Question

According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is $14\%.$ Suppose that a survey is conducted to determine this year’s rate. Nine out of $70$ randomly chosen N.Y. City residents reply that they smoke. Conduct a hypothesis test to determine if the rate is still $14\%$ or if it has decreased.

Short answer

The hypothesis test has resulted in ap-value greater than the alpha value and the null hypothesis is not rejected and the rate is still at $14\%$.

Step-by-step solution

Given information

The adult smoking rate$=14\%$

Explanation

State the hypothesis:

The null hypothesis states that $14\%$of adults do smoke and the alternate hypothesis states that not more than $14\%$of adults do smoke.

$$\begin{gathered} H_0:p=0.14 \\ {H}_0:p<0.14 \end{gathered}$$

The normal distribution is:

$N\left(0.14,\sqrt{\frac{\left(0.14\right)\left(0.86\right)}{70}}\right)$

The Z test statistic is:

$$\begin{gathered} z=\frac{p^{\wedge }-p}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ {P}^{\wedge }=\frac{x}{n} \\ =\frac{9}{70} \\ =0.129 \end{gathered}$$

Substitute the $p^{\wedge }$value in the Z statistic formula

$$\begin{gathered} z=\frac{p^{\wedge }p}{\sqrt{{\displaystyle \frac{p\left(1-p\right)}{n}}}} \\ =\frac{-0.01143}{0.0415} \\ =0.2756 \end{gathered}$$