A 99% confidence interval (in inches) for the mean height of a population is $65.7

Question

A  99%  confidence interval (in inches) for the mean height of a population is $65.7 < \mu < 67.3$
This result is based on a sample of size 144 . Construct the  95%  confidence interval. (Hint: you will first need to find the sample mean and sample standard deviation).
A) $65.7 \text { in. } < \mu < 67.3 \text { in. }$
B) $65.9 \text { in. } < \mu < 67.1 \text { in. }$
C) $65.6 \text { in. } < \mu < 67.4 \text { in. }$
D) $66.0 \text { in. } < \mu < 67.0 \text { in. }$

Quizplus · Accepted Answer

The sample mean is the midpoint of the 99% confidence interval, which is $(65.7 + 67.3) / 2 = 66.5$ inches. The margin of error for the 99% confidence interval can be calculated as $(67.3 - 65.7) / 2 = 0.8$ inches. To find the sample standard deviation ($s$), we use the formula for the margin of error at 99% confidence, which involves the Z-score for 99% confidence (approximately 2.576) and the square root of the sample size ($\sqrt{144} = 12$). Solving for $s$ gives us $s = 0.8 \times 12 / 2.576 = 3.72$ inches. To construct the 95% confidence interval, we use the Z-score for 95% confidence (approximately 1.96). The new margin of error is $1.96 \times 3.72 / 12 = 0.606$. Subtracting and adding this from the sample mean gives us a 95% confidence interval of $66.5 \pm 0.606$, which is $65.9$ to $67.1$ inches.

A 99% Confidence Interval (In Inches) for the Mean