Question

health insurance benefits vary by the size of the company (Atlanta Business chronicle, December 31, 2010). 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 com- panies have 100 to 999 employees, and large companies have 1000 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.
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? using a .05 level of significance, what is your conclusion?
b.   a newspaper article indicated employees of small companies are more likely to lack health insurance

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

Contingency Table						ANSWERS
	Health Insurance
Size of Company	Yes	No	TOTALS			H0:
Small						Ha:
Medium
Large						P-VALUE
TOTALS						Conclusion:
Expected Frequency Table
	Health Insurance
Size of Company	Yes	No	TOTALS			Percentages
Small						Small
Medium						Medium
Large						Large
TOTALS
Test Statistic Calculation
	Health Insurance
Size of Company	Yes	No	TOTALS
Small
Medium
Large
TOTALS

Answer 1

The frequency table of the health insurance status is:

	Small	Medium	Large	Total
Yes	36	65	88	189
No	14	10	12	36
Total	50	75	100	225

The hypothesis under the problem are:

H0: Health insurance coverage is independent of the size of the company.

H1: Health insurance coverage is dependent on the size of the company

The expected frequency table is given as:

	Small	Medium	Large	Total
Yes	42	63	84	189
No	8	12	16	36
Total	50	75	100	225

The test statistic under the null hypothesis is:

with 2 degrees of freedom

chi = 6.94

p-value = 0.031

Since the p-value is less than 0.05 we reject the null hypothesis at 5% level of significance and conclude that the health insurance coverage is dependent on the size of the company.

b) It can be seen that the expected number of employees of the small companies is 42 and the actual number of employees who have insurance is 36. There is difference of 6 employees which is almost 16.67% and thus, the claim of the article is valid small companies are more likely to lack health insurance.