Question

Sample satisfaction scores about the two retailers by the customers are shown below. In the report, 100 means the respondent is completely satisfied. Assume that a population standard deviation of 12 is a reasonable assumption for both retailers. Conduct the hypothesis test and report the p-value. At a .05 level of significance what is your conclusion? Include the solution process.

Retailer A Sample size = 25 Sample mean = 79
Retailer B sample size = 30 sample mean = 71

Answer 1

`t=\frac{(79 -71)-(0)}{12\sqrt{(1)/(25)+(1)/(30)}}=2.462`

Now we can calculate the degrees of freedom given by:

`df=25+30-2=53`

And now we can calculate the p value using the altenative hypothesis:

`p_v =2*P(t_(53)>2.462) =0.0171`

For this case the p value is lower than the significance so then we have enough evidence to reject the null hypothesis and we can conclude that the true means are different

Explanation:

We assume that the population deviation is the same for both cases

`\sigma^2_A =\sigma^2_B =\sigma^2`

And the statistic is given by this formula:

`t=\frac{(\bar X_A -\bar X_B)-(\mu_(A)-\mu_B)}{\sigma \sqrt{(1)/(n_A)+(1)/(n_B)}}`

Where t follows a t distribution with
`n_A+n_B -2` degrees of freedom

The system of hypothesis on this case are:

Null hypothesis:
`\mu_A = \mu_B`

Alternative hypothesis:
`\mu_A \\eq \mu_B`

The info given is:

`n_A =25` represent the sample size for group A

`n_B =30` represent the sample size for group B

`\bar X_A =79` represent the sample mean for the group A

`\bar X_B =71` represent the sample mean for the group B

And now we can calculate the statistic:

`t=\frac{(79 -71)-(0)}{12\sqrt{(1)/(25)+(1)/(30)}}=2.462`

Now we can calculate the degrees of freedom given by:

`df=25+30-2=53`

And now we can calculate the p value using the altenative hypothesis:

`p_v =2*P(t_(53)>2.462) =0.0171`

For this case the p value is lower than the significance so then we have enough evidence to reject the null hypothesis and we can conclude that the true means are different