Question

At an alpha level of 0.05, test to determine if the means of the three populations are equal. The data below is based on samples taken from three populations. Sample 1 Sample 2 Sample 3 60 84 60 78 78 57 72 93 69 66 81 66

Answer 1

The data is :

Sample 1	Sample 2	Sample 3
60	84	60
78	78	57
72	93	69
66	81	66

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ₁ = μ₂ = μ₃

Ha: Not all means are equal

The above hypotheses will be tested using an F-ratio for a One-Way ANOVA.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df1​=2 and df2​=2, therefore, the rejection region for this F-test is R={F:F>Fc​=4.256}

(3) Test Statistics

(4) Decision about the null hypothesis

Since it is observed that F=10.636>Fc​=4.256, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0043, and since p=0.0043<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 not all 3 population means are equal, at the α=0.05 significance level.

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