Question

Health insurance benefits vary by the size of the company (the Henry J. Kaiser Family Foundation website, June 23, 2016). The sample data below show the number of companies providing health insurance for small, medium, and large companies. For the purposes of this study, small companies are companies that have fewer than  employees. Medium-sized companies have  to  employees, and large companies have  or more employees. The questionnaire sent to  employees asked whether or not the employee had health insurance and then asked the enployee to indicate the size of the company.

		Health Insurance
	Size of Company	Yes	No	Total
	Small	36	14	50
	Medium	66	9	75
	Large	87	13	100

a. Conduct a test of independence to determine whether health insurance coverage is independent of the size of the company. What is the -value?

Compute the value of the  test statistic (to 2 decimals).

The p value is _______ (less than .005, between .005 and .01, between .01 and .025, between .025 and .05, between .05 and .10, greater than .10)

Item 2

Using  level of significance, what is your conclusion?

________________

(Cannot reject the assumption that health insurance coverage and size of the company are independent, Conclude health insurance coverage is not independent of the size of the company)

Item 3

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).

Small
Medium
Large

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

__________(Large companies have a higher percentage of no coverage than medium and small companies, Medium companies have a higher percentage of no coverage than large and small companies, Small, medium, and large companies all have roughly the same percentage of no coverage, Small companies have a higher percentage of no coverage than large and medium companies)

Answer 1

Solution:

a. Null Hypothesis (Ho): Health insurance coverage is independent of the size of the company.

Alternative Hypothesis (Ha): Health insurance coverage is dependent on the size of the company.

Observed Frequency Table (O)

		Health Insurance
Size of company		Yes	No	Total
Small		36	14	50
Medium		66	9	75
Large		87	13	100
Total		189	36	225

Expected frequency (E) is calculated using the formula

Expected frequency = (Row total x Column total)/Overall total

		Health Insurance
Size of company		Yes	No	Total
Small		42.00	8.00	50
Medium		63	12	75
Large		84.00	16.00	100
Total		189	36	225

Test Statistics

6.92

Using chi-square tables, the p-value with degrees of freedom , df = (number of rows - 1) (number of columns -1) = (3 -1)(2-1)=2 falls between 0.025 and 0.05

Since p-value is less than 0.05 significance level, we reject the null hypothesis.

Conclude health insurance coverage is not independent of the size of the company

------------------------------------------------------------------------------------------------------------------------------------------

b.

Small 14/50 = 0.28

Medium 9/75 = 0.12

Large 13/100 = 0.13

Small companies have a higher percentage of no coverage than large and medium companies