Answer:
80% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases ( ) is [-9.132 , 23.332].
) is [-9.132 , 23.332].
Step-by-step explanation:
We are given that a random sample of 7 sales receipts for mail-order sales results in a mean sale amount of $81.70 with a standard deviation of $18.75. 
A random sample of 11 sales receipts for internet sales results in a mean sale amount of $74.60 with a standard deviation of $28.25.
Firstly, the Pivotal quantity for 80% confidence interval for the difference between population means is given by;
                             P.Q. =   ~
  ~ 
where,  = sample mean sales receipts for mail-order sales = $81.70
 = sample mean sales receipts for mail-order sales = $81.70
 = sample mean sales receipts for internet sales = $74.60
 = sample mean sales receipts for internet sales = $74.60
 = sample standard deviation for mail-order sales = $18.75
 = sample standard deviation for mail-order sales = $18.75
 = sample standard deviation for internet sales = $28.25
 = sample standard deviation for internet sales = $28.25
 = size of sales receipts for mail-order sales = 7
 = size of sales receipts for mail-order sales = 7
 = size of sales receipts for internet sales = 11
 = size of sales receipts for internet sales = 11
Also,  =
 =  = 25.11
 = 25.11
<em>Here for constructing 80% confidence interval we have used Two-sample t test statistics as we don't know about population standard deviations.</em>
<em />
So, 80% confidence interval for the difference between population means, ( ) is ;
) is ;
P(-1.337 <  < 1.337) = 0.80  {As the critical value of t at 16 degree
 < 1.337) = 0.80  {As the critical value of t at 16 degree
                                          of freedom are -1.337 & 1.337 with P = 10%}  
P(-1.337 <  < 1.337) = 0.80
 < 1.337) = 0.80
P(  <
 <  <
 <  ) = 0.80
 ) = 0.80
P(  < (
 < ( ) <
) <  ) = 0.80
 ) = 0.80
<u>80% confidence interval for</u> ( ) =
) =
[  ,
 ,  ]
 ]
= [  ,
 ,  ]
 ]
= [-9.132 , 23.332]
Therefore, 80% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases ( ) is [-9.132 , 23.332].
) is [-9.132 , 23.332].