Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained. Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.Using the Normal approximation,what is the value of the two-sided P-value for the test?
A)0.0102
B)0.0125
C)0.0203
D)0.0250

Question

Some researchers have conjectured that stem-pitting disease in peach-tree seedlings might be controlled with weed and soil treatment.An experiment was conducted to compare peach-tree seedling growth with soil and weeds treated with one of two herbicides.In a field containing twelve seedlings,six were randomly selected throughout the field and assigned to receive Herbicide A.The remaining six were to receive Herbicide B.Soil and weeds for each seedling were treated with the appropriate herbicide,and at the end of the study period the height in centimeters was recorded for each seedling.The following results were obtained.  Let W be the Wilcoxon rank sum test statistic equal to the sum of the ranks assigned to the Herbicide A group.Using the Normal approximation,what is the value of the two-sided P-value for the test?
A)0.0102 
B)0.0125 
C)0.0203 
D)0.0250

Quizplus · Accepted Answer

First, we need to calculate the value of the test statistic W.

For the Herbicide A group, the ranks assigned are:
3, 5, 6, 8, 9, 10

For the Herbicide B group, the ranks assigned are:
1, 2, 4, 7, 11, 12

The sum of ranks for the Herbicide A group is:
3 + 5 + 6 + 8 + 9 + 10 = 41

So, W = 41.

Now, we can calculate the two-sided P-value using the Normal approximation. Since we have 6 observations in each group, the total sample size is n = 12.

The mean rank for each group is n(n+1)/4 = 36, and the standard deviation is √(n(n+1)(2n+1)/24) = 6. This allows us to calculate the z-score for W:

z = (W - n(n+1)/2) / √(n(n+1)(2n+1)/24) = (41 - 36) / 6 = 0.8333

Using a standard normal distribution table or software, we find that the two-sided P-value is approximately 2*Φ(-|z|) = 2*Φ(-0.8333) = 0.0203, where Φ is the cumulative distribution function of the standard normal distribution.

Therefore, the best choice is C.

Some Researchers Have Conjectured That Stem-Pitting Disease in Peach-Tree Seedlings