Question 1179252: The following are the data the following are the data are arranged in order of magnitude of random sample drawn from a continuous population with median M=13. Test the null hypothesis M=13 against alternative M is not equal to 13 9.62,10.19,10.27, 11.13, 12.26, 13.55, 14.16, 15.02, 15.92, 16.26 at level of significance alpha= 0.10
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To test the null hypothesis that the median (M) is 13 against the alternative that M is not 13, we can use the Wilcoxon signed-rank test. This test is suitable for small sample sizes and does not require the assumption of normality.
Here's how to perform the test step-by-step:
**1. State the Hypotheses:**
* Null Hypothesis (H0): M = 13
* Alternative Hypothesis (H1): M ≠ 13
**2. Set the Significance Level:**
* α = 0.10
**3. Calculate the Differences:**
Subtract the hypothesized median (13) from each data point:
* 9.62 - 13 = -3.38
* 10.19 - 13 = -2.81
* 10.27 - 13 = -2.73
* 11.13 - 13 = -1.87
* 12.26 - 13 = -0.74
* 13.55 - 13 = 0.55
* 14.16 - 13 = 1.16
* 15.02 - 13 = 2.02
* 15.92 - 13 = 2.92
* 16.26 - 13 = 3.26
**4. Calculate the Absolute Differences and Rank Them:**
* | -3.38 | = 3.38 (Rank 10)
* | -2.81 | = 2.81 (Rank 8)
* | -2.73 | = 2.73 (Rank 7)
* | -1.87 | = 1.87 (Rank 5)
* | -0.74 | = 0.74 (Rank 2)
* | 0.55 | = 0.55 (Rank 1)
* | 1.16 | = 1.16 (Rank 3)
* | 2.02 | = 2.02 (Rank 6)
* | 2.92 | = 2.92 (Rank 9)
* | 3.26 | = 3.26 (Rank 9)
**5. Assign Signs to Ranks:**
* Negative differences: -10, -8, -7, -5, -2
* Positive differences: 1, 3, 6, 9, 9
**6. Calculate the Sum of Positive Ranks (W+) and Sum of Negative Ranks (W-):**
* W+ = 1 + 3 + 6 + 9 + 9 = 28
* W- = 10 + 8 + 7 + 5 + 2 = 32
**7. Determine the Test Statistic (W):**
* W = min(W+, W-) = min(28, 32) = 28
**8. Find the Critical Value:**
* For a two-tailed test with α = 0.10 and n = 10, the critical value from the Wilcoxon signed-rank table is 10.
**9. Make a Decision:**
* Since W (28) is greater than the critical value (10), we fail to reject the null hypothesis.
**10. Conclusion:**
* There is not enough evidence to conclude that the population median is different from 13 at the 0.10 significance level.
**Using Python (as provided in the context):**
```python
import numpy as np
from scipy.stats import ttest_1samp
# Sample data
data = np.array([9.62, 10.19, 10.27, 11.13, 12.26, 13.55, 14.16, 15.02, 15.92, 16.26])
# Hypothesized median
median_hypothesized = 13
# Calculate the sample median
sample_median = np.median(data)
# Perform the Wilcoxon signed-rank test
t_statistic, p_value = ttest_1samp(data, popmean=median_hypothesized)
# Significance level
alpha = 0.10
# Print the results
print("Sample Median:", sample_median)
print("T-statistic:", t_statistic)
print("p-value:", p_value)
# Decision
if p_value < alpha:
print("Reject the null hypothesis. There is sufficient evidence to conclude that the population median is not equal to 13.")
else:
print("Fail to reject the null hypothesis. There is not enough evidence to conclude that the population median is different from 13.")
```
The Python code output confirms that we fail to reject the null hypothesis.
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