Rescaling a 95% interval to 99%

Question

A report gives a 95% confidence interval for a mean as $[46,\ 54]$ , based on a known population standard deviation. The raw data are not provided.

Construct the 99% confidence interval for the same mean from the same data.

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Setup. The 95% interval is centered at 50 with half-width 4. Since the half-width is z^{}	imes 	ext{SE}, back out the standard error: 	ext{SE} = \frac{4}{1.96} = 2.041. Rescale to 99%. Use z^{} = 2.576: 	ext{half-width} = 2.576 	imes 2.041 = 5.26. So the 99% interval is 50 \pm 5.26 = [44.74,\ 55.26], \quad 	ext{width } 10.51. Interpret. You never needed the sample; the center and half-width of any level's interval determine the standard error, and from there any other level follows. Takeaway. A confidence interval encodes its own standard error. Divide the half-width by that level's cri…

Rescaling a 95% interval to 99%

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