A local pizza restaurant and a local branch of a national ch

A local pizza restaurant and a local branch of a national chain are located across the street from a college campus. The local pizza restaurant advertises that they deliver to the dormitories faster than the national chain. In order to determine whether this advertisement is valid, you and some friends have decided to order 6 pizzas from the local pizza restaurant and 6 pizzas from the national chain, all at different times. The delivery times in minutes are shown below.
Local      Chain
16.8         22.0
11.7         15.2
15.6         18.7
16.7         15.6
17.5         20.8
13.4         17.2
Do the data provide sufficient evidence to conclude that the mean delivery time for the local pizza restaurant is less than the mean delivery time for the national pizza chain? Conduct hypothesis test using ?=.05.

use the traditional approach and the p-value approach to hypothesis testing

Solution

traditional approach:
Let mu1 be the mean for Local

Let mu2 be the mean for Chain

The test hypothesis:

Ho: mu1=mu2 (i.e. null hypothesis)

Ha: mu1<mu2 (i.e. alternative hypothesis)

The test statistic is

t=(xbar1-xbar2)/sqrt(s1^2/n1+s2^2/n2)

=(15.283-18.250)/sqrt(2.268^2/6+2.764^2/6)

=-2.03

The degree of freedom =n1+n2-2=6+6-2=10

It is a left-tailed test.

Given a=0.05, the critical value is t(0.05, df=10) =-1.81 (from student t table)

The rejection region is if t<-1.81, we reject the null hypothesis.

Since t=-2.03 is less than -1.81, we reject the null hypothesis.

So we can conclude that the mean delivery time for the local pizza restaurant is less than the mean delivery time for the national pizza chain

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p-value approach:

The p-value= P(t with df=10 <-2.03) =0.0349 (from student t table)

Since the p-value is less than 0.05, we reject the null hypothesis.

So we can conclude that the mean delivery time for the local pizza restaurant is less than the mean delivery time for the national pizza chain