{p}_O) \pm ME. The new lower bound is −(old center)−ME-(\text{old center}) - ME and the new upper bound is −(old center)+ME-(\text{old center}) + ME. This corresponds to negating the original endpoints and swapping them. For the interval (0.08, 0.22), the new interval is (-0.22, -0.08).

Question 2

A 95% confidence interval for the difference between the proportion of men and women who favor a certain policy (pmen−pwomenp_{men} - p_{women}) is (-0.02, 0.08). If a statistician wanted to re-evaluate the data using a 90% confidence level, what would the new interval be?

  1. Narrower than (-0.02, 0.08) and centered at 0.03. (correct answer)
  2. Wider than (-0.02, 0.08) and centered at 0.03.
  3. Narrower than (-0.02, 0.08), but the center cannot be determined.
  4. The interval cannot be recalculated without knowing the sample sizes and proportions.
Explanation: The center of the confidence interval is the point estimate, p^men−p^women\hat{p}_{men} - \hat{p}_{women}, which is independent of the confidence level. The center of the original interval is −0.02+0.082=0.03\frac{-0.02 + 0.08}{2} = 0.03. The new interval will have the same center. The width of the interval is determined by the margin of error, which is z∗×SEz^* \times SE. Decreasing the confidence level from 95% to 90% decreases the critical value z∗z^* (from 1.96 to 1.645). Since the standard error (SE) depends only on the sample data, it remains unchanged. A smaller z∗z^* results in a smaller margin of error and thus a narrower interval. Therefore, the new interval will be narrower and still centered at 0.03.

Question 3

A medical researcher is planning a study to compare the effectiveness of two different flu vaccines. They want to construct a 95% confidence interval for the difference in proportions of individuals contracting the flu (Vaccine A - Vaccine B) with a margin of error no greater than 0.05. Assuming the sample size for each group will be equal (nA=nB=nn_A = n_B = n) and using the conservative estimate of p^A=p^B=0.5\hat{p}_A = \hat{p}_B = 0.5, what is the minimum sample size nn required for each vaccine group?

  1. 385
  2. 542
  3. 769 (correct answer)
  4. 1537
Explanation: The formula for the margin of error (ME) for the difference between two proportions is ME=z∗p^A(1−p^A)nA+p^B(1−p^B)nBME = z^* \sqrt{\frac{\hat{p}_A(1-\hat{p}_A)}{n_A} + \frac{\hat{p}_B(1-\hat{p}_B)}{n_B}}. For a 95% confidence level, z∗=1.96z^* = 1.96. Using nA=nB=nn_A = n_B = n, ME = 0.05, and the conservative estimates p^A=p^B=0.5\hat{p}_A = \hat{p}_B = 0.5, we solve for nn: 0.05=1.960.5(0.5)n+0.5(0.5)n=1.960.5n0.05 = 1.96 \sqrt{\frac{0.5(0.5)}{n} + \frac{0.5(0.5)}{n}} = 1.96 \sqrt{\frac{0.5}{n}}. Rearranging gives n=0.5(1.960.05)2=0.5(39.2)2=0.5(1536.64)=768.32n = 0.5 \left( \frac{1.96}{0.05} \right)^2 = 0.5(39.2)^2 = 0.5(1536.64) = 768.32. Since the sample size must be an integer, we round up to the next whole number, which is 769.

Question 4

A 95% confidence interval for the difference in proportions pA−pBp_A - p_B is constructed from samples of size nA=100n_A=100 and nB=100n_B=100. Which of the following procedural changes would result in the narrowest confidence interval?

