2022-05-09

An MP3 manufacturer claimsthat 65% of teenagers 13 to 16 years old have their own MP3 players.

A researcher wishes to test the claim and selects a random sample of 80 teenagers. She finds that

57 have their own MP3 players. At � = 0.05, should the claim be rejected?

nick1337

Expert2022-05-18Added 777 answers

The following information is provided: The sample size is $N=80$, the number of favorable cases is $X=57$,

and the sample proportion is $\overline{p}=\frac{X}{N}=\frac{57}{80}=0.7125$ ,

and the significance level is $\alpha =0.05$

__(a) ____Null and Alternative Hypotheses__

The following null and alternative hypotheses need to be tested:

$Ho:p=0.65\phantom{\rule{0ex}{0ex}}H\alpha :p\ne 0.65$

This corresponds to a two-tailed test, for which a $z$-test for one population proportion needs to be used.

__Rejection Region__

Based on the information provided, the significance level is $\alpha =0.05$ , and the critical value for a two-tailed test is ${z}_{c}=1.96$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}$

__(b) ____Test Statistics__

The $z$-statistic is computed as follows:

$z=\frac{\overline{p}-{p}_{0}}{\sqrt{{p}_{0}(1-{p}_{0})/n}}=\frac{0.7125-0.65}{\sqrt{0.65(1-0.65)/80}}=1.172$

__Decision about the null hypothesis__

Since it is observed that $\left|z\right|=1.172\le {z}_{c}=1.96$ , it is then concluded that *the null hypothesis is not rejected.*

(c) Using the $P$-value approach: The $p$-value is $p=0.2412$, and since $p=0.2412\ge 0.05$ , it is concluded that the null hypothesis is not rejected.

__(d) ____Conclusion__

It is concluded that the null hypothesis Ho is *not rejected.* Therefore, there is not enough evidence to claim that the population proportion $p$ is different than ${p}_{0}$, at the $\alpha =0.05$ significance level.

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