91Ó°ÊÓ

Over the last 3 years, Art's Supermarket has observed the following distribution of modes of payment in the express lines: cash (C) \(41 \%\), check (CK) \(24 \%\), credit or debit card (D) \(26 \%\), and other (N) \(9 \% .\) In an effort to make express checkout more efficient, Art's has just begun offering a \(1 \%\) discount for cash payment in the express checkout line. The following table lists the frequency distribution of the modes of payment for a sample of 500 express-line customers after the discount went into effect. $$ \begin{array}{l|cccc} \hline \text { Mode of payment } & \text { C } & \text { CK } & \text { D } & \text { N } \\ \hline \text { Number of customers } & 240 & 104 & 111 & 45 \\ \hline \end{array} $$ Test at a \(1 \%\) significance level whether the distribution of modes of payment in the express checkout line changed after the discount went into effect.

Step by step solution

01

Calculate the Expected Frequencies

Using the previous distribution percentages and the total sample size: \(N=500\), the expected frequencies are calculated as follows: \(E_C = 0.41*500 = 205\), \(E_{CK} = 0.24*500 = 120\), \(E_D = 0.26*500 = 130\), and \(E_N = 0.09*500 = 45\).

02

Calculate the Chi-Square Test Statistic

The formula used to calculate the Chi-square statistic is: \(\chi^2 = \sum[(O_i - E_i)^2 / E_i]\), where O is the observed frequencies and E is the expected frequencies. Substituting observed and expected values for each payment mode: \(\chi^2 = (240 - 205)^2/205 + (104 - 120)^2/120 + (111 -130)^2 /130 + (45 - 45)^2 /45 = 6.79 + 2.13 + 2.78 + 0 = 11.7.\)

03

Find the Critical Value

The critical value can be obtained from the Chi-square distribution table. With the degrees of freedom being \(df = k - 1 = 4 - 1 = 3\) and the significance level equal to \(0.01\), the critical value is approximately \(11.34\).

04

Make a Decision

Comparing the test statistic with the critical value (i.e., \(\chi^2 > \text{Critical Value}\)), we realize \(11.7 > 11.34\). Hence, the distribution of payments shows a significant change; the null hypothesis is rejected at a \(1 \%\) level of significance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Frequencies

Expected Frequencies help predict what the data should ideally look like based on historical or theoretical distributions. They are calculated using the known probabilities or proportions and the total number of observations.
In our exercise, we calculate expected frequencies for different payment modes based on the initial percentages.
These percentages were cash (41%), check (24%), credit/debit card (26%), and other (9%) with a total of 500 customers.
The formula for each expected frequency is:

• Cash: \( E_C = 0.41 \times 500 = 205 \)
• Check: \( E_{CK} = 0.24 \times 500 = 120 \)
• Credit/Debit Card: \( E_D = 0.26 \times 500 = 130 \)
• Other: \( E_N = 0.09 \times 500 = 45 \)

These values represent what we would expect if nothing had changed after implementing the discount strategy.

Observed Frequencies

Observed Frequencies are the actual data collected from observations or experiments. In this context, they show the real number of customers choosing each payment mode after the initiative.
In our example with Art's Supermarket, once the discount was put into effect, we recorded the following observed frequencies for the 500 customers:

• Cash: 240 customers
• Check: 104 customers
• Credit/Debit Card: 111 customers
• Other: 45 customers

By comparing these numbers to the expected frequencies, we can determine if there is a significant shift in behavior or if the variations occurred purely by chance.

Significance Level

The significance level is a critical concept in hypothesis testing, determining the probability of rejecting the null hypothesis when it is, in fact, true.
Typically set at 1%, 5%, or 10%, it reflects the level of confidence you require in your results. A smaller significance level means stricter criteria for concluding a significant effect.
For the exercise, we are using a 1% significance level, which suggests high confidence in deciding whether the payment distribution changed post-discount. If the test statistic exceeds the critical value at this level, we infer that the change is unlikely due to random fluctuations alone.

Degrees of Freedom

Degrees of Freedom (df) in statistical tests determine the number of independent values or quantities that can vary.
In a Chi-Square test, degrees of freedom are calculated as the number of categories minus one: \( df = k - 1 \). This accounts for the constraints in the data.
For Arts' Supermarket, there are four categories of payment methods, so the degree of freedom is:

• \( df = 4 - 1 = 3 \)

This value is essential for determining the test's critical Chi-Square value at the chosen significance level. It helps decide if the observed change is significant or a random occurrence.