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A large group of workers is divided into three departments: Special Operations, Main Floor, and Management. Strep throat has been spreading, and a health inspector wants to determine if there is a relationship between department and having strep throat. She selects a random sample of 100 workers and classifies each one according to their department and whether they have strep throat. The responses are displayed in the table:

[tex]
\[
\begin{array}{|c|c|c|c|}
\cline{2-4}
\multicolumn{1}{c|}{} & \text{Special Operations} & \text{Main Floor} & \text{Management} \\
\hline
\text{Strep} & 6 & 13 & 5 \\
\hline
\text{No Strep} & 40 & 30 & 6 \\
\hline
\end{array}
\]
[/tex]

The health inspector wants to know if these data provide convincing evidence that the distribution of strep throat status differs across the departments in the population of all workers in this large company. Are the conditions for inference met?

A. No, the random condition is not met.
B. No, the [tex]$10\%$[/tex] condition is not met.
C. No, the Large Counts condition is not met.
D. Yes, all three conditions for inference are met.

Answer :

To determine if the data provides convincing evidence that the distribution of strep throat status differs across departments, we need to check if the conditions for inference in a chi-square test are met. There are three main conditions to consider:

1. Random Sampling Condition: The data must be collected through a random sample. According to the problem, a random sample of 100 workers is taken, so this condition is satisfied.

2. 10% Condition: The sample size should be less than 10% of the population to ensure independence. Since the sample consists of 100 workers, it is assumed that the total number of workers is significantly larger than 1000 (which is 10 times the sample size), so this condition is satisfied.

3. Large Counts Condition: Each expected frequency should be at least 5 to use the chi-square test properly. We calculate the expected frequencies for each cell under the null hypothesis that there is no difference in distributions.

Let's review the table and expected frequencies:

- Observed counts:
- Strep:
- Special Operations: 6
- Main Floor: 13
- Management: 5
- No Strep:
- Special Operations: 40
- Main Floor: 30
- Management: 6

- Expected frequencies calculated based on proportions:
- Expected Strep:
- Special Operations: 11.04
- Main Floor: 10.32
- Management: 2.64
- Expected No Strep:
- Special Operations: 34.96
- Main Floor: 32.68
- Management: 8.36

The expected frequencies for the "Strep" in the Management department is less than 5 (2.64), and there might be other expected numbers that do not meet the Large Counts Condition of being at least 5.

Since not all expected frequencies are at least 5, the Large Counts Condition is not met. This means inference conditions are not fully satisfied for conducting a chi-square test.

Hence, the answer is: No, the Large Counts condition is not met.

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