Question

Health insurance benefits vary by the size of the company. The sample data below show the number of companies providing health insurance for small, medium, and large companies. For purposes of this study, small companies are companies that have fewer than 100 employees. Medium-sized companies have 100 to 999 employees, and large companies have 1,000 or more employees. The questionnaire sent to 225 employees asked whether or not the employee had health insurance and then asked the employee to indicate the size of the company.

Health Insurance
Size of the Company Yes No Total
Small 36 14 50
Medium 65 10 75
Large 88 12 100
A. Conduct a test of independence to determine whether health insurance coverage is independent of the size of the company. What is the p-value? Compute the value of the χ2 test statistic (to 2 decimals). Using α .05 level of significance, what is your conclusion?
B. A newspaper article indicated employees of small companies are more likely to lack health insurance coverage. Calculate the percentages of employees without health insurance based on company size (to the nearest whole number). Based on the percentages calculated above, what can you conclude?

Answer 1

A) The critical value found is 5.99 which is < the test statistic value ( 6.94 ) hence we will reject the null hypothesis (H₀)

The P-value associated is : 0.031 which is < 0.05 ( level of significance )

B) from the calculation it can be sen that employees of small companies are more likely without health insurance

Explanation:

A) To conduct a test of independence

we apply the hypothesis test

Null hypothesis (H₀) = The health insurance coverage is independent of the size of the company

Alternative hypothesis (H₁) = The health insurance coverage is not independent of the size of the company

The test statistic is calculated as attached below

also the expected values are attached below indicating the formula used

E = expected value

O = observed value

Test statistic value ( as calculated ) = 6.94

The critical value found is 5.99 which is < the test statistic value ( 6.94 ) hence we will reject the null hypothesis (H₀)

The P-value associated is : 0.031 which is < 0.05 ( level of significance )

B) percentage of employees without health insurance

Small company : 14/50 * 100 = 28%

medium company: 10/75 * 100 = 13%

large company : 12/100 * 100 = 12%

from the above calculation it can be sen that employees of small companies are more likely without health insurance