Question

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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 30 59 82
No 20 16 18
(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 not independent of the size of the company.
Ha: Having health insurance is independent of the size of the company. H0: Having health insurance is not independent of the size of the company.
Ha: The proportion of companies with health insurance benefits is not equal for small, medium and large companies. H0: Having health insurance is independent of the size of the company.
Ha: The proportion of companies with health insurance benefits is equal for small, medium and large companies.H0: Having health insurance is independent of the size of the company.
Ha: Having health insurance is not independent of the size of the company.
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?
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. 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.
(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 more equally 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.

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

χ²= 9.284

The corresponding p-value for the test is = 0.0096

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

For small companies, 40% do not provide health insurance. For medium companies, 11.4% do not provide health insurance. For large companies,18 % do not provide health insurance.

These percentages support the conclusion that small companies are less more equally likely to provide health insurance coverage when compared to medium and large companies.

Explanation:

small medium large Total

yes 30 59 82 171

no 20 16 18 54

Total: 50 75 100 225

Expected Values small medium large Total

yes { 50* 171/225} = 38 57 76 171

no 12 18 24 54

Total 50 75 100 225

Squared Distances small medium large

yes {(30 - 38)^2}/38

= 1.684 0.07 0.474

no {(20 - 12)^2} /12

= 5.333 0.22 1.5

1) The null and alternative hypotheses can be formulated as

H0: Having health insurance is independent of the size of the company.

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

(2) the significance level is alpha =0.05 ,

the number of degrees of freedom is df = (2 - 1) * (3 - 1) = 2df

so then the rejection region for this test is χ 2 :χ 2 >5.991.

3) Test Statistics under H0 is

χ²= ∑ (O - E)²/ E where O is the observed and E is the expected frequency

4) χ²= 1.684+5.333+0.07+0.222+0.474+1.5

= 9.284

Since it is observed that χ²=9.284 >χ²= 5.991 it is concluded that the null hypothesis is rejected.

5) Conclusion

It is concluded that the null hypothesis H0 is rejected. Therefore, there is not enough evidence to claim that the two variables are dependent at 0.05 significance level.

The corresponding p-value for the test is = 0.0096

b) The percentages of

Small %= 20/50= 0.4= 40%

Medium %= 16/75 = 0.786= 11.4%

Large %= 18/100= 0.18= 18%

For small companies, 40% do not provide health insurance. For medium companies, 11.4% do not provide health insurance. For large companies,18 % do not provide health insurance.

These percentages support the conclusion that small companies are less more equally likely to provide health insurance coverage when compared to medium and large companies.