  1. Decrease the confidence level to 90% and keep the sample sizes at 100.
  2. Increase the confidence level to 99% and double both sample sizes to 200.
  3. Keep the confidence level at 95% and double both sample sizes to 200.
  4. Decrease the confidence level to 90% and double both sample sizes to 200. (correct answer)
Explanation: The width of a confidence interval is determined by the margin of error, ME=z∗×SEME = z^* \times SE. The standard error (SE) is inversely proportional to the square root of the sample sizes. To make the interval narrower, we need to decrease the margin of error. This can be achieved by either decreasing the confidence level (which decreases z∗z^*) or increasing the sample sizes (which decreases SE). Option D does both: it decreases the confidence level from 95% to 90% (z∗z^* from 1.96 to 1.645) and increases the sample sizes. This combined effect will produce a greater reduction in width than any of the other options.

Question 5

A researcher constructs a 95% confidence interval for p1−p2p_1 - p_2 using the sample data n1=100,p^1=0.80n_1=100, \hat{p}_1=0.80 and n2=100,p^2=0.20n_2=100, \hat{p}_2=0.20. When calculating the standard error, the researcher mistakenly uses the pooled proportion, p^c=100(0.80)+100(0.20)100+100=0.50\hat{p}_c = \frac{100(0.80)+100(0.20)}{100+100} = 0.50, which is appropriate for a hypothesis test but not for a confidence interval. How will the resulting incorrect interval compare to the correctly calculated interval?

  1. The incorrect interval will be wider than the correct interval. (correct answer)
  2. The incorrect interval will be narrower than the correct interval.
  3. The center of the incorrect interval will be different from the correct interval's center.
  4. The incorrect interval will be identical to the correct interval because the sample sizes are equal.
Explanation: The standard error for a confidence interval for p1−p2p_1 - p_2 should be calculated using the unpooled formula: SEunpooled=p^1(1−p^1)n1+p^2(1−p^2)n2SE_{unpooled} = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}. The researcher incorrectly used the pooled formula: SEpooled=p^c(1−p^c)(1n1+1n2)SE_{pooled} = \sqrt{\hat{p}_c(1-\hat{p}_c)(\frac{1}{n_1} + \frac{1}{n_2})}. Let's calculate both. Correct: SEunpooled=0.8(0.2)100+0.2(0.8)100=0.0016+0.0016=0.0032≈0.0566SE_{unpooled} = \sqrt{\frac{0.8(0.2)}{100} + \frac{0.2(0.8)}{100}} = \sqrt{0.0016+0.0016} = \sqrt{0.0032} \approx 0.0566. Incorrect: SEpooled=0.5(0.5)(1100+1100)=0.25(0.02)=0.005≈0.0707SE_{pooled} = \sqrt{0.5(0.5)(\frac{1}{100} + \frac{1}{100})} = \sqrt{0.25(0.02)} = \sqrt{0.005} \approx 0.0707. Since SEpooled>SEunpooledSE_{pooled} > SE_{unpooled}, the margin of error will be larger, and the resulting confidence interval will be wider than the correct one. The center, p^1−p^2\hat{p}_1 - \hat{p}_2, is unaffected.

Question 6

A researcher constructs a 90% confidence interval for the difference in the proportion of men and women who are left-handed, pM−pWp_M - p_W. The interval is (0.01, 0.05). Which statement provides a correct interpretation of the 90% confidence level?

  1. There is a 90% probability that the true difference in the proportion of left-handed men and women is between 0.01 and 0.05.
  2. If this sampling procedure were repeated many times, approximately 90% of the resulting confidence intervals would capture the true difference in proportions. (correct answer)
  3. We are 90% confident that the sample difference, p^M−p^W\hat{p}_M - \hat{p}_W, is between 0.01 and 0.05.
  4. In 90% of the population, the difference in the propensity to be left-handed between men and women is between 0.01 and 0.05.
Explanation: The confidence level (e.g., 90%) refers to the long-run performance of the statistical method, not to a specific interval. It means that if we were to take many random samples and construct a confidence interval from each one, we would expect about 90% of those intervals to contain the true population parameter (in this case, the true difference in proportions). Choice A is a common misinterpretation; the true parameter is a fixed value, so it is either in the interval or not (the probability is 0 or 1). Choice C is incorrect because we know the sample difference with certainty; it is the center of the interval (0.03). Choice D misinterprets the confidence level as applying to individuals in the population.

