# Investigating Egg Purchase Preferences in a Large City

## Are the data from the sample consistent with the distributor's claim?

To investigate this claim, researchers collected data from a random sample of the distributor's customers in a large city. The results were 274 white, 159 brown, 126 organic, and 41 free-range egg purchases.

## Statistical Test Analysis

The conclusion based on the statistics is that 45% of all individuals who purchase eggs from its grocery store choose white eggs. The hypothesis testing will be conducted to determine the consistency of the sample data with the distributor's claim.

Before conducting the statistical test, we need to formulate the null and alternative hypotheses. Let's assume:

H0: The proportions of white, brown, organic, and free-range egg purchases match the distributor's claim.

Ha: The proportions of egg purchases do not match the distributor's claim.

Next, we need to check the conditions for conducting the chi-square test:

- Random Sample: The data was collected from a random sample of the distributor's customers.
- Expected Cell Frequency: All expected cell frequencies are greater than 5.
- Independence: The categories of egg purchases are independent of each other.

Now, let's calculate the test statistic and degree of freedom for the chi-square test:

Test Statistic = Σ [(Observed - Expected)^2 / Expected]

From the data provided and assuming the distributor's claim percentages, we can calculate the expected frequencies for each type of egg purchase.

After calculating the test statistic, we determine the critical value at the 5% significance level with degrees of freedom equal to 3. The critical value from the chi-square table is 7.815.

Comparing the test statistic to the critical value, if the test statistic is less than the critical value, we fail to reject the null hypothesis. Therefore, the conclusion is that the sample data is consistent with the distributor's claim that 45% of all individuals choose white eggs.