Question

question 11
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

As the sample size $n$ increases, the term $\frac{\sigma}{\sqrt{n}}$ decreases. Since the width of the confidence interval is directly proportional to $\frac{\sigma}{\sqrt{n}}$, increasing the sample size will make the confidence interval narrower. A sample size of 400 ($n = 400$) is larger than 81, so using a sample of size 400 instead of 81 will make the confidence interval narrower. A sample size of 36 ($n = 36$) is smaller than 81, so it will make the confidence interval wider. Using a different sample of size 81 will not systematically make the interval narrower as it has the same sample - size and the same $z_{\alpha/2}$ for the same confidence level.

Step3: Analyze the effect of confidence level

The value of $z_{\alpha/2}$ decreases as the confidence level decreases. For a 95% confidence level, $z_{\alpha/2}\approx1.96$, for a 90% confidence level, $z_{\alpha/2}\approx1.645$, and for a 99% confidence level, $z_{\alpha/2}\approx2.576$. Using a 90% confidence level instead of 95% will decrease the value of $z_{\alpha/2}$, making the confidence interval narrower. Using a 99% confidence level instead of 95% will increase the value of $z_{\alpha/2}$, making the confidence interval wider.

Answer:

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