Question

A trucking company wants to find out if their drivers are still alert after driving long hours. So, they give a test for alertness to two groups of drivers. They give the test to 395 drivers who have just finished driving 4 hours or less and they give the test to 565 drivers who have just finished driving 8 hours or more. The results of the tests are given below. Passed Failed Drove 4 hours or less 290 105 Drove 8 hours or more 350 215 Is there is a relationship between hours of driving and alertness? (Do a test for independence.) Test at the 1 % level of significance.

Answer 1

Solution:-

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

H₀: Hours of driving and alertness are independent.
H_a: Hours of driving and alertness 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) * (2 - 1)
D.F = 1
E_r,c = (n_r * n_c) / n

?² = ? [ (O_r,c - E_r,c)² / E_r,c ]
?² = 13.76

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 1 degrees of freedom is more extreme than 13.76.

We use the Chi-Square Distribution Calculator to find P(?² > 13.76) = 0.00021

Interpret results. Since the P-value (0.00021) is less than the significance level (0.01), we cannot accept the null hypothesis. Thus, we conclude that there is a relationship between Hours of driving and alertness.