Problem 11 - Basic Statistics for Business and Economics with Student

Question

The bank credit card department of Carolina Bank knows from experience that 5 percent of its card holders have had some high school, 15 percent have completed high school, 25 percent have had some college, and 55 percent have completed college. Of the 500 card holders whose cards have been called in for failure to pay their charges this month, 50 had some high school, 100 had completed high school, 190 had some college, and 160 had completed college. Can we conclude that the distribution of card holders who do not pay their charges is different from all others? Use the .01 significance level.

Step-by-Step Solution

Answer

The distribution of defaulters is significantly different from the overall cardholder distribution.

Step 1: Understand the Problem

We are provided with two distributions: the educational background of typical card holders and the educational background of card holders who have defaulted this month. We need to determine if the distribution of defaults significantly differs from the overall distribution.

Step 2: Set Up the Hypotheses

We will use the Chi-Square test for independence. The null hypothesis ( H_0 ) is that the default distribution matches the overall distribution. The alternative hypothesis ( H_a ) is that the default distribution is different from the overall distribution.

Step 3: Calculate Expected Frequencies

The expected number of defaulters in each educational category is calculated based on the overall proportions. Multiply each overall proportion by the total number of defaulters (500):\[\begin{align*}\text{Some High School: } & 0.05 \times 500 = 25, \\text{Completed High School: } & 0.15 \times 500 = 75, \\text{Some College: } & 0.25 \times 500 = 125, \\text{Completed College: } & 0.55 \times 500 = 275.\end{align*}\]

Step 4: Compute the Chi-Square Statistic

Use the formula for the Chi-Square statistic:\[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i},\]where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency.\[\begin{align*}\chi^2 & = \frac{(50-25)^2}{25} + \frac{(100-75)^2}{75} + \frac{(190-125)^2}{125} + \frac{(160-275)^2}{275} \& = \frac{625}{25} + \frac{625}{75} + \frac{4225}{125} + \frac{13225}{275} \& = 25 + 8.33 + 33.8 + 48.09 = 115.22\end{align*}\]

Step 5: Determine the Critical Value

The Chi-Square statistic is compared against a critical value from the Chi-Square distribution table with 3 degrees of freedom (since there are 4 categories - 1) at \(\alpha = 0.01\). The critical value is approximately 11.34.

Step 6: Make the Decision

Since the calculated \(\chi^2\) value of 115.22 is much greater than the critical value of 11.34, we reject the null hypothesis. This means the distribution of defaulters is significantly different from the expected distribution based on education.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that helps us decide if there's enough evidence to support a specific claim about a population. In this exercise, we're using hypothesis testing to see if the distribution of educational backgrounds among credit card defaulters is different from what we expect. We start by formulating our null hypothesis, which assumes no difference between observed and expected distributions. Then, we check if evidence (our data) is strong enough to reject this assumption. This process involves calculating a test statistic and comparing it to a critical value from a statistical distribution.

Expected Frequency Calculations

Expected frequency calculations are crucial in a chi-square test. They represent what we would anticipate if the null hypothesis were true. In our problem, we want to know if the level of education affects the likelihood of someone defaulting on their credit card. To find the expected frequencies, we use the proportions of the overall population and apply them to the defaulters.
For example:

• For those with some high school education, we multiply 5% by the total defaulters (500), giving us 25.
• Similarly, we calculate 75 for high school graduates, 125 for some college, and 275 for college graduates.
This tells us what the defaulter distribution should look like if it's the same as the overall cardholder population. We'll compare these expected values against our observed values.

Statistical Significance

Statistical significance in hypothesis testing tells us whether our findings are likely due to chance or if they reflect true differences in the population. We assess this by comparing our calculated test statistic to a critical value.
In our exercise, we compute the chi-square statistic and compare it to a critical value from the chi-square distribution table. We use a 0.01 significance level, meaning there's a 1% chance of observing such extreme values by random chance, assuming the null hypothesis is true. Since our calculated value (115.22) is much greater than the critical value (11.34), this suggests a significant difference—that the distribution of defaulters isn't purely by chance.

Null and Alternative Hypothesis

Formulating the null and alternative hypotheses is the foundation of hypothesis testing. The null hypothesis ( H_0 ) asserts that there's no difference between our observed data and what we expect, based on a known distribution or model. Here, it suggests that the educational distribution of defaulters matches that of all cardholders.
The alternative hypothesis ( H_a ), on the other hand, suggests there's a difference—a mismatch between the expected and observed distribution. The decision to reject or not reject the null hypothesis depends on our test statistic. If our statistic exceeds a critical value, we reject the null hypothesis in favor of the alternative. In this scenario, our data supports the alternative hypothesis, indicating a significant difference in default distribution related to educational level.