Question

Teenage Tobacco Use: In a random sample of 1100 teenagers, 207 had used tobacco of some form in the last year. The managers of an anti-tobacco campaign want to claim that less than 20% of all teenagers use tobacco. Test their claim at the 0.10 significance level.

Answer 1

The P-value (0.159) is higher than the significance level (0.1), so we can not reject the null hypothesis.

The claim of the anti-tobacco campaign could be right about their estimation.

Explanation:

This is a problem of hypothesis testing. In this case, about the teenage population's proportion of smokers.

Hypothesis

The null hypothesis, that needs to be tested, is that the proportion of teenage that use tobacco is less than 20%:

`H_0: \mu<0.2`

The alternative hypothesis is that this proportion is higher than 20%

`H_0: \mu>0.2`

The significance level is 0.10 and it is used a one-sample z-test.

Analysis

The proportion for the sample is:

`p=207/1100=0.188`

Then, we calculate the standard deviation of the sample:

`\sigma=\sqrt{(p(1-p))/(n) } =\sqrt{(0.188(1-0.188))/(1100) }= 0.012`

The z-value can now be calculated as:

`z=(p-\mu)/(\sigma)=(0.188-0.2)/(0.012)  =-1`

The P-value for
`z=-1` is

`P(z<-1)=0.15866`

The P-value (0.159) is higher than the significance level (0.1), so we can not reject the null hypothesis.