Question 7

A marketing firm compares the proportion of customers who click on two different ad designs, A and B. They construct a 95% confidence interval for pA−pBp_A - p_B and get (-0.09, 0.05). A senior manager notes that the sample proportion for B was higher than for A and states, "The sample data shows Ad B is better, so we should use Ad B." What is the most statistically sound response to the manager?

  1. The manager is correct; since the sample proportion for Ad B was higher, it is the more effective ad, and further statistical analysis is unnecessary.
  2. Although the sample proportion for Ad B was higher, the confidence interval contains zero, indicating there is no statistically significant evidence of a difference in effectiveness. (correct answer)
  3. A new study should be conducted with a higher confidence level, such as 99%, to provide a more definitive conclusion about which ad is better.
  4. The manager is incorrect; the confidence interval is centered at -0.02, which is so close to zero that the ads can be considered equally effective in the population.
Explanation: The point estimate of the difference is the center of the interval, (p^A−p^B)=−0.09+0.052=−0.02(\hat{p}_A - \hat{p}_B) = \frac{-0.09+0.05}{2} = -0.02. This confirms the manager's observation that the sample proportion for B was higher than for A. However, the key conclusion comes from the interval itself. Since the 95% confidence interval (-0.09, 0.05) contains the value 0, we cannot conclude that there is a statistically significant difference between the proportions at the α=0.05\alpha=0.05 level. Therefore, the data do not provide convincing evidence that one ad is superior to the other.

Question 8

A political analyst constructs a 95% confidence interval for the difference in the proportion of voters supporting a candidate in two different districts (District A - District B). The resulting interval is (0.03, 0.11). Based on this interval, which of the following is the most appropriate conclusion?

  1. There is a 95% probability that the true difference in the proportion of support for the candidate between the two districts is between 0.03 and 0.11.
  2. The proportion of voters supporting the candidate is guaranteed to be higher in District A than in District B.
  3. At the 5% significance level, there is evidence of a difference in support for the candidate between the two districts, with support likely being higher in District A. (correct answer)
  4. Since the interval contains more values closer to 0.03 than to 0.11, it is more likely that the true difference is small.
Explanation: The confidence interval (0.03, 0.11) does not contain 0. This indicates a statistically significant difference between the two proportions at the corresponding significance level (α=1−0.95=0.05\alpha = 1 - 0.95 = 0.05). Since all values in the interval are positive, it provides evidence that pA−pB>0p_A - p_B > 0, meaning the proportion of support is higher in District A. Choice A is an incorrect interpretation of the confidence level. Choice B uses definitive language ('guaranteed'), which is inappropriate for statistical inference. Choice D incorrectly assumes that values near the center of the interval are not as plausible as those near the endpoints; all values within the interval are considered plausible.

Question 9

A manufacturer develops a new technique that is believed to reduce the proportion of defective products. A test is run on the old process (Sample 1) and the new process (Sample 2). A 95% confidence interval is constructed for pold−pnewp_{old} - p_{new}. Which of the following intervals would provide the strongest evidence that the new technique is effective?

  1. (-0.01, 0.09)
  2. (-0.10, -0.02)
  3. (0.02, 0.18)
  4. (0.10, 0.12) (correct answer)
Explanation: When you see a confidence interval for the difference between two proportions, you're looking at whether one group significantly differs from another. Here, we want evidence that the new technique reduces defects, meaning pold−pnewp_{old} - p_{new} should be positive (old process has more defects than new). The strongest evidence comes from option D: (0.10, 0.12). This interval tells us we're 95% confident that the old process produces between 10-12 percentage points more defective products than the new process. Not only is this difference positive (supporting our hypothesis), but the entire interval is well above zero, indicating a substantial, consistent improvement. The narrow range (0.02 wide) also suggests high precision in our estimate. Option A (-0.01, 0.09) includes negative values, meaning we can't rule out that the new process might actually be worse. Option B (-0.10, -0.02) suggests the new process produces more defects than the old one – the opposite of what we want. Option C (0.02, 0.18) does show the new process is better, but the wide range (0.16) indicates much less precision than option D, making it weaker evidence. Remember: when comparing proportions, you want the confidence interval to be entirely on the side that supports your hypothesis, with values as far from zero as possible and a narrow range for precision. The combination of being completely positive, substantial in magnitude, and precise makes D the strongest evidence.

