A 97% confidence interval for the difference
𝜇
1
−
𝜇
2
is (4.2, 5.8). If you were to change the confidence level to 93% instead, which of the following could represent a plausible interval?

Question

A 97% confidence interval for the difference 
𝜇
1
−
𝜇
2
 is (4.2, 5.8). If you were to change the confidence level to 93% instead, which of the following could represent a plausible interval?

BlackTom AI · Accepted Answer

To analyze the question, note that a 97% confidence interval for the difference μ1 − μ2 is (4.2, 5.8). The center of this interval is (4.2 + 5.8)/2 = 5.0, and the half-width is (5.8 − 4.2)/2 = 0.8.

With a lower confidence level (93%), the interval typically becomes narrower, assuming the same estimation method and sampling variability. This means the plausible 93% interval should be centered near 5.0 and have a half-width smaller than 0.8 (i.e., less than 0.8). Any option that satisfies these criteria is plausible.

Option A: (4.3, 5.7)
- Center is (4.3 + 5.7)/2 = 5.0, exactly the original center.
- Half-width is (5.7 − 4.3)/2 = 0.7, which is smaller than 0.8. This aligns with the expectation of a narrower 93% interval, making it plausible.

Option B: (4.35, 5.75)
- Center is (4.35 + 5.75)/2 = 5.05, very close to 5.0 but not exactly the original center; slight shift can occur due to sampling variability at a different confidence level.
- Half-width is (5.75 − 4.35)/2 = 0.70, which is also smaller than 0.8. This is plausible as well, given small shifts in the estimate can occur.

Option C: (-5.6, -4.2)
- Center is (-5.6 + -4.2)/2 = -4.9, far from the original estimate of 5.0 and implies a wildly different effect direction.
- This interval would not be a plausible 93% interval for the same data, as it contradicts the original estimate and confidence interval logic.

Option D: (4.1, 5.9)
- Center is (4.1 + 5.9)/2 = 5.0, matching the original center.
- Half-width is (5.9 − 4.1)/2 = 0.9, which is larger than 0.8. A 93% interval is typically not wider than the 97% interval, so this is unlikely and not plausible.

In summary, the interval that best fits the expectation of a narrower 93% confidence interval centered near the same estimate is (4.3, 5.7). The other options either shift the center more than expected or widen the interval beyond the 97% width, which is inconsistent with moving to a lower confidence level.

A 97% confidence interval for the difference 𝜇 1 − 𝜇 2 is (4.2, 5.8). If you were to change the confidence level to 93% instead, which of the following could represent a plausible interval?单项选择题

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