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 CompanyYesNoTotal Small311950 Medium70575 Large8812100 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 x^2 test statistic (to 2 decimals).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

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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	31	19	50
Medium	70	5	75
Large	88	12	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 x^2 test statistic (to 2 decimals).

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

Expert Solution

Step 1

NOTE: As the significance level, i.e, $α$ is not mentioned, I am taking it as 5%, i.e, 0.05.

a).

The null hypothesis (H₀) states that health insurance coverage is independent of the size of the company.
The alternative hypothesis (H_a) states that health insurance coverage is dependent of the size of the company.

First we will depict the information in a proper manner. The given frequencies are the observed frequencies. It is shown as:

Observed Frequencies (O)
	Yes	No	Row Total
Small	31	19	31+19=50
Medium	70	5	70+5=75
Large	88	12	88+12=100
Column Total	31+70+88=189	19+5+12=36	(189+36) or (50+75+100)=225

Now, we have to calculate the expected frequencies. The formula for expected frequencies is:

$E x p e c t e d f r e q u e n c y = \frac{R o w t o t a l * C o l u m n t o t a l}{G r a n d t o t a l}$

The following table shows the calculations of the expected freqncuies:

Expected Frequencies (E)
	Yes	No
Small	50*189/225=42	50*36/225=8
Medium	75* 189/225=63	75*36/225=12
Large	100*189/225=84	100*36/225=16

The following table shows the required calculations for the $χ^{2}$ test statistic.

O	E	(O-E)²	(O-E)²/E
31	42	121	2.881
19	8	121	15.125
70	63	49	0.7778
5	12	49	4.0833
88	84	16	0.1905
12	16	16	1

The formula and calculations for the $χ^{2}$ test statistic is as follows;

$χ^{2} = \sum_{}^{} \frac{{(O - E)}^{2}}{E} = 2.881 + 15.125 + 0.7778 + 4.0833 + 0.1905 + 1 = 24.0576 \approx 24.06$

Thus, the $χ^{2}$ test statistic=24.06

The degree of freedom for the $χ^{2}$ test is calculated as:

Df=(Number of rows-1)*(Number of column-1)
=(3-1)*(2-1)
=2

Now, observe $χ^{2} = 24.06$ and df=2 in the $χ^{2}$ table to obtain the p-value.
The p-value obtained is less than 0.00001

The decision rule states that if the p-value is less than the significance level, we Reject the null hypothesis.

In this case, 0.00001<0.05, Hence We Reject the null hypothesis.
Therefore, It is concluded that health insurance coverage is dependent of the size of the company.