Question 1191683
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.