Question

Suppose the data below represent, in thousands, the type of health insurance coverage of people of a certain age.

a. Determine P(< 18 years old | no health insurance).

b. Are the events "< 18 years old" and "no health insurance" Independent?

Age
<18 18-44 45-64 >64 Total
Private health insurance 43,167 74,871 58,228 29,420 205,686
Government health insurance 24,201 13,552 13,247 39,286 90,285
No Health insurance 8,867 29,451 14,108 731 53,157
Total 76,235 117,874 85,583 69,436 349,128

Answer 1

a.
`\( P( < 18 \text{ years old} | \text{no health insurance}) \approx 0.1667 \).`

b. The events are dependent as
`\( P( < 18 \text{ years old} \cap \text{no health insurance}) \\eq P( < 18 \text{ years old}) * P(\text{no health insurance}) \).`

a. To determine
`\( P( < 18 \text{ years old} | \text{no health insurance}) \)`, we use the conditional probability formula:

`\[ P( < 18 \text{ years old} | \text{no health insurance}) = \frac{\text{Number of people < 18 with no health insurance}}{\text{Total number of people with no health insurance}} \]`

From the table:

`\[ P( < 18 \text{ years old} | \text{no health insurance}) = (8,867)/(53,157) \approx 0.1667 \]`

b. To check independence, we compare
`\( P( < 18 \text{ years old}) \)` and
`\( P(\text{no health insurance}) \):`

`\[ P( < 18 \text{ years old}) = \frac{\text{Number of people < 18}}{\text{Total number of people}} = (76,235)/(349,128) \]`

`\[ P(\text{no health insurance}) = \frac{\text{Number of people with no health insurance}}{\text{Total number of people}} = (53,157)/(349,128) \]`

If
`\( P( < 18 \text{ years old}) * P(\text{no health insurance}) = P( < 18 \text{ years old} \cap \text{no health insurance}) \)`, the events are independent. Otherwise, they are dependent. Calculate this to determine independence.