An MP3 manufacturer claimsthat 65% of teenagers 13

Answered question

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?

Answer & Explanation

nick1337

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 p¯=XN=5780=0.7125 ,

and the significance level is α=0.05

(a) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:p=0.65 Hα:p0.65

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 α=0.05 , and the critical value for a two-tailed test is zc=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=p¯-p0p0(1-p0)/n=0.7125-0.650.65(1-0.65)/80=1.172

Decision about the null hypothesis

Since it is observed that |z|=1.172zc=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=0.24120.05 , 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 p0​, at the α=0.05 significance level.

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