Question

a large group of workers is divided into three departments: those who work in special operations, those who work on the main floor, and those who work in management. strep throat has been going around and a health inspector would like to know 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.

the health inspector would like 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 of this large company. are the conditions for inference met?

no, the random condition is not met.

no, the 10% condition is not met.

no, the large counts condition is not met.

yes, all three conditions for inference are met.

Explanation:

Step1: Check Random Condition

The health inspector selects a random sample of workers. So the random condition is met.

Step2: Check 10% Condition

The population is all workers of the large company. A sample of 100 workers is less than 10% of a large population. So the 10% condition is met.

Step3: Check Large Counts Condition

First, calculate the row totals: Strep row total \(6 + 13+5=24\), No - Strep row total \(40 + 30+6 = 76\). Column totals: Special Operations \(6 + 40=46\), Main Floor \(13+30 = 43\), Management \(5+6=11\).
Expected count formula \(E_{ij}=\frac{\text{Row total}\times\text{Column total}}{\text{Grand total}}\)
Grand total \(=100\)
For Special Operations - Strep: \(E=\frac{24\times46}{100}=11.04\)
For Special Operations - No Strep: \(E=\frac{76\times46}{100}=34.96\)
For Main Floor - Strep: \(E=\frac{24\times43}{100}=10.32\)
For Main Floor - No Strep: \(E=\frac{76\times43}{100}=32.68\)
For Management - Strep: \(E=\frac{24\times11}{100}=2.64\)
For Management - No Strep: \(E=\frac{76\times11}{100}=8.36\)
Since \(2.64<5\), the Large Counts condition is not met.

Answer:

No, the Large Counts condition is not met.