Question

3. With double- digit annual percentage increases in the cost of health insurance, more and more workers are likely to lack health insurance coverage. The following sample data provide a comparison of workers with and without health insurance coverage for small, medium, and large companies. For the purposes of this study, small companies are companies that have fewer than 100 employees. Medium companies have 100 to 999 employees, and large companies have 1000 or more employees. Sample data is reported as follows: Health Insurance Size of Company Yes No Total Small 50 25 75 Medium 80 20 100 Large 115 10 125 Total 245 55 300 a. Conduct a test of independence using critical-value approach to determine whether employee health insurance coverage is independent of the size of the company. State the Hypotheses and the conclusion. Use α = .005. b. What is the p-value? What is your conclusion based on p-value approach? c. The USA Today article indicated employees of small companies are more likely to lack health insurance coverage. Use percentages based on the preceding data to support this conclusion.

Answer 1

a.

The sample data is,

Size of Company \ Health Insurance	Yes	No	Total
Small	50	25	75
Medium	80	20	100
Large	115	10	125
Total	245	55	300

Null Hypothesis H0: Employee health insurance coverage is independent of the size of the company.

Alternative Hypothesis Ha: Employee health insurance coverage is dependent of the size of the company.

For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

DF is the degrees of freedom, r is the number of levels of company size, c is the number of levels of the health insurance, nr is the number of observations from level r of company size, nc is the number of observations from level c of health insurance, n is the number of observations in the sample, Er,c is the expected frequency count when company size is level r and health insurance is level c, and Or,c is the observed frequency count when company size is level r health insurance is level c.

DF = (r - 1) * (c - 1) = (3 - 1) * (2 - 1) = 2

E_r,c = (n_r * n_c) / n
E_1,1 = (245 * 75) / 300 = 61.25
E_1,2 = (55 * 75) / 300 = 13.75
E_2,1 = (245 * 100) / 300 = 81.67
E_2,2 = (55 * 100) / 300 = 18.33
E_3,1 = (245 * 125) / 300 = 102.08
E_3,2 = (55 * 125) / 300 = 22.92

Χ² = Σ [ (O_r,c - E_r,c)² / E_r,c ]
Χ² = (50 - 61.25)²/61.25 + (25 - 13.75)²/13.75 + (80 - 81.67)²/81.67 + (20 - 18.33)²/18.33 + (115 - 102.08)²/102.08 + (10 - 22.92)²/22.92 = 20.37542

b.

p-value = P[ > 20.375, DF = 2) = 0.000038

Since p-value is less than the the significance level, we reject null hypothesis H0 and conclude that there is significant evidence that Employee health insurance coverage is dependent of the size of the company.

c.

Percentage of employees of small companies to lack health insurance coverage = (25/75) * 100 = 33.33%

Percentage of employees of medium companies to lack health insurance coverage = (20/100) * 100 = 20%

Percentage of employees of large companies to lack health insurance coverage = (10/125) * 100 = 8%

Since the percentage of employees of small companies to lack health insurance coverage is largest among all company sizes, employees of small companies are more likely to lack health insurance coverage.