SOLUTION: Using the data below, suppose we focus on the proportions of patients who show improvement. Is there a statistically significant difference in the proportions of patients who show

Question 1191683: Using the data below, suppose we focus on the proportions of patients who show improvement. Is there a statistically significant difference in the proportions of patients who show improvement between treatments 1 and 2. Run the test at a 5% level of significance.

Symptoms
Worsened
No Effect
Symptoms Improved
Total
Treatment 1
22
14
14
50
Treatment 2
14
15
21
50
Treatment 3
9
12
29
50

Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to test for a statistically significant difference in improvement proportions between Treatment 1 and Treatment 2:
**1. State the Hypotheses:**
* **Null Hypothesis (H0):** There is no difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2.
* **Alternative Hypothesis (H1):** There is a difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2.
**2. Calculate Proportions:**
* **Treatment 1:** Proportion of improvement = 14/50 = 0.28
* **Treatment 2:** Proportion of improvement = 21/50 = 0.42
**3. Perform a Two-Proportion Z-Test:**
This test is appropriate for comparing two proportions from independent samples.
* **Calculate the pooled proportion:**
p = (14 + 21) / (50 + 50) = 35/100 = 0.35
* **Calculate the standard error:**
SE = sqrt{ p * (1-p) * [(1/50) + (1/50)] }
SE = sqrt{ 0.35 * 0.65 * 0.04 } ≈ 0.095
* **Calculate the Z-statistic:**
Z = (0.42 - 0.28) / 0.095 ≈ 1.47
**4. Determine the p-value:**
Using a Z-table or calculator, find the p-value corresponding to a Z-statistic of 1.47 (two-tailed test). The p-value is approximately 0.14.
**5. Make a Decision:**
* **Significance Level (α):** 5% (0.05)
Since the p-value (0.14) is greater than the significance level (0.05), we fail to reject the null hypothesis.
**Conclusion:**
There is not enough evidence to conclude that there is a statistically significant difference in the proportions of patients who show improvement between Treatment 1 and Treatment 2 at a 5% level of significance.
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