Question

Driver error can be listed as the cause of approximately $54\%$of all fatal auto accidents, according to the American Automobile Association. Thirty randomly selected fatal accidents are examined, and it is determined that $14$were caused by driver error. Using $\alpha =0.05,$ is the AAA proportion accurate?

Short answer

There is sufficient evidence that $54\%$of all fatal auto accidents.

Step-by-step solution

Given information

$x=14,n=30,p_0=0.54,a=0.05$

Explanation

State the hypothesis:

The null hypothesis states that $54\%$of all fatal auto accidents. In symbols:

$H_0:p=0.54$

The alternative hypothesis states that not $54\%$of all fatal auto accidents. In symbols:

$H_a:p\ne 0.54$

The sample proportion is:

$$\begin{gathered} \stackrel{⏜}{p}=\frac{x}{n} \\ =\frac{14}{30} \\ =0.467 \end{gathered}$$

The test statistic is:

$$\begin{gathered} z=\frac{\stackrel{⏜}{p}-p_0}{\sqrt{{\displaystyle \frac{p_0\left(1-p_0\right)}{n}}}} \\ =\frac{0.467-0.54}{\sqrt{{\displaystyle \frac{0.54\left(1-0.54\right)}{30}}}} \\ =\frac{-0.073}{0.0910} \\ =-0.81 \end{gathered}$$

The p-value is:

$$\begin{gathered} p-value=P\left(z<-0.81\right)+P\left(z>0.81\right) \\ =0.2090+0.2090 \\ =0.4180 \end{gathered}$$

Since $p-value=0.4180>0.05$, do not reject the null hypothesis.