Answer: Reject \(H_0\). At the \(10\%\) significance level, there is sufficient evidence to suggest that performance in the driving test is associated with age.

The test statistic is \(\chi^2=5.21\), with \(2\) degrees of freedom, and the critical value is \(4.605\).

Let \(H_0\) be that performance in the driving test is independent of age, and let \(H_1\) be that performance in the driving test is not independent of age.

The observed frequencies, with expected frequencies in brackets, are:

For example, the expected frequency for pass in the first age group is \(\frac{81\times 50}{145}=27.93\). The other expected frequencies are found similarly using \(\frac{\text{row total}\times \text{column total}}{\text{grand total}}\).

The chi-squared test statistic is

\(\chi^2=\sum \frac{(O-E)^2}{E}\).

So

\(\chi^2=\frac{(34-27.93)^2}{27.93}+\frac{(41-44.69)^2}{44.69}+\frac{(6-8.38)^2}{8.38}+\frac{(16-22.07)^2}{22.07}+\frac{(39-35.31)^2}{35.31}+\frac{(9-6.62)^2}{6.62}\).

This gives \(\chi^2\approx 5.21\).

The number of degrees of freedom is \((2-1)(3-1)=2\). At the \(10\%\) significance level, the critical value for \(\chi^2_2\) is \(4.605\).

Since \(5.21\gt 4.605\), reject \(H_0\). There is sufficient evidence to suggest that driving-test performance and age are not independent.