Question

The following data represent a sample of people's smoking habits and their usage of seat belts while in a car. A researcher wants to determine whether smoking habits and seat belt usage are related. Test the researcher's claim at the

level of significance.

No Seat Belt Seat Belt
Smoke 67 448
Do not Smoke 327 2,187
determine t statistic

Answer 1

The calculated χ² value is approximately 0.004445. Compare this with the critical χ² value at the desired level of significance to make a decision about the null hypothesis.

Certainly! Let's go through the detailed calculations for the chi-square test of independence.

Given data:

-------------------------------------

| No Seat Belt | Seat Belt | Total

-----------------|--------------|-----------|-------

Smoke | 67 | 448 | 515

Do not Smoke | 327 | 2,187 | 2,514

-----------------|--------------|-----------|-------

Total | 394 | 2,635 | 3,029

-------------------------------------

Calculate the expected frequencies (Ei):

- E (Smoke, No Seat Belt) ≈ 66.48

- E (Smoke, Seat Belt) ≈ 448.52

- E (Do not Smoke, No Seat Belt) ≈ 326.52

- E (Do not Smoke, Seat Belt) ≈ 2,186.48

Now, calculate the chi-square test statistic (χ²):

`\[ \chi^2 \approx ((67 - 66.48)^2)/(66.48) + ((448 - 448.52)^2)/(448.52) + ((327 - 326.52)^2)/(326.52) + ((2,187 - 2,186.48)^2)/(2,186.48) \]`

`\[ \chi^2 \approx 0.00346 + 0.000061 + 0.000828 + 0.000105 \]`

`\[ \chi^2 \approx 0.004445 \]`

Now, compare this χ² value with the critical χ² value at the desired level of significance (using a chi-square distribution table). If the calculated χ² is greater than the critical value, reject the null hypothesis.

Note: Degrees of freedom (df) for a 2x2 table is 1.