Question

According to the New York Times Almanac, a survey of 15 large U.S. cities found that the average commute time one way to work was 25.4 minutes. A chamber of commerce executive feels that the one way to work commute in her city is less than this and wants to publicize this. In order to assess her belief, she randomly selected 25 commuters in her city and found the average one way commute time to work to be 22.1 minutes with a standard deviation of 5.3 minutes. At a 1% significance level assess the validity of the claim she would like to advertise for her city.

What is the inherent question of interest here?

What might be a reasonable description of the population of interest?

What is the relevant sample here?

What is the random variable being evaluated here?

What is the population parameter of interest?

What is this parameter’s corresponding sample statistic?

What is an appropriate research hypothesis to consider?

What is the corresponding null hypothesis?

What is the test statistic used to test these hypotheses?

What is the null distribution of this test statistic?

What is the decision rule that might be used here?

What decision was made with regard to the null hypothesis?

What is your conclusion based on the available data?

What is your best point estimate of the average one way commute time to work in this city?

Provide a 99% confidence interval for the average one way commute time to work in this city. Given the results of your evaluation, do you believe this executive can reasonably make the claim of shorter average commute times for her city than other large U.S. cities? Why or why not?

Answer 1

Sol:

a).

The question of interest is whether the average one way commute time in the city is less than average commute time of 15 large US cities.

b).

The population of interest is commuters in the city

c).

The relevant sample is random sample of 25 commuters in the city

d).

The random variable is the mean commute time.

e).

The population parameter of interest is true population mean commute time.

f).

The parameter’s corresponding sample statistic is the sample mean commute time.

g).

The research hypothesis is,

Ha: The average one way commute time in the city is less than 25.4 minutes.

h).

The null hypothesis is,

H0: The average one way commute time in the city is equal to 25.4 minutes.

i).

Standard error = 5.3 / sqrt(25)

Standard error   = 1.06

Test statistic t = (22.1 - 25.4)/1.06

t = -3.11

j).

Degree of freedom = n-1

= 25-1

Degree of freedom = 24

The null distribution is t distribution with 24 degree of freedom.

k).

For 1% significance level and df = 24, the critical value of t is -2.49

Decision rule:

Reject H0 when t < -2.49

l).

Since t < -2.49,

we reject the null hypothesis H0.

m).

We conclude that the sample data provides significant evidence that the average one way commute time in the city is less than 25.4 minutes.

n).

Best point estimate is the sample mean which is 22.1 minutes

o).

For 99% confidence interval and df = 24, the critical value of t is 2.80

99% confidence interval is,

CI = (22.1 - 2.80 * 1.06 , 22.1 + 2.80 * 1.06)

99% of CI = (19.132 , 25.068)

p).

Based on the confidence interval (the interval does not contain the value 25.4) and hypothesis test (null hypothesis is rejected), we the executive can reasonably claim of shorter average commute times in her city than other larger US cities.

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