Question

Teens and sex The Gallup Youth Survey asked a random sample of U.S. teens aged 13 to 17 whether they thought that young people should wait until marriage to have sex.14 The Minitab output shows the results of a significance test and a 95% confidence interval based on the survey data.

a. Define the parameter of interest.

b. Check that the conditions for performing the significance test are met in this case.

c. Interpret the P-value.

d. Do these data give convincing evidence that the actual population proportion differs from $0.5$? Justify your answer with appropriate evidence.

Short answer

Part a) Population proportion$p$of all U.S. teens aged 13 to 17 who suppose that young people ought to wait to have sex until wedding.

Part b) All conditions are satisfied.

Part c) 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 $0.50$

Part d) Yes, there is enough proof to help the claim.

Step-by-step solution

Part a) Step 1: The objective is to explain the parameter of interest.

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.

Part b) Step 1: Given information

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

Part b) Step 2: The objective is to explain that the conditions for performing the significance test are met in this case.  

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.

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

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

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.

Part c) Step 1: The objective is to explain theP value 

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 $0.50.$

Part d) Step 1: The objective is to explain that the data give convincing evidence that the population proportion differs from0.5 and justify the answer with appropriate evidence. 

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

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

$$\begin{gathered} \alpha =1-95\% \\ =1-0.95 \\ =0.05 \end{gathered}$$

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.

$0.041<0.05\Rightarrow \text{Reject}{H}_0$

There is sufficient evidence to support the claim.