Question

Joan Nguyen recently claimed that the proportion of college-aged males with at least one pierced ear is as high as the proportion of college-aged females. She conducted a survey in her classes. Out of 116 males, 21 had at least one pierced ear. Out of 83 females, 49 had at least one pierced ear. Do you believe that the proportion of males with at least one pierced ear is different from the proportion of females with at least one pierced ear? (Use α = 0.05.)(NOTE: If you are using a Student's t -distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. Using TI 83 Calculator if possible.

Answer 1

Could not use TI 83 Calculator, as do not own one. I have solved it mathematically step wise instead:-

For sample 1, we have that the sample size is N1​=116, the number of favorable cases is X1​=21, so then the sample proportion is

For sample 2, we have that the sample size is N2​=83, the number of favorable cases is X2​=49, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted.

(2) 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}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣z∣=5.962>zc​=1.96, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0, and since p=0<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is different than p2​, at the 0.05 significance level. So the proportion of males and females with one pierced ear is different.

Graphically

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