Question

From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of 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?

Answer 1

Answer:

If we compare the p value and the significance level given $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis, so on this case we have enough evidence to conclude that the true mean is significantly less than 19.  

Step-by-step explanation:

Data given and notation  

$\bar X=18.1$ represent the mean height for the sample  

$s=1.3$ represent the sample standard deviation for the sample

$\sigma=2.1$ represent the population standard deviation for the sample  

$n=40$ sample size  

$\mu_o =68$ represent the value that we want to test

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

t would represent the statistic (variable of interest)  

$p_v$ represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean starting age is at least 19, the system of hypothesis would be:  

Null hypothesis:$\mu \geq 19$  

Alternative hypothesis:$\mu < 19$  

We don't know the population deviation so is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

$z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}$  (1)  

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

$z=\frac{18.1-19}{\frac{2.1}{\sqrt{40}}}=-2.71$    

P-value

Since is a one side test the p value would be:  

$p_v =P(Z$  

Conclusion  

If we compare the p value and the significance level given $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis, so on this case we have enough evidence to conclude that the true mean is significantly less than 19.