Question

if the 90% confidence limits for the population mean are 33 and 47, which of the following could be the 98% confidence limits a) (30, 50) b) (36, 41) c) (38, 42) d) (38, 45) e) (39, 43) f) none of the above

Answer 1

The 98% confidence limit will be wider than the 90% confidence limits. Thus, a) (30, 50) have a wider interval that would be expected for a 98% confidence level.

Step-by-step explanation:

The 90% confidence limits for the population mean are 33 and 47. To determine which of the given 98% confidence limits could be possible, we need to consider the relationship between the confidence level and the width of the interval.

We don't know the sample mean or the standard error, but we do know the 90% confidence limits. We can use these to estimate the standard error:

Standard error = (90% confidence limits upper bound - 90% confidence limits lower bound) / (1.645 * 2)

= (47 - 33) / (1.645 * 2)

= 4.73

Now we can use this to estimate the 98% confidence limits:

98% confidence limits = sample mean ± (2.33) * (4.73)

We know that the range of the 90% confidence limits is 14 (47 - 33). The range of the 98% confidence limits will be wider than this, so we can eliminate options b), c), and e). Checking which of the remaining options gives us a range wider than 14 by subtracting the lower limit from the upper limit:

a) (30, 50) -> range = 20

d) (38, 45) -> range = 7

Answer 2

None of the listed options (a-e) could be the 98% confidence limits for the population mean.

Step-by-step explanation:

A higher confidence level (98% vs. 90%) generally translates to a wider confidence interval.

Logic: Imagine drawing confidence intervals like "umbrellas" around the true population mean. A wider umbrella (larger interval) signifies a higher probability of capturing the true mean, but with less precision.

Therefore, the 98% confidence limits should be wider than the 90% limits (33 and 47).

Analysis:

Option a) (30, 50) and e) (39, 43) are narrower than the 90% limits, making them impossible for 98% confidence.

Options b) (36, 41), c) (38, 42), and d) (38, 45) fall within the 90% limits or have one limit overlapping, making them infeasible for a separate 98% interval.

Therefore, the correct answer is f) none of the above.