Question

g "Motor vehicle accidents are rare events, but it is always better to have fewer. Vehicle speed may be related to the number of accidents, and in general it is believed that slower speeds will lead to fewer accidents. On a stretch of highway 42, the average number of accidents per day was 0.23. A month ago, the speed limit was changed from 60 miles per hour to 50 miles per hour. In the one month period following the change in the speed limit, three accidents were observed. Explain in detail how you would test whether this is a significant decrease (3pts). Include a description of your null and alternative hypotheses."

Answer 1

Answer:

To test wether this is a significant decrease we have to perform a hypothesis test on the proportions.

The null hypothesis represents the past condition (the proportion of 0.23 accidents/day is equal or bigger) and the alternative hypothesis is what we claim that is happening now (the proportion have lowered).

We want to perform the test in order to know if there is enough evidence that it has changed. The result can be:

- The null hypothesis is rejected: there is enough evidence with this sample that the rate of accidents has lowered from 0.23.

- The null hypothesis failed to be rejected: there is not enough evidence to say that the rate of accidents has lowered, although the sample proportion is lower.

Step-by-step explanation:

Numerical solution:

We have to perform a hypothesis test on proportions. We want to know if there is enough evidence to claim that the number of accidents per day has lowered from 0.23.

The null and alternative hypothesis are:

$H_0: \pi\geq0.23\\\\H_1:\pi<0.23$

The significance level assumed is 0.05.

The proportion of the sample is:

$p=\frac{3}{30}=0.1$

The standard deviation is calculated from the population proportion

$\sigma=\sqrt{\pi(1-\pi)/N} =\sqrt{0.23*(1-0.23)/30} =0.077$

The z-value now can be calculated as

$z=\frac{p-\pi+0.5/N}{\sigma}=\frac{0.10-0.23+0.5/30}{0.077}  =-1.475$

The P-value for z=-1.475 is P=0.07011. The P-value is greater than the significant level, so the effect is not significant and it failed to reject the null hypothesis.