Question 10

A 95% confidence interval for pA−pBp_A - p_B is calculated as (0.02, 0.10) based on sample sizes of nA=400n_A = 400 and nB=400n_B = 400. Suppose the data for group A remained identical, but the data for group B had yielded the same sample proportion (p^B\hat{p}_B) but was based on a smaller sample of nB=100n_B = 100. How would the new confidence interval compare to the original?

  1. The new interval would be narrower and centered at the same value.
  2. The new interval would be wider and centered at the same value. (correct answer)
  3. The new interval would be the same width but centered at a different value.
  4. The new interval would be wider and centered at a different value.
Explanation: The center of the confidence interval is the point estimate p^A−p^B\hat{p}_A - \hat{p}_B. Since p^A\hat{p}_A and p^B\hat{p}_B are unchanged, the center of the new interval remains the same. The width of the interval depends on the standard error, SE=p^A(1−p^A)nA+p^B(1−p^B)nBSE = \sqrt{\frac{\hat{p}_A(1-\hat{p}_A)}{n_A} + \frac{\hat{p}_B(1-\hat{p}_B)}{n_B}}. By decreasing the sample size nBn_B from 400 to 100, the term p^B(1−p^B)nB\frac{\hat{p}_B(1-\hat{p}_B)}{n_B} increases. This increases the overall standard error, which in turn increases the margin of error and makes the confidence interval wider.

Question 11

A political analyst constructs a 95% confidence interval for the difference in the proportion of voters supporting a candidate in two different districts (District A - District B). The resulting interval is (0.03, 0.11). Based on this interval, which of the following is the most appropriate conclusion?

  1. There is a 95% probability that the true difference in the proportion of support for the candidate between the two districts is between 0.03 and 0.11.
  2. The proportion of voters supporting the candidate is guaranteed to be higher in District A than in District B.
  3. At the 5% significance level, there is evidence of a difference in support for the candidate between the two districts, with support likely being higher in District A. (correct answer)
  4. Since the interval contains more values closer to 0.03 than to 0.11, it is more likely that the true difference is small.
Explanation: The confidence interval (0.03, 0.11) does not contain 0. This indicates a statistically significant difference between the two proportions at the corresponding significance level (α=1−0.95=0.05\alpha = 1 - 0.95 = 0.05). Since all values in the interval are positive, it provides evidence that pA−pB>0p_A - p_B > 0, meaning the proportion of support is higher in District A. Choice A is an incorrect interpretation of the confidence level. Choice B uses definitive language ('guaranteed'), which is inappropriate for statistical inference. Choice D incorrectly assumes that values near the center of the interval are not as plausible as those near the endpoints; all values within the interval are considered plausible.

Question 12

A quality control specialist compares two production lines. On Line 1, 15 out of 200 items are defective. On Line 2, 8 out of 100 items are defective. The specialist wishes to construct a 90% confidence interval for the difference in the proportion of defective items (Line 1 - Line 2). Which of the following conditions for the validity of the standard z-interval procedure has not been met?

  1. The data are from two independent random samples.
  2. The sample sizes are large enough for the normal approximation. (correct answer)
  3. The samples are less than 10% of their respective populations.
  4. All conditions for the validity of the procedure have been met.
Explanation: The large counts condition requires that np^n\hat{p} and n(1−p^)n(1-\hat{p}) be at least 10 for both samples. For Line 1: p^1=15/200=0.075\hat{p}_1 = 15/200 = 0.075. The number of successes (defectives) is 15 (which is ≥10\ge 10), and the number of failures is 200−15=185200-15=185 (which is ≥10\ge 10). This sample meets the condition. For Line 2: p^2=8/100=0.08\hat{p}_2 = 8/100 = 0.08. The number of successes (defectives) is 8. Since 8 is less than 10, the large counts condition is not met for the second sample, making the normal approximation potentially unreliable.

