Question

A market research firm supplies manufacturers with estimates of the retail sales of their products from samples of retail stores. Marketing managers are prone to look at the estimates and ignore sampling error. An SRS of 75 stores this month shows mean sales of 52 units of a small appliance, with standard deviation 13 units. During the same month last year, an SRS of 53 stores had a mean sales of 49 units, with standard deviation 11 units. An increase from 49 to 52 is a rise of 6%. The marketing manager is happy because sales went up 6%.

Construct a 95% confidence interval estimate of the difference μ1−μ2, where μ1 is the mean of this year's sales and μ2 is the mean of last year's sales. What is the margin of error?

Answer 1

`(52-49) -4.300= -1.300`

`(52-49) +4.300= 7.300`

And the 95% confidence would be :

`-1.300 \leq \mu_1 -\mu_2 \leq 7.300`

Explanation:

We have the following info given from the problem

`\bar X_1 = 52` sample mean for this year

`s_1= 13` sample deviation for this year

`n_1 = 75` random sample selected for this year

`\bar X_2 = 49` sample mean for last year

`s_2= 11` sample deviation for last year

`n_1 = 53` random sample selected for last year

And we want to construct a 95% confidence interval estimate of the difference μ1−μ2, where μ1 is the mean of this year's sales and μ2 is the mean of last year's sales

For this case the formula that we need to use is:

`(\bar X_1 -\bar X_2) \pm t_(\alpha/2) \sqrt{(s^2_1)/(n_1) +(s^2_2)/(n_2)}`

The degrees of freedom are given by:

`df= n_1 +n_2 -2 = 75+53-2= 126`

The confidence level is 0.95 and the significance would be
`\alpha=0.05` and
`\alpha/2 =0.025` so then the critical value for this case is :

`t_(\alpha/2)= 1.979`

The margin of error would be:

`ME = 1.979 \sqrt{(13^2)/(75) +(11^2)/(49)}= 4.300`

And the confidence interval would be given by:

`(52-49) -4.300= -1.300`

`(52-49) +4.300= 7.300`

And the 95% confidence would be :

`-1.300 \leq \mu_1 -\mu_2 \leq 7.300`