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
	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.

State the null and alternative hypotheses.

H₀: Having health insurance is independent of the size of the company.
H_a: Having health insurance is not independent of the size of the company.H₀: Having health insurance is independent of the size of the company.
H_a: The proportion of companies with health insurance benefits is equal for small, medium and large companies.    H₀: Having health insurance is not independent of the size of the company.
H_a: Having health insurance is independent of the size of the company.H₀: Having health insurance is not independent of the size of the company.
H_a: The proportion of companies with health insurance benefits is not equal for small, medium and large companies.

Find the value of the test statistic. (Round your answer to three decimal places.)

What is the p-value? (Round your answer to four decimal places.)

p-value =

Using a 0.05 level of significance, what is your conclusion?

Reject H₀. We conclude health insurance coverage is independent of the size of the company.Reject H₀. We conclude health insurance coverage is not independent of the size of the company.     Do not reject H₀. We cannot conclude health insurance coverage and the size of the company are not independent.Do not reject H₀. We cannot conclude health insurance coverage is independent of the size of the company.

(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 this conclusion. (Round your answers to the nearest integer.)

For small companies,  % do not provide health insurance. For medium companies,  % do not provide health insurance. For large companies,  % do not provide health insurance. These percentages support the conclusion that small companies are  ---Select--- less equally more likely to provide health insurance coverage when compared to medium and large companies.

Answer 1

H₀: Having health insurance is independent of the size of the company.
H_a: Having health insurance is not independent of the size of the company.

Applying chi square test of independence:

Expected	Ei=row total*column total/grand total	small	medium	large	Total
	Yes	42.0000	63.0000	84.0000	189
	No	8.0000	12.0000	16.0000	36
	total	50	75	100	225
chi square    χ2	=(Oi-Ei)2/Ei	small	medium	large	Total
	Yes	0.857	0.063	0.190	1.1111
	No	4.500	0.333	1.000	5.8333
	total	5.3571	0.3968	1.1905	6.9444
test statistic X2 =		6.944

p value =

0.0310

Reject H₀. We conclude health insurance coverage is not independent of the size of the company

b) For small companies 28.0 % do not provide health insurance. For medium companies, 13.3 % do not provide health insurance. For large companies 12 % do not provide health insurance. These percentages support the conclusion that small companies are less ikely to provide health insurance coverage when compared to medium and large companies.