Answer to Question #200948 in Statistics and Probability for Zach

Question

In some states, the law requires drivers to turn on their headlights when driving in the rain. A highway patrol officer believes that less than one-quarter of all drivers follow this rule. As a test, he randomly samples 200 cars driving in the rain and counts the number whose headlights are turned on. He finds this number to be 41. Does the officer have enough evidence at the 10% significance level to support 

his belief? 

Answer 1

We calculate the p-value

Null hypothesis (H_o) is p=0.25 and alternate hypothesis (H₁)p<0.25

Estimated p-value$(\tilde{p})$ can be calulated as follows

$\tilde{p} =\frac{event _{successful}}{event _{total}}$

$=\frac{41}{200}$

$=0.205$

The z-value can be calculated as follows

$Z=\frac{\tilde{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

$Z=\frac{{0.205}-0.25}{\sqrt{\frac{0.25(1-0.25)}{200}}}$

$=-1.47$

The p value can be calculated as follows

$P=p(z<-1.47)$

$=0.0708$

The p-value is 0.0708. Since the p-value is very lower, it accepts the alternative hypothesis.