Problem

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Problem 16 Hard Difficulty

The American Bankers Association collects data on the use of credit cards, debit cards,
personal checks, and cash when consumers pay for in-store purchases (The wall Street
Journal, December $16,2003 ) .$ In 1999 , the following usages were reported.
$$ \begin{array}{ll}{\text { In-Store Purchase }} & {\text { Percentage }} \\ {\text { Credit card }} & {21} \\ {\text { Debit card }} & {21} \\ {\text { Personal check }} & {\text { is }} \\ {\text { Cash }} & {39}\end{array}
$$
A sample taken in 2003 found that for 220 in-stores purchases, 46 used a credit card, 67 used
a debit card, 33 used a personal check, and 74 used cash.
a. $\quad$ At $\alpha=.01,$ can we conclude thange occurred in how customers paid for in-store
purchases over the four-year period from 1999 to 2003$?$ What is the $p$ -value?
b. Compute the percentage of use foreach method of payment using the 2003 sample data.
What appears to have been the major change or changes over the four-year period?
c. In $2003,$ what percentage of payments was made using plastic (credit card or debit card)?

Answer

a. There is sufficient evidence to reject the claim of the specific distribution.
b. $20.91 \%, 30.45 \%, 15 \%, 33.64 \%,$ Debit card
c. 51.36$\%$

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Intro Stats / AP Statistics

Essentials of Modern Business Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

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Video Transcript

So we want to check if there has been a change in the payment method for in store purchases over the last four year period. So our null hypothesis is going to be that there has been no cherry change, So the proportion of people that use card is still 0.22 Um, actually, let me change this, too. Portion of people accuse credit is still 0.22 Proportion of people that use debit is still point to one proportion of people that used check is still 0.18 and the proportion of people that use cash is still 0.39 And now the alternative hypothesis would be that deep population proportions are not all the same as they were four years ago. So we're given the following information, and with this we have to fill out the rest of the table. The first thing that we will fill out is our expected frequency. So the expected frequency is simply the the number of the proportion of payment method for each payment method times our total end. So you should be 74. Our end is equal to the sum of this column here so are N is equal to 220. So we're going to multiply 0.22 by 2 20 and we're going to get the following values for 46.21 times 2 20 Oh, sorry. We're going to get 48.4 and then 0.21 times 2 20 which is 46.2 and then 0.18 times to 20 which is 39.6 and then point 39 times to 2085.8. And now we're going to take the difference between these two columns, and our difference is going to be the actual frequency minus the expected frequency. So this is going to be negative 2.4. This is 20.8 need of 6.6 and negative 11.8. Now we're going to square these differences, and the reason we're doing this is because we are looking to find a test statistic in order to find a P value to come up with a conclusion. And we're looking at a chi square Goodness of fit. The chi square goodness of fit test is equal to, um the following formula. Some from J equals one. Okay, Where Kay is the number of categories that we have of the actual frequency minus the expected frequency squared over the expected frequency for each category. So the first thing we need to do is find the difference between the expected frequency or between the actual frequency and the ex effective frequency, and then square it. So that's what we did. And now we're looking to square these values. So this is equal to 5.76 equally 4 32.64 is equal 43.56 and this is equal to 1 39 point 24 And now we have to find the different squared over expected frequency, which is what this column here represents. And we get the following values and these are gonna be truncated 0.119 9.36 1.1 and 1.62 And now we have to find out where Kai Squared is on our chi squared. As we figured out from this formula is just the sum of this column. So once we add all these values up, we get a chi squared of 12.21 approximately 12.21 And with a chi squared of 12.21 we have to find our degrees of freedoms that we can compute A p value. And the use of freedom is just the number of categories that we have minus one. And from here, we have four categories. One for credit, one for debit, one for check, one for cash. So this is Eagle Forum minus one, which is three. The probability that Kai squared is, um, that the chi squared the probability with acai squared of 12.21 with a degrees of freedom. Of the three is between is a member of the set 0.14 point 5.1 So this is the range of our P value, and we are going to compare this against our significance level of 0.1 So, um, 0.5 is less than an hour p value, which is less than 0.1 just less than hour. Alfa, that equals 0.1 Therefore we reject and the no. So that means that we have sufficient evidence to reject the claim that um, the populations population proportions have not changed, so we can reject this. And now, based on this, we have to find the rest of the information in this question. So this was the answer to part a comport B. We're asked to calculate the percentage of each payment method use in the 2003 sample. So, um, just using this data here we are given Let me move this. That was not smart. Okay, um, give me one second. Pull 0.21 and 12. 21. And now we have to find out what the proportion of the actual frequencies are. So, um, proportion of the actual frequencies. Um, So we are given that you could calculate the proportion of the actual frequency, we're just going to take our actual frequency and divide it by our total are end of 2 20 So here we get. Um, so for credit, our proportion would be 46 over to 20. Um, for Debbie, it would be 67 over to 20 for check. It would be 33 over to 20. And for cash, it would be 74 divided by 2 20 So, in a new page um, credit proportion of people that use credit in 2003 is approximately equal to 21%. Proportion of people that use debit is approximately equal to 30% proportion of people that use check is approximately 15% and the portion of people that use cash is approximately 34%. And after this, you're asked to find, um, the percentage of payment made using plastic, which is credit or debit. So the percent of payments made using plastic would be the percent of credit plus percent or the proportion of debit, so that would be 21 plus 30 21% plus 30% which is equal to 51%.

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