Question 13

A 99% confidence interval for the difference in the proportion of students who pass a certification exam after using two different study guides (Guide A vs. Guide B) is (0.04, 0.26). What is the margin of error for this confidence interval?

  1. 0.11 (correct answer)
  2. 0.15
  3. 0.22
  4. 0.26
Explanation: The margin of error (ME) is half the width of the confidence interval. The width of the interval is the upper bound minus the lower bound: 0.26−0.04=0.220.26 - 0.04 = 0.22. The margin of error is half of this width: ME=0.222=0.11ME = \frac{0.22}{2} = 0.11. The value 0.22 is the full width of the interval. The value 0.15 is the point estimate (p^A−p^B\hat{p}_A - \hat{p}_B), which is the center of the interval: 0.04+0.262=0.15\frac{0.04 + 0.26}{2} = 0.15.

Question 14

In a clinical trial, 150 patients received a new drug and 200 received a placebo. Of those receiving the drug, 105 showed significant improvement. A 95% confidence interval for the difference in the proportion of improvement (Drug - Placebo) was found to be (0.16, 0.34). Based on this information, approximately how many patients who received the placebo showed improvement?

  1. 50
  2. 80
  3. 90 (correct answer)
  4. 100
Explanation: First, find the sample proportion for the drug group: p^drug=105/150=0.70\hat{p}_{drug} = 105/150 = 0.70. Next, find the point estimate of the difference in proportions, which is the center of the confidence interval: p^drug−p^placebo=0.16+0.342=0.502=0.25\hat{p}_{drug} - \hat{p}_{placebo} = \frac{0.16 + 0.34}{2} = \frac{0.50}{2} = 0.25. Now, substitute the known p^drug\hat{p}_{drug} into this equation: 0.70−p^placebo=0.250.70 - \hat{p}_{placebo} = 0.25. Solving for p^placebo\hat{p}_{placebo} gives p^placebo=0.70−0.25=0.45\hat{p}_{placebo} = 0.70 - 0.25 = 0.45. Finally, calculate the number of placebo patients who showed improvement: Number=nplacebo×p^placebo=200×0.45=90Number = n_{placebo} \times \hat{p}_{placebo} = 200 \times 0.45 = 90.

Question 15

A hypothesis test is conducted to assess if there is a difference in the proportion of defective items produced by two machines (H0:p1=p2H_0: p_1 = p_2 vs. Ha:p1≠p2H_a: p_1 \ne p_2). The test yields a p-value of 0.03. If a 95% confidence interval for the difference p1−p2p_1 - p_2 were constructed using the same data, which of the following statements must be true?

  1. The confidence interval would contain the value 0.
  2. The confidence interval would not contain the value 0. (correct answer)
  3. The center of the confidence interval would be 0.03.
  4. The width of the confidence interval would be 0.03.
Explanation: There is a direct correspondence between a two-sided hypothesis test and a confidence interval. A hypothesis test with significance level α\alpha will reject the null hypothesis H0:p1=p2H_0: p_1 = p_2 if and only if the 100(1−α)%100(1-\alpha)\% confidence interval for p1−p2p_1 - p_2 does not contain 0. In this case, the p-value is 0.03. For a 95% confidence interval, the corresponding significance level is α=0.05\alpha = 0.05. Since the p-value (0.03) is less than α\alpha (0.05), we would reject the null hypothesis. Therefore, the 95% confidence interval for the difference must not contain 0.

Question 16

An experiment was conducted to determine if a new fertilizer (F) increases the proportion of seeds that germinate compared to an old fertilizer (O). From two independent random samples, the 95% confidence interval for pF−pOp_F - p_O was calculated to be (0.08, 0.22). The research team wants to report the 95% confidence interval for the difference pO−pFp_O - p_F. What is this interval?

