The American Bankers Association collects data on the use of...

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)?

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)?

Key Concepts

Significance Level (alpha)

The significance level, denoted as alpha, is the pre-determined threshold for deciding when to reject the null hypothesis. It represents the maximum probability of committing a Type I error, where a true null hypothesis is incorrectly rejected.

Hypothesis Testing

Hypothesis testing is a statistical methodology for making inferences about a population by comparing a null hypothesis, which typically represents no change or effect, against an alternative hypothesis that represents a change or effect. It provides a framework to assess whether observed data deviates significantly from what would be expected under the null hypothesis.

Chi-Square Goodness-of-Fit Test

The Chi-square goodness-of-fit test is used to determine how well an observed frequency distribution matches an expected distribution. It calculates a test statistic by comparing observed counts to expected counts across categorical outcomes, enabling statisticians to assess whether any discrepancies are due to random variation or indicate a significant change.

p-value

The p-value is the probability, under the null hypothesis, of observing a test statistic as extreme as or more extreme than the statistic computed from the sample data. It quantifies the strength of the evidence against the null hypothesis; a small p-value indicates that the observed data are unlikely under the null hypothesis.

Degrees of Freedom

Degrees of freedom refer to the number of values in the final calculation of a statistic that are free to vary. In the context of a chi-square goodness-of-fit test, it is typically calculated as the number of categories minus one, which is used to determine the appropriate chi-square distribution for evaluating the test statistic.

Calculation of Sample Proportions

Calculating sample proportions involves dividing the count of observations in each category by the total number of observations in the sample. This process converts raw counts into percentages or proportions, allowing for meaningful comparisons between observed sample data and theoretical or previous distributions.

Transcript

00:01 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.

00:07 So our null hypothesis is going to be that there has been no change.

00:12 So the proportion of people that use card is still 0 .22.

00:17 Actually, let me change this to a portion of people that use credit is still 0 .22.

00:25 Proportion of people that use debit is still 0 .21.

00:28 The proportion of people that use check is still 0 .18, and the proportion of people that use cash is still 0 .39.

00:39 And now the alternative hypothesis would be that the population proportions are not all the same as they were four years ago.

01:08 So we're given the following information, and with this we have to fill out the rest of the table.

01:14 The first thing that we will fill out is our expected frequency.

01:18 So the expected frequency is simply the number of, or the proportion of payment method for each payment method times our total n.

01:30 So there should be 74.

01:32 Our n is equal to the sum of this column here.

01:37 So our n is equal to 220.

01:40 So we're going to multiply 0 .22 by 220.

01:42 And we're going to get the following values, 46 and .21 times 220.

01:49 Oh, sorry.

01:52 We're going to get 48 .4, and then 0 .21 times 220, which is 46 .2, and then 0 .18 times 220, which is 39 .6, and then 0 .39 times 220, which is 85 .8.

02:11 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.

02:20 So this is going to be negative 2 .4, and this is 20 .8, negative 6 .6, and negative 11 .8.

02:31 Now we're going to square these differences.

02:33 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.

02:42 And we are looking at a kye squared goodness of fit.

02:46 And the kye squared goodness of fit test is equal to the following formula, the sum from j equals 1 to k, where k is the number of categories that we have of the actual frequency minus the expected frequency squared over the expected frequency for each category.

03:12 So the first thing we need to do is find the difference between the expected frequency or between the actual frequency and the expected frequency and then square it.

03:20 So that's what we did.

03:21 And now we're looking to square these values.

03:23 So this is equal to 5 .76.

03:32 It's equal to 432 .64.

03:36 This is equal to 43 .56.

03:40 And this is equal to 139 .24.

03:48 And now we have to find the difference squared over our expected frequency, which is what this column here represents.

03:55 And we get the following values.

03:59 And these are going to be truncated...