Chapter 9.2, Problem 58E

(a)

To determine

To Explain: the parameter of interest.

Answer to Problem 58E

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.

Explanation of Solution

The parameter of interest is the population value of which requires knowing the value or other information.

PAREMETER OF INTEREST =Population proportion $p$ of all U.S. teens aged 13 to 17 who suppose that young people ought to wait to own sex till wedding.

(b)

To determine

To Explain: that the conditions for performing the significance test are met in this case.

Answer to Problem 58E

All conditions satisfied.

Explanation of Solution

Given:

  $\begin{array}{c}n=439\\ p=0.5\end{array}$

Conditions for carrying out a one-sample $z$ test: Random, Normal and independent

Random: Satisfied, because the sample is a random sample. Normal: the distribution can be assumed to be normal if the number of failures $n\left(1-p\right)$ and the number of successes $np$ are both greater than 10.

  $\begin{array}{c}np=439\times 0.5=219.5\\ n\left(1-p\right)=439\times \left(1-0.5\right)=219.5\end{array}$

Both are greater than 10, therefore the normal requirement is satisfied.

Independent: can be assumed, the reason is that the sample size (439) is less than 10% of the population size.

Thus all conditions satisfied.

(c)

To determine

To Explain: the P-value.

Answer to Problem 58E

There is a 0.011 possibility that the sample proportion of U.S. teens who think that young people ought to wait to have sex until wedding is 0.56 or more extreme, when the population proportion of U.S. teens who suppose that young people ought to wait to have sex until wedding is 0.50.

Explanation of Solution

The Minitab output is having “Test of $p=0.5$ vs. $p\text{ not}=0.5$ ”, which implies that the hypotheses are:

  $\begin{array}{c}{H}_0:p=0.5\\ H_1:p\ne 0.5\end{array}$

The sample proportion is

  $\begin{array}{c}\widehat{p}=0.560364\\ =0.56\end{array}$

The P-value is mentioned in the Minitab output underneath “P-value”:

  $P=0.011$

The P-value is that the likelihood of getting the worth of the sample proportion or a worth more extreme, once the null hypothesis is true.

Interpretation P-value, there is a 0.011 chance that the sample proportion of U.S. teens who think that young people ought to wait to have sex until wedding is 0.56 or more extreme, when the population proportion of U.S. teens who think that young people ought to wait to have sex until wedding is 0.50.

(d)

To determine

To Explain: that the data give convincing evidence that the population proportion differs from 0.5, justify answer with appropriate evidence.

Answer to Problem 58E

Yes, there is enough proof to help the claim.

Explanation of Solution

  $\begin{array}{c}{H}_0:p=0.5\\ H_1:p\ne 0.5\end{array}$

The significance level is 1 decreased by the confidence level:

  $\alpha =1-95\%=1-0.95=0.05$

P-value given in the output:

  $P=0.041$

If the P-value is less than the significance level, then reject the null hypothesis.

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

There is enough proof to help the claim.