Question

u suspect that the true proportion of teens who smoke is different than 15%. Conduct a hypothesis test to test your theory. Use a level of significance of LaTeX: \alpha = 0.05α = 0.05. Calculate the point estimator LaTeX: \hat{p}p ^, the test statistic, and the P-value for this test.

Answer 1

Answer:

There is enough statistical evidence that the proportion of teenagers that smoke is different from 15%.

Step-by-step explanation:

The question is incomplete:

"Gallup survey interviewed a nationally representative sample of 785 teenagers aged 13 to 17. Seventy-one teenagers in the survey acknowledged having smoked at least once in the past week."

We have this null and alternative hypothesis:

$H_0:p=0.15\\\\H_a: p\neq 0.15$

The significance level is

$\alpha=0.05$

The sample proportion is:

$\hat{p}=X/n=71/785=0.09$

The standard deviation is

$\sigma=\sqrt{\frac{p(1-p)}{n}} =\sqrt{\frac{0.15*0.85}{785}}=0.013$

The z-statistic can be calculated as:

$z=\frac{\hat{p}-p+0.5/n}{\sigma} =\frac{0.09-0.15+0.00}{0.013}=\frac{-0.06}{0.013}= -4.61$

The P-value of z=-4.61 for this test is P=0

$2P(|z|>-4.61)=0$

The P-value is smaller than the significance level, so the effect is significant.

Then, the null hypothesis is rejected.

There is enough statistical evidence that the proportion of teenagers that smoke is different from 15%.