$$L_n = a\phi^n + b\widehat{\phi}^n.$$
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Answer:
We conclude that the mean commute time in the U.S. is less than half an hour.
Step-by-step explanation:
We are given that a random sample of 500 people from the 2000 U.S. Census is selected who reported a non-zero commute time.
In this sample the mean commute time is 27.6 minutes with a standard deviation of 19.6 minutes.
Let = <u><em>mean commute time in the U.S..</em></u>
So, Null Hypothesis, :
30 minutes {means that the mean commute time in the U.S. is more than or equal to half an hour}
Alternate Hypothesis, :
< 30 minutes {means that the mean commute time in the U.S. is less than half an hour}
The test statistics that would be used here <u>One-sample t-test statistics</u> as we don't know about population standard deviation;
T.S. = ~
where, = sample mean commute time = 27.6 minutes
s = sample standard deviation = 19.6 minutes
n = sample of people from the 2000 U.S. Census = 500
So, <u><em>the test statistics</em></u> = ~
= -2.738
The value of t test statistic is -2.738.
Since, in the question we are not given with the level of significance so we assume it to be 5%. <u>Now, at 5% significance level the t table gives critical values of -1.645 at 499 degree of freedom for left-tailed test.</u>
Since our test statistic is less than the critical value of t as -2.378 < -1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which <u>we reject our null hypothesis.</u>
Therefore, we conclude that the mean commute time in the U.S. is less than half an hour.