Question

From generation to generation, the mean age when smokers first start to smoke varies. However , the standard deviation pf that age remains constant of around $2.1$years . A survey of $40$smokers of this generation was done to see if the mean starting age is at least $19$. the sample mean was $18.1$ with a sample standard deviation of$1.3$do the data support the claim at the $5\%$level?

Short answer

The true mean is significantly less than $19$.

Step-by-step solution

Given Information

From the question itself we are given,

$\overline{X}=18.1$represents the mean height for the sample .

$s=1.3$represents the sample standard deviation for the sample.

$\sigma =2.1$represents the population standard deviation for the sample .

$n=40$represents the sample size .

${\mu }_0=68$represents the value that we want to test.

$\alpha =0.05$represents the significance level for the hypothesis test.

$t$would represent the statistic ( variable of interest)

$p_v$represents the p value for the test

Explanation

$$\begin{gathered} H_0=\mu \ge 19 \\ {H}_0=\mu <19 \end{gathered}$$

The statistic is given by:

$$\begin{gathered} z=\frac{\overline{X}-{\mu }_o}{{\displaystyle \frac{\sigma }{\sqrt{n}}}} \\ z=\frac{18.1-19}{{\displaystyle \frac{2.1}{\sqrt{40}}}} \\ z=-2.71 \end{gathered}$$

Since it is a one sided test the p value would be ;

$$\begin{gathered} p_v=P(z<-2.71) \\ {p}_v=0.0033 \end{gathered}$$