  1. (0.08, 0.22)
  2. (-0.22, -0.08) (correct answer)
  3. (-0.08, -0.22)
  4. The interval cannot be determined without the original sample sizes and proportions.
Explanation: If a confidence interval for a difference A−BA-B is (L,U)(L, U), then the confidence interval for the difference B−AB-A is (−U,−L)(-U, -L). This is because the point estimate changes sign (from p^F−p^O\hat{p}_F - \hat{p}_O to p^O−p^F\hat{p}_O - \hat{p}_F) and the margin of error remains the same. The original interval is (p^F−p^O)±ME(\hat{p}_F - \hat{p}_O) \pm ME. The new interval is (p^O−p^F)±ME=−(p^F−p^O)±ME(\hat{p}_O - \hat{p}_F) \pm ME = -(\hat{p}_F - \hat{p}_O) \pm ME. The new lower bound is −(old center)−ME-(\text{old center}) - ME and the new upper bound is −(old center)+ME-(\text{old center}) + ME. This corresponds to negating the original endpoints and swapping them. For the interval (0.08, 0.22), the new interval is (-0.22, -0.08).

Question 17

A study on social media habits found that 50% of 400 teenagers surveyed use Platform X, while 40% of 400 young adults surveyed use it. Let pTp_T be the proportion of all teenagers and pAp_A be the proportion of all young adults who use Platform X. A 95% confidence interval for pT−pAp_T - p_A is constructed. What is the standard error for the difference in sample proportions?

  1. 0.00123
  2. 0.0245
  3. 0.0350 (correct answer)
  4. 0.1000
Explanation: The standard error for the difference in sample proportions is SE=p^T(1−p^T)nT+p^A(1−p^A)nASE = \sqrt{\frac{\hat{p}_T(1-\hat{p}_T)}{n_T} + \frac{\hat{p}_A(1-\hat{p}_A)}{n_A}}. With p^T=0.50\hat{p}_T = 0.50, nT=400n_T = 400, p^A=0.40\hat{p}_A = 0.40, and nA=400n_A = 400: SE=0.50(0.50)400+0.40(0.60)400=0.25+0.24400=0.49400=0.720=0.035SE = \sqrt{\frac{0.50(0.50)}{400} + \frac{0.40(0.60)}{400}} = \sqrt{\frac{0.25 + 0.24}{400}} = \sqrt{\frac{0.49}{400}} = \frac{0.7}{20} = 0.035.

Question 18

A 95% confidence interval for the difference in proportion of satisfied customers between two services, ponline−pinstorep_{online} - p_{instore}, is (-0.25, 0.15). Which of the following statements is the most appropriate conclusion based on this interval?

  1. Online and in-store services definitely have different levels of customer satisfaction, but the direction is unknown.
  2. It is more likely that in-store service is better, because more of the plausible values in the interval are negative.
  3. The proportion of satisfied in-store customers is higher than for online customers, but the difference is not significant.
  4. The estimate is not precise, and it is plausible that there is no difference in customer satisfaction between the two services. (correct answer)
Explanation: When interpreting confidence intervals for the difference between two proportions, you need to consider what the interval tells you about the plausible range of true differences and whether zero (no difference) falls within that range. The interval (-0.25, 0.15) contains zero, which means it's plausible that there's no actual difference between online and in-store customer satisfaction (ponline−pinstore=0p_{online} - p_{instore} = 0). Since zero is included in the range of plausible values, we cannot conclude there's a significant difference between the services. The relatively wide interval (spanning 0.40 units) also indicates low precision in our estimate. Answer D correctly captures both of these key insights. Answer A is wrong because when a confidence interval contains zero, we cannot conclude the groups "definitely" have different satisfaction levels. The interval suggests the opposite—that no difference is plausible. Answer B misinterprets confidence intervals. You don't determine likelihood by comparing the proportion of negative versus positive values in the interval. The entire interval represents plausible values with equal credibility. Answer C makes a contradictory statement. If the difference "is not significant" (which is correct since zero is in the interval), then you cannot simultaneously claim that "the proportion of satisfied in-store customers is higher." Statistical significance and the direction of difference go hand-in-hand. Study tip: When a confidence interval for a difference includes zero, remember this key rule: no significant difference exists between the groups. The wider the interval, the less precise your estimate.

