(a)
To explain is there evidence that the presence of nine weeds per meter reduces corn yields when compared with weed-free corn and use Wilcoxon rank sum test with the data given and part of the data from example
(a)
Answer to Problem 27.8AYK
We have sufficient evidence to conclude that the presence of nine weeds per meter reduces corn yields when compared with weed-free corn.
Explanation of Solution
In the question, it is given that the corn yield study of example
0-weed | 9-weed |
166.7 | 162.8 |
172.2 | 162.7 |
165 | 162.4 |
176.9 | 142.4 |
Now, let us use the software to conduct the Wilcoxon test. The hypotheses are defined as: Null hypothesis: There is no difference between them and Alternative hypothesis: The zero-weed field increases corn yield than nine-weed yield. Thus, we have the result as:
n | sum of ranks | |
4 | 26 | 0-weed |
4 | 10 | 9-weed |
8 | 36 | total |
18.00 | expected value |
3.46 | standard deviation |
2.17 | z |
.0152 | p-value (one-tailed, upper) |
No. | Label | Data | Rank |
1 | 0-weed | 166.7 | 6 |
2 | 0-weed | 172.2 | 7 |
3 | 0-weed | 165 | 5 |
4 | 0-weed | 176.9 | 8 |
5 | 9-weed | 162.8 | 4 |
6 | 9-weed | 142.4 | 1 |
7 | 9-weed | 162.7 | 3 |
8 | 9-weed | 162.4 | 2 |
Thus, we have test statistics vale and P-value as:
As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,
Thus, we have sufficient evidence to conclude that the presence of nine weeds per meter reduces corn yields when compared with weed-free corn.
(b)
To compare the results from part (a) with those from the two-sample t test for these data.
(b)
Answer to Problem 27.8AYK
We have sufficient evidence to conclude that the presence of nine weeds per meter reduces corn yields when compared with weed-free corn.
Explanation of Solution
In the question, it is given that the corn yield study of example
0-weed | 9-weed |
166.7 | 162.8 |
172.2 | 162.7 |
165 | 162.4 |
176.9 | 142.4 |
Thus, the hypotheses will be defined as:
Thus, for testing the hypothesis we will use the calculator
Thus, by using the calculator
As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,
Thus, we have sufficient evidence to conclude that the presence of nine weeds per meter reduces corn yields when compared with weed-free corn. So, we can see that the P-value for both the tests are less than the level of significance and thus, the conclusion for both is same. Also the test statistics value is approximately equal.
(c)
To repeat the Wilcoxon test and t analyses by removing the outlier
(c)
Explanation of Solution
In the question, it is given that the corn yield study of example
0-weed | 9-weed |
166.7 | 162.8 |
172.2 | 162.7 |
165 | 162.4 |
176.9 |
Now, we have to conduct both the test by removing the outlier
n | sum of ranks | |
4 | 22 | 0-weed |
3 | 6 | 9-weed |
7 | 28 | total |
16.00 | expected value |
2.83 | standard deviation |
1.94 | z |
.0259 | p-value (one-tailed, upper) |
No. | Label | Data | Rank |
1 | 0-weed | 166.7 | 5 |
2 | 0-weed | 172.2 | 6 |
3 | 0-weed | 165 | 4 |
4 | 0-weed | 176.9 | 7 |
5 | 9-weed | 162.8 | 3 |
6 | 9-weed | 162.7 | 2 |
7 | 9-weed | 162.4 | 1 |
Now, if we compare it with the above result in part (a), we can see that both the P-values are less than the level of significance so the conclusion will be the same but the mean is decreased by two and the standard deviation is decreased by:
Now, let us conduct the two-sample t test, thus, for testing the hypothesis we will use the calculator
Thus, by using the calculator
Now, if we compare it with the above result in part (b), we can see that both the P-values are less than the level of significance so the conclusion will be the same but the standard deviation is decreased by two and the mean is decreased by:
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Chapter 27 Solutions
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