Question

a study was conducted to estimate μ, the mean number of weekly hours that u.s. adults use computers at home. suppose a random sample of 81 u.s. adults gives a mean weekly computer usage time of 8.5 hours and that from prior studies, the population standard deviation is assumed to be σ = 3.6 hours. the 95% confidence interval for the mean, μ, is (7.7, 9.3). which of the following will provide a more informative (i.e., narrower) confidence interval than the 95% confidence interval? check all that apply. a. using a sample of size 400 (instead of 81) b. using a sample of size 36 (instead of 81) c. using a different sample of size 81 d. using a 90% level of confidence (instead of 95%) e. using a 99% level of confidence (instead of 95%)

Explanation:

Step1: Recall confidence - interval formula

The formula for a confidence interval for the population mean $\mu$ when the population standard deviation $\sigma$ is known is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - score, $\sigma$ is the population standard deviation, and $n$ is the sample size. The width of the confidence interval is $2z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$.

Step2: Analyze the effect of sample size

The width of the confidence interval is inversely proportional to the square - root of the sample size $n$. That is, as $n$ increases, the width of the confidence interval decreases. When $n = 81$, and if we change $n$ to $400$ (since $\sqrt{400}=20$ and $\sqrt{81}=9$), $\frac{\sigma}{\sqrt{n}}$ will decrease, resulting in a narrower confidence interval. A smaller sample size ($n = 36$, $\sqrt{36}=6$) will make the interval wider. A different sample of the same size $n = 81$ will not systematically make the interval narrower.

Step3: Analyze the effect of confidence level

The value of $z_{\alpha/2}$ is related to the confidence level. For a 95% confidence level, $z_{\alpha/2}=1.96$. For a 90% confidence level, $z_{\alpha/2}=1.645$, and for a 99% confidence level, $z_{\alpha/2}=2.576$. Decreasing the confidence level (from 95% to 90%) will decrease the value of $z_{\alpha/2}$, resulting in a narrower confidence interval. Increasing the confidence level (from 95% to 99%) will make the interval wider.

Answer:

A. Using a sample of size 400 (instead of 81)
D. Using a 90% level of confidence (instead of 95%)