The management of a department store is interested in estimating the difference between the mean
credit purchases made by customers using the store’s credit card versus those customers using a
national major credit card. Independent samples of credit sales are shown below.
Store’s Card
𝑛1 = 64
𝑥̅1 = $140
𝑠1 = $10
Major Credit Card
𝑛2 = 49
𝑥̅2 = $125
𝑠2 = $8
(a) Develop a point estimate for the difference between the mean purchases of the users of the
two credit cards.
(b) Construct and interpret a 95% confidence interval to estimate the difference between the mean
purchases of the customers using the two different credit cards.
(c) Does this interval give supporting evidence that there one type of credit card is used more
prevalently?
In order to determine if there is a difference between the hourly wages of two companies, the following
information was gathered.
Company A
𝑛1 = 80
𝑥̅1 = $6.75
𝑠1 = $1.00
Company B
𝑛2 = 60
𝑥̅2 = $6.25
𝑠2 = $0.95
Is there a difference between the hourly wages of the two companies?
The management of a department store is interested in estimating the difference between the mean
credit purchases made by customers using the store’s credit card versus those customers using a
national major credit card. Independent samples of credit sales are shown below.
Store’s Card
𝑛1 = 64
𝑥̅1 = $140
𝑠1 = $10
Major Credit Card
𝑛2 = 49
𝑥̅2 = $125
𝑠2 = $8
(a) Develop a point estimate for the difference between the mean purchases of the users of the
two credit cards.
1  2 : The difference in the means between Store Card and Major Credit Card purchases.
(b) Construct and interpret a 95% confidence interval to estimate the difference between the mean
purchases of the customers using the two different credit cards.
Independence: It is reasonable to assume that whether one customer uses a Store Card or a
Major Credit Card is independent of each other.
No need to verify with Randomization and 10% conditions since they are independent.
Independent samples: the two samples are independent (as stated in the problem).
Nearly Normal Condition: By the CLT, both sampling distributions will be approximately normal
because both sample sizes are sufficiently large ( n1  64, n2  49 ), likewise, the sampling
distribution for the difference in means will also be approximately normal.
Construct a 2-sample t-interval with 95% confidence.
x1  x2   tdf* 
s12 s22
10 2 82

 140  125  t df*

 11.64,18.36
n1 n2
64 49
We are 95% confident that the average Store Card purchase is between $11.64 and $18.36 more
than a purchase using a Major Credit Card.
(c) Does this interval give supporting evidence that one type of credit card is used more
prevalently?
Yes, since 0 is not within the interval, there is evidence to suggest that Store Cards are used for
larger purchases than Major Credit Cards.
In order to determine if there is a difference between the hourly wages of two companies, the following
information was gathered.
Company A
𝑛1 = 80
𝑥̅1 = $6.75
𝑠1 = $1.00
Company B
𝑛2 = 60
𝑥̅2 = $6.25
𝑠2 = $0.95
Is there a difference between the hourly wages of the two companies?
Ho: The mean hourly wage at Company A is the same as the mean hourly wage at Company B.
Ha: The mean hourly wage at Company A is different than the mean hourly wage at Company B.
Independence: Is it reasonable to assume the hourly wage for one worker is independent of another
worker? Probably.
Randomization: We don’t know the samples were drawn randomly, but assume the samples are
representative of all hourly wages for workers at these two companies.
10% condition: If Company A has at least 800 hourly wage workers and company B has 600 hourly wage
workers, then we are okay.
Independent samples: It is reasonable to think the two samples are independent of each other.
Since the conditions are satisfied, we can conduct a 2-sample t-test.
t
x1  x2   0  6.75  6.25  0  3.0128
2
s1
s2
 2
n1 n2
1.00 2 0.952

80
60
2 * Pt130  3.0128  0.0031
If there was no difference in the mean hourly wages between Company A and Company B, the
probability of observing a difference of $0.50 or more extreme is 0.31%. Since the p-value is so small,
there is strong evidence to suggest there is a difference in the mean hourly wages between Company A
and Company B. Workers at Company A have a average hourly wage higher than workers at Company
B.