A public health team surveys two regions about willingness to receive a new vaccine.

Region A: 54 of 180 randomly sampled adults indicate they are willing.

Region B: 77 of 220 randomly sampled adults indicate they are willing.

Let
p
A
and
p
B
be the population proportions willing in Regions A and B, respectively. Construct a 95% confidence interval for the difference of proportions
p
A
−
p
B

Question

A public health team surveys two regions about willingness to receive a new vaccine.

Region A: 54 of 180 randomly sampled adults indicate they are willing.

Region B: 77 of 220 randomly sampled adults indicate they are willing.

Let 
p
A
 and 
p
B
 be the population proportions willing in Regions A and B, respectively. Construct a 95% confidence interval for the difference of proportions 
p
A
−
p
B

BlackTom AI · Accepted Answer

First, let's restate the key data: Region A has 54 willing out of 180, giving p̂_A = 54/180 = 0.30. Region B has 77 willing out of 220, giving p̂_B = 77/220 = 0.35. The quantity of interest is p_A − p_B, which in this sample is −0.05.

Option A: (−0.120, 0.020). This interval would imply a smaller margin around the observed difference than what the standard error calculation yields. To judge accuracy, we should compute the standard error properly and compare. The standard error for the difference in sample proportions is SE = sqrt(p̂_A(1 − p̂_A)/n_A + p̂_B(1 − p̂_B)/n_B) = sqrt(0.30×0.70/180 + 0.35×0.65/220) ≈ sqrt(0.0011667 + 0.0010341) ≈ sqrt(0.0022008) ≈ 0.0469. The 95% margin would be about 1.96×0.0469 ≈ 0.092. If we take the observed difference −0.05 and add/subtract 0.092, we get roughly (−0.142, 0.042). Thus A’s narrower interval is not consistent with the standard 95% CI; it would be too tight.

Option B: (−0.142, 0.042). This aligns with the calculation above: −0.05 ± 1.96×0.0469 ≈ −0.05 ± 0.092, giving approximately (−0.142, 0.042). This interval contains zero and reflects the uncertainty from both samples.

Option C: (−0.160, 0.060). This interval is wider than the one obtained from the standard error, suggesting a margin of about 0.05 larger on each side than the computed 0.092. Since the computed margin is about 0.092, this interval would not be supported by the standard error with a 95% level.

Option D: (−0.108, 0.008). This interval is centered near −0.05 but has a much smaller margin (about 0.041 on each side) than the calculated 0.092, making it inconsistent with the 95% CI using the standard approach.

In summary, using the standard error for the difference in two proportions and a 95% confidence level yields a margin of roughly 0.092 around the observed difference of −0.05, which produces an interval close to (−0.142, 0.042). The other options deviate from this margin by being too narrow or too wide and thus do not match the correct calculation.

MATH1062_MATH1005_MATH1023 MATH1062/1005 Data Quiz 8

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MATH1062_MATH1005_MATH1023 MATH1062/1005 Data Quiz 8

View Explanation

Log in for full answers

Similar Questions

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 ​ ​ Refer to Exhibit 10-10. The standard error of is _____.

Questions 8 and 9 continued: Report the upper limit of the confidence interval to three decimal places. Make sure your answer is a difference of proportions, not a difference of percentages.

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 ​ ​ Refer to Exhibit 10-10. The standard error of is _____.

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Making Your Study Simpler

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.

Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product Shoppers Surveyed Shoppers Favoring This Product A 800 560 B 900 612 Refer to Exhibit 10-10. The standard error of is _____.