A researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate $$${\hat{p}}_1-{\hat{p}}_2=0.09$ with a margin of error of 0.07. Should they reject or fail to reject the claim that there is no difference in these two proportions?

• A. Reject
• B. Fail to reject
• C. There is not enough information to answer the question.

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The researcher is testing whether there is a difference in two proportions. The null hypothesis (H₀) is that there is no difference between the two proportions, i.e., p₁ - p₂ = 0. The alternative hypothesis (Hₐ) is that there is a difference, i.e., p₁ - p₂ ≠ 0.
2. Step 2: Recall the interpretation of a confidence interval. A 90% confidence interval provides a range of values within which the true difference in proportions (p₁ - p₂) is likely to fall, with 90% confidence. If the interval does not include 0, we reject the null hypothesis.
3. Step 3: Use the given information to construct the confidence interval. The point estimate for the difference in proportions is p̂₁ - p̂₂ = 0.09, and the margin of error is 0.07. The confidence interval is calculated as: [ (p̂₁ - p̂₂) - margin of error, (p̂₁ - p̂₂) + margin of error ].
4. Step 4: Substitute the values into the formula. The confidence interval becomes: [ 0.09 - 0.07, 0.09 + 0.07 ]. Simplify the bounds of the interval to determine whether 0 is included.
5. Step 5: Analyze the interval. If 0 is not within the confidence interval, reject the null hypothesis (H₀). If 0 is within the interval, fail to reject the null hypothesis. If the interval cannot be determined from the given information, conclude that there is not enough information to answer the question.