位置4的问题 If the margin of error in an interval estimate of μ is 3.8, the interval estimate equals 1.9 (that is half of 3.8).TrueFalse

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BlackTom AI · Accepted Answer

Question restatement: The prompt asks whether the statement 'If the margin of error in an interval estimate of μ is 3.8, the interval estimate equals 1.9 (that is half of 3.8)' is True or False. Answer options:
- True
- False

Option 1: True. This claims that the interval estimate equals 1.9. Remember that in interval estimation, the margin of error (MOE) is the amount added to and subtracted from the point estimate to form the interval. The interval estimate itself is the central point (the point estimate of μ), not the half-width. If the MOE is 3.8, then the half-width of the interval is 3.8, and the entire interval would be (point estimate − 3.8) to (point estimate + 3.8). The statement that the interval estimate equals 1.9 would imply that the center is 1.9, but nothing in the given data ties the center to 1.9. More importantly, the MOE being 3.8 has nothing to do with the interval estimate equaling 1.9; the MOE is the distance from the center to either endpoint, not the value of the center itself. Therefore, this option misstates the relationship between the margin of error and the interval estimate, making it incorrect.

Option 2: False. This option asserts that the statement in Option 1 is not correct. Since the margin of error is 3.8, the interval estimate is the center value (the point estimate of μ), and the endpoints are the center minus 3.8 and the center plus 3.8. The center is not fixed at 1.9 by the given MOE; 1.9 would only occur if the center were 1.9 and the MOE were 0.0 or if additional information specified that the center equals 1.9, which is not provided. In short, 1.9 is not the interval estimate when MOE is 3.8; the interval estimate is the point estimate (center), and 3.8 is the half-width. Thus, the statement is indeed false, making Option 2 the appropriate choice.

Summary of reasoning: The idea that a margin of error of 3.8 makes the interval estimate equal to 1.9 conflates the half-width with the interval center. The MOE determines how far the endpoints are from the center, not the center’s value itself, so the claim is incorrect. The correct interpretation is that the interval estimate is the point estimate, and the interval extends 3.8 to either side of that point; consequently, the statement in Option 1 is false, and Option 2 correctly represents that assessment.

位置4的问题 If the margin of error in an interval estimate of μ is 3.8, the interval estimate equals 1.9 (that is half of 3.8).TrueFalse

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