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	33	62	85
No	17	13	15

(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.

(a)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.

(b)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.  

(c)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.

(d)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.

(a-1)Find the value of the test statistic. (Round your answer to three decimal places.)

________________

(a-2)What is the p-value? (Round your answer to four decimal places.)

p-value = ______________

(a-3) Using a 0.05 level of significance, what is your conclusion?

(a)Reject H₀. We conclude health insurance coverage is independent of the size of the company.

(b)Do not reject H₀. We cannot conclude health insurance coverage is independent of the size of the company.

(c)Do not reject H₀. We cannot conclude health insurance coverage and the size of the company are not independent.

(d)Reject H₀. We conclude health insurance coverage is not 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___________(equally(or)more(or)less) likely to provide health insurance coverage when compared to medium and large companies.

Answer 1

A)

ANSWER: (C)

Ho: Having health insurance is independent of the size of the company.
Ha: Having health insurance is not independent of the size of the company.

a- 1)

Observed Frequency Table:

Health Insurance	Size of the Company
	small	medium	large	Total
Yes	33	62	85	180
No	17	13	15	45
Total	50	75	100	225

Now We have to find out Expected Frequencies:

Ei = (Corresponding row total * Corresponding column total) / sample size

For First cell:

E1 = (180 * 50) / 225 = 40

Similarly for all other cells, we have calculated for expected frequencies and presented in the following table:

The Expected Frequencies Table:

Health Insurance	Size of the Company			Total
	small	medium	large
Yes	40	60	80	180
No	10	15	20	45
Total	50	75	100	225

Using the above Observed and Expected Frequency table, we can find the value of Chi-square.

The calculation of chi-square is given in the following table:

	Oi	Ei	(Oi-Ei)	(Oi-Ei)2	(Oi-Ei)2/Ei
	33	40	-7	49	1.2250
	17	10	7	49	4.9000
	62	60	2	4	0.0667
	13	15	-2	4	0.2667
	85	80	5	25	0.3125
	15	20	-5	25	1.2500
Total	225	225	0	156	8.0208

Thus the value of Chi square statistic is given by,

a-2)

Df = (r-1) * (c-1)

Where, r = Number of rows = 2

c = Number of columns = 3

Df = (2-1)*(3-1) = 1*2 = 2

P-value: 0.0181

a-3)

P-value < , i.e. 0.0181 < 0.05, There is sufficient evidence to Reject Ho at 5% level of significance.

(d) Reject Ho. We conclude health insurance coverage is not independent of the size of the company.

B)

  

		yes	No	Total
small	Observed	33	17	50
	% of column	0.1833	37.78%
Medium	Observed	62	13	75
	% of column	34.44%	28.89%
Large	Observed	85	15	100
	% of column	47.22%	33.33%
Total	Observed	180	45	225
	% of column	100%	100%	100%

For small companies, 38 %do not provide health insurance. For medium companies, 29% do not provide health insurance. For large companies, 34% do not provide health insurance. These percentages support the conclusion that small companies are more likely to provide health insurance coverage when compared to medium and large companies