A meteorologist has measured the amount of rain in four cities for the last six months.She wants to know if there are different amounts of rain in the four cities.The table shows the data.
Conduct a Kruskal-Wallis test to see if there is a significant difference in the amount of rain in the four cities.Use a χ2 approximation.Select the appropriate critical value (CV), test statistic (TS), and decision.(Use α = 0.025.)
A) CV = 7.3778, TS = 14.04, reject H₀
B) CV = 9.2103, TS = 15.98, reject H₀
C) CV = 9.3484, TS = 15.98, reject H₀
D) CV = 9.3484, TS = 9.22, fail to reject H₀

Question

A meteorologist has measured the amount of rain in four cities for the last six months.She wants to know if there are different amounts of rain in the four cities.The table shows the data. ​  ​
Conduct a Kruskal-Wallis test to see if there is a significant difference in the amount of rain in the four cities.Use a χ2 approximation.Select the appropriate critical value (CV), test statistic (TS), and decision.(Use α = 0.025.)
A) CV = 7.3778, TS = 14.04, reject H₀ 
B) CV = 9.2103, TS = 15.98, reject H₀ 
C) CV = 9.3484, TS = 15.98, reject H₀ 
D) CV = 9.3484, TS = 9.22, fail to reject H₀

Quizplus · Accepted Answer

The Kruskal-Wallis test is a non-parametric test used to determine if there is a significant difference in the median of two or more groups. We can use the following steps to conduct the test:

1. State the null hypothesis: H₀: There is no significant difference in the amount of rain in the four cities.
2. State the alternative hypothesis: H₁: There is a significant difference in the amount of rain in the four cities.
3. Determine the level of significance: α = 0.025 (two-tailed).
4. Rank all the observations from lowest to highest.
5. Calculate the sum of the ranks for each city.
6. Calculate the test statistic:

$H = \frac{12}{N(N+1)}\sum_{j=1}^k\frac{R_j^2}{n_j} - 3(N+1)$

where:
k = number of groups (cities)
N = total number of observations
n₁, n₂, n₃, n₄ = sample sizes
R₁, R₂, R₃, R₄ = sum of ranks for each group

In this case, k = 4, N = 24, n₁ = n₂ = n₃ = n₄ = 6, and we can find the sum of ranks and plug into the formula.

7. Find the critical value: We need to use a chi-squared approximation for this test. The degrees of freedom (df) is k-1 = 3. Using a chi-squared distribution table with df=3 and α=0.025, we get the critical value to be 9.3484.

8. Make a decision: If the test statistic is greater than the critical value, we reject the null hypothesis. If it is less than the critical value, we fail to reject the null hypothesis.

Using the formula, we get the test statistic to be H = 15.98. Since 15.98 is greater than the critical value of 9.3484, we reject the null hypothesis. Therefore, we can conclude that there is a significant difference in the amount of rain in the four cities. The correct choice is C.

A Meteorologist Has Measured the Amount of Rain in Four