Question

Suppose that a random sample of n = 5 was selected from the apple orchard properties for sale in Chilton County, Alabama, in each of three years. The following data are consistent with summary information on price per acre for disease-resistant apple orchards in Chilton County. Carry out an ANOVA to determine whether there is evidence to support the claim that the mean price per acre for vineyard land in Chilton County was not the same for each of the three years considered. Test at the 0.05 level and at the 0.01 level.

2002: 30000 34000    36000    38000    40000

2003: 30000 35000 37000 38000 40000

2004: 40000 41000 43000 44000 50000

Answer 1

(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.

WHEN level of significance is 5%

(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​=3.885}.

(3) Test Statistics

(4) Decision about the null hypothesis

Since it is observed that F=6.834>Fc​=3.885, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0104, and since p=0.0104<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.

WHEN level of significance is 1%

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the degrees of freedom are df1​=2 and df2​=2, therefore, the rejection region for this F-test is R={F:F>Fc​=6.927}.

(3) Test Statistics

(4) Decision about the null hypothesis

Since it is observed that F=6.834≤Fc​=6.927, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0104, and since p=0.0104≥0.01, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that not all 3 population means are equal, at the α=0.01 significance level.

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