Question

Do 9-year-old boys and girls have different average height? Please answer the question by performing an appropriate hypothesis test at the 10% significance level based on the sample results below:

Boys: n1= 60, 1x=123, S1= 10

Girls: n2= 50, 1x=126, S2= 1

Answer 1

(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 t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.10, and the degrees of freedom are df=60.414. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal.

Hence, it is found that the critical value for this two-tailed test is tc​=1.67, for α=0.10 and df=60.414.

The rejection region for this two-tailed test is R={t:∣t∣>1.67}.

(3) Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

(4) The decision about the null hypothesis

Since it is observed that ∣t∣=2.31>tc​=1.67, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0243, and since p=0.0243<0.10, 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 mean μ1​ is different than μ2​, at the 0.10 significance level.

Graphically

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