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.


Size of the company
Health Insurance Small Medium Large
Yes 36 65 88
No 14 10 12

a. Conduct a test of independence to determine whether health insurance coverage is independent of the size of the company. Using a .05 level of significance. Compute the value of the 2 test statistic (to 2 decimals).

The p-value is:_______
What is your conclusion?

b. A newspaper article indicated employees of small companies are more likely to lack health insurance coverage. Use percentages based on the above data to support or reject this conclusion. (Round to whole numbers.)

Small:__________ %
Medium:_______ %
Large:__________ %

Based on the percentages calculated above, what can you conclude?

Answer 1

Answer:

Step-by-step explanation:

a. The hypothesis of testing are

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 statistics is calculated as

X² =  ∑(Ei - Oi)²  ≅X²(r-1)(c-1)

        i      Ei

where Oi's are the observed values and Ei's are the expected values,

E= row total × column total

            total

The expected values are

        Sizeof the Company  

Health Insurance    Small   Medium Large Total

Yes                             42     63          84           189

No                              8             12          16           36

Total                      50    75                  100           225

Therefore, the test statistics is

X² =∑ (Ei -Oi)² =6.94 ≅  x²₂

      i       Ei

The critical value is found to be 5.99. Since test statistics > critical value, we reject H0 and conclude that the health insurance coverage is not independent of the size of the company.

The associated p-value is 0.031 (<0.05).

b. Percentage of employees lacking health insurance coverage is

Small 14/50 x 100 = 28%

Medium 10/75 x 100 = 13%

Large 12/100 x 100 = 12%

Hence depending on the above percentage, we can support the claim that the employees of small companies are more likely to lack health insurance coverage.