Question

a study was conducted to estimate $mu$, 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 $sigma = 3.6$ hours.
a similar study conducted a year earlier estimated that $mu$, the mean number of weekly hours that u.s. adults use computers at home, was 8 hours. we would like to test (at the usual significance level of 5%) whether the current study provides significant evidence that this mean has changed since the previous year.
using a 95% confidence interval of (7.7, 9.3), which of the following is an appropriate conclusion?
a. the current study does provide significant evidence that the mean number of weekly hours has changed over the past year, since 8 falls outside the confidence interval.
b. the current study does not provide significant evidence that the mean number of weekly hours has changed over the past year, since 8 falls outside the confidence interval.
c. the current study does provide significant evidence that the mean number of weekly hours has changed over the past year, since 8 falls inside the confidence interval.
d. the current study does not provide significant evidence that the mean number of weekly hours has changed over the past year, since 8 falls inside the confidence interval.
e. you cannot draw a conclusion because the only way to reach a conclusion is to find the p - value of the test.

Explanation:

Step1: Understand the confidence interval concept

A confidence interval gives a range of values within which the population parameter (here, the population mean \(\mu\)) is likely to lie. If the hypothesized value (in this case, the previous mean of \(8\) hours) lies outside the confidence interval, we reject the null hypothesis (that the mean has not changed). If it lies inside, we fail to reject the null hypothesis.

Step2: Check the position of \(8\) in the confidence interval \((7.7,9.3)\)

Since \(7.7<8 < 9.3\), the value \(8\) lies inside the \(95\%\) confidence interval \((7.7,9.3)\). When the hypothesized value lies inside the confidence interval, we do not have significant evidence to reject the null hypothesis (that the mean has not changed).

Answer:

D. The current study does not provide significant evidence that the mean number of weekly hours has changed over the past year, since 8 falls inside the confidence interval.