Question

With double-digit annual percentage increases in the cost of health insurance, more and more workers are likely to lack health insurance coverage (USA Today, January 23, 2004). The following sample data provide a comparison of workers with and without health insurance coverage for small, medium, and large companies. For the purposes of this study, small companies are companies that have fewer than 100 employees. Medium companies have 100 to 999 employees, and large companies have 1000 or more employees. Sample data are reported for 50 employees of small companies, 75 employees of medium companies, and 100 employees of large companies. Health Insurance Size of Company Yes No Total Small 31 19 50 Medium 68 7 75 Large 90 10 100 Conduct a test of independence to determine whether employee health insurance coverage is independent of the size of the company. Use = .05. Use Table 12.4. Compute the value of the 2 test statistic (to 2 decimals). The p value is What is your conclusion? The USA Today 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 % Based on the percentages calculated above, what can you conclude?

Answer 1

To test the independence between the given categorical variables:Health insurance coverage and size of the company, chi suare test for independence is the mos appropriate.

The chi square test statistic used for this purpose is given by,

where, O_i is the observed values and E_i denotes the expected values obtained from the cross tab.

Expected frequency for a cell =[ (Its row total)*(corresponding column total)] / Grand total

I.e. For cell 1 containing frequency '31', the expected frquency = 189 * 50 / 225

= 9450 / 225 = 42.

Similarly, computing the rest, we get,

Substituting in the formula for chi square, we obtain the test statistic value,

= 23.165 at (r-1)(c-1) = (2-1)(3-1) = 2 df

Using SPSS, by the following steps,

The p value obtained for the chi suare test (pearson's) is significant (0.000 > 0.05) at 5% level.This suggests that there is little evidence to support the null hypothesis H₀ : Health insurance coverage and Size of the company are independent. can be rejected.

Hence it may be concluded that Health insurance coverage and Size of the company are assosiated.

This can be supported by the percentages obtained based on these frequencies, here 52.28%.

The percentages of employees without health insurance based on company size is obtained as,

a. Percentages of employees without health insurance in small companies = 19 / 36 = 52.277 52%

b. In medium companies = 7 / 36 = 20%

c. In large companies = 10 / 36 = 28%

These percentages provides quite an evidence on the information provided by the USA Today article which states that employees of small companies are more likely to lack health insurance coverage compared to the others.