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Given

$x_1=5690$

$n_1=12931$

$x_2=1051$

$n_2=3285$

Determine the hypothesis

$H_0:p_1-p_2=0$

$H_a:p_1-p_2鈮�0$

This is important for the data of each car to be independent of the other cars.

If a car slows down, because other cars slow down, then this car is not independent of the other cars that slowed down too.

Given

$x_1=5690$

$n_1=12931$

$x_2=1051$

$n_2=3285$

Determine the hypothesis

$H_0:p_1-p_2=0$

$H_a:p_1-p_2鈮�0$

The sample proportion is the number of successes divided by the sample size:

${\hat{p}}_1=\frac{x_1}{n_1}=\frac{5690}{12931}鈮�0.440$

${\hat{p}}_2=\frac{x_2}{n_2}=\frac{1051}{3285}鈮�0.320$

${\hat{p}}_p=\frac{x_1+x_2}{n_1+n_2}=\frac{5690+1051}{12931+3285}=\frac{6741}{16216}鈮�0.416$

Determine the value of the test statistic:

localid="1650451405878" $$\begin{gathered} z=\frac{{\hat{p}}_1-{\hat{p}}_2}{\sqrt{{\hat{p}}_p\left(1-{\hat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \\ =\frac{0.440-0.320}{\sqrt{0.416(1-0.416)}\sqrt{\frac{1}{12931}+\frac{1}{3285}}} \\ 鈮�12.46 \end{gathered}$$

The $p$-value is the probability of obtaining the value of the test statistic, or a value more extreme. Determine the $p$-value using table A:

localid="1650451422257" $$\begin{gathered} P=P(Z<-12.46\text{or}Z>12.46) \\ =2脳P(Z<-12.46) \\ =2脳0.0001 \\ =0.0002 \end{gathered}$$

If the $p$-value is smaller than the significance level, reject the null hypothesis:

$P<0.05鈬�\text{Reject}{H}_0$