Question

The U-Plant’um Nursery must determine if there is a difference in the growth rate of saplings that have been treated with four different chemical formulas. The resulting growth rates over a given period are shown here. Does a difference appear to exist in the growth factor of the formulas? Set alpha = 0.01.

                                                                        FORMULA

  

                                                10                    8                      5                      7

                                                12                    15                    17                    14

                                                17                    16                    15                    15

Answer 1

With the information provided we can now easily compute the sum of ranks for each of the samples:

R₁ = 4 + 5 + 11.5 = 20.5

R₂ = 3 + 8 + 10 = 21

R₃ = 1 + 8 + 11.5 = 20.5

R₄ = 2 + 6 + 8 = 16

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: The samples come from populations with equal medians

Ha: The samples come from populations with medians that are not all equal

The above hypotheses will be tested using the Kruskal-Wallis test.

(2) Rejection Region

The significance level is α=0.01, and the number of degrees of freedom is df=4−1=3. But, since at least one sample size is less than 5, we cannot use normal approximation, so then, the H-statistic cannot longer be approximated by the Chi-Square distribution.

Using the appropriate for table for H values when normal approximation cannot be used, it is found that the critical H-value is Hc=8.538. The rejection region for this test is R={H≥8.538}.

(3) Test Statistics

The H statistic is computed as shown in the following formula:

(4) Decision about the null hypothesis

Since it is observed that H=0.423<8.538, it is then 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 some of the population medians are unequal, at the 0.01 significance level.