Question

A researcher investigates whether there is a relationship between the number of cigarettes a person smokes and whether the person wears a seat belt when driving. His thinking is that people who smoke more are less concerned about their health and safety and therefore may be less inclined to wear seat belts. He collects the following data:

Number of Cigarettes Smoked per Day

                                                0          1-14        15-34            35 and over

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

            Wear seat belts            175        20           42                 41

            Don’t wear seat belts  150        17           53                 64

At a 0.01 significance level test the claim that there is a relationship between the number of cigarettes a person smokes and whether the person wears a seat belt when driving.

Answer 1

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

H₀: Number of cigarettes a person smokes and whether the person wears a seat belt when driving are independent.
H_a: Number of cigarettes a person smokes and whether the person wears a seat belt when driving are not independent.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (4 - 1)
D.F = 3
E_r,c = (n_r * n_c) / n


Χ² = 8.4150

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 8.415.

We use the Chi-Square Distribution Calculator to find P(Χ² > 8.415) = 0.038.

Interpret results. Since the P-value (0.038) is greater than the significance level (0.01), we have to accept the null hypothesis.

Thus, we conclude that there is a relationship between number of cigarettes a person smokes and whether the person wears a seat belt when driving.