91Ó°ÊÓ

The population value is the parameter of interest, and knowing the value or other information is required. PARAMETER OF INTEREST =Population proportion$p$of all $13$- to$17$-year-olds in the United States who believe that young people should wait until marriage to have sex.

$$\begin{gathered} n=439 \\ p=0.5 \end{gathered}$$

Random, Normal, and Independent conditions for performing a one-sample$z$test.

Random: Satisfied because the sample was chosen at random.

Normal: If the number of failures$n(1-p)$and the number of successes$np$is both greater than$10$, the distribution is assumed to be normal.

$$\begin{gathered} np=439×0.5 \\ =219.5 \end{gathered}$$

$$\begin{gathered} n(1-p)=439(1-0.5) \\ =219.5 \end{gathered}$$

Both are greater than$10$, so the normal requirement is met.

Independent: can be assumed because the sample size$\left(439\right)$) is less than $10\%$of the population size.

Therefore, all conditions are satisfied.

The Minitab output is having "Test of $p=0.5$vs. $pnot=0.5$", which implies that the hypotheses are:

$$\begin{gathered} H_0:p=0.5 \\ {H}_1:pnotequalto0.5 \end{gathered}$$

The sample proportion is as follows:

$$\begin{gathered} \hat{p}=0.560364 \\ =0.56 \end{gathered}$$

The P-value appears in the Minitab output under "P-value":

$P=0.011$

The P-value represents the probability of receiving the sample proportion or a more extreme value if the null hypothesis is true.

Interpretation P value, there is a $0.011$chance that the sample proportion of US teens who believe young people should wait until marriage is $0.56$or higher than the population proportion of US teens who believe young people should wait until marriage is $0.50.$

$$\begin{gathered} H_0:p=0.5 \\ {H}_1:pisnotequalto0.5 \end{gathered}$$

The confidence level reduces the significance level to $1$:

$$\begin{gathered} \mathrm{Î\pm }=1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

The following P-value is provided in the output:

$P=0.041$

If the P-value is less than the significance level, the null hypothesis should be rejected.

$0.041<0.05⇒\text{Reject}{H}_0$

There is sufficient evidence to support the claim.