Question 19

A city is considering a ban on plastic bags. A survey of 500 residents is taken. Then, a city-wide informational campaign is launched. Afterward, a new survey of 600 different residents is taken. The city wants to construct a confidence interval to see if the proportion of residents supporting the ban changed. Why is a two-proportion z-interval potentially inappropriate for this situation?

  1. The sample sizes are not equal, which violates an assumption of the two-proportion z-interval.
  2. A paired-data procedure, such as a matched-pairs t-test, should have been used instead.
  3. The procedure is invalid because the second sample was taken after an intervention, which could bias the results.
  4. The two samples may not be independent because they are from the same city and exposed to the same campaign. (correct answer)
Explanation: When evaluating whether a two-proportion z-interval is appropriate, you need to check several key assumptions: independence of samples, normal approximation conditions, and random sampling. The most critical assumption here is independence. The correct answer is D because the two samples violate the independence assumption in a subtle but important way. While the samples don't overlap (different people were surveyed), both groups come from the same population and were exposed to the same informational campaign between surveys. This shared exposure creates a dependency between the samples because any population-wide effect from the campaign influences both the baseline measurement and the post-campaign measurement. The samples are measuring the same underlying population parameter under two different conditions, which violates the assumption that the two proportions come from independent populations. Option A is incorrect because two-proportion z-intervals don't require equal sample sizes. Unequal sample sizes are perfectly acceptable as long as both samples meet the normal approximation conditions. Option B misidentifies the appropriate procedure. A matched-pairs t-test requires the same individuals measured twice, but here we have completely different people in each survey. This is correctly designed as a two-sample situation, not paired data. Option C incorrectly suggests the intervention itself invalidates the procedure. The intervention is actually the point of the study—the city wants to measure whether opinions changed after the campaign. Remember: Independence violations are often the most subtle assumption violations in two-sample procedures. Always consider whether samples might be connected through shared experiences, environments, or influences, even when they don't overlap in membership.

Question 20

Two different polling organizations survey voters about a ballot measure. Organization A reports a 95% confidence interval for the proportion supporting the measure as (0.52, 0.58). Organization B, surveying a different group of voters, reports a 95% confidence interval for the same measure as (0.47, 0.53). A student wants to combine the results to estimate the difference in proportions found by the two organizations (pA−pBp_A - p_B). Which statement is correct?

  1. The confidence interval for the difference is (0.52 - 0.53, 0.58 - 0.47), which is (-0.01, 0.11).
  2. Since the intervals overlap, there is no significant difference between the results of the two organizations.
  3. The appropriate method is to construct a two-proportion z-interval using the point estimates and standard errors from each poll. (correct answer)
  4. A confidence interval for the difference cannot be constructed because the polls were conducted independently on different populations.
Explanation: The student is treating the results of two separate polls as two independent samples and wants to compare them. This is a valid application of the two-proportion z-interval procedure. The point estimates are the centers of the intervals: p^A=0.55\hat{p}_A = 0.55 and p^B=0.50\hat{p}_B = 0.50. The standard errors can be derived from the margins of error. Then, the standard error for the difference is SEdiff=SEA2+SEB2SE_{diff} = \sqrt{SE_A^2 + SE_B^2}, and a new confidence interval can be constructed. Choice A is incorrect; confidence intervals cannot be subtracted this way. Choice B is a common misconception; while overlapping intervals often suggest no significant difference, it is not a formally correct test. The correct procedure is to construct the CI for the difference. Choice D is incorrect; independence is a requirement for the test, not a barrier to it.