J. J.Bean sells a wide variety of outdoor equipment and clothing. The company sells both through mail order and via the internet. Random samples of sales receipts were studied for mail-order sales and internet sales, with the total purchase being recorded for each sale. A random sample of 16 sales receipts for mail-order sales results in a mean sale amount of $74.50 with a standard deviation of $17.25. A random sample of 9 sales receipts for internet sales results in a mean sale amount of $84.40 with a standard deviation of $21.25. Using this data, find the 99% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases. Assume that the population variances are not equal and that the two populations are normally distributed. Step 2 of 3 : Find the margin of error to be used in constructing the confidence interval. Round your answer to six decimal places.

99% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is [$(-31.82) , $12.02].

Step-by-step explanation:

We are given that a random sample of 16 sales receipts for mail-order sales results in a mean sale amount of $74.50 with a standard deviation of $17.25.

A random sample of 9 sales receipts for internet sales results in a mean sale amount of $84.40 with a standard deviation of $21.25.

The pivotal quantity that will be used for constructing 99% confidence interval for true mean difference is given by;

                      P.Q.  =    ~

where, = sample mean for mail-order sales = $74.50

= sample mean for internet sales = $84.40

= sample standard deviation for mail-order purchases = $17.25

= sample standard deviation for internet purchases = $21.25

= sample of sales receipts for mail-order purchases = 16

= sample of sales receipts for internet purchases = 9

Also,    =   = 18.74

The true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is represented by ().

Now, 99% confidence interval for () is given by;

             =

Here, the critical value of t at 0.5% level of significance and 23 degrees of freedom is given as 2.807.

          =

          = [$-31.82 , $12.02]

Hence, 99% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is [$(-31.82) , $12.02].

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step-by-step explanation: