Carpal tunnel syndrome is a painful wrist condition that can be treated with surgery or less invasively with wrist splints. In a study of patients with the condition, 88 had surgery and 76 used wrist splints. In the surgery group, 62 patients showed improvement after three months while 39 of those who used wrist splints improved. Is surgery more effective than the use of wrist splints for improving symptoms of the condition?

Determine the z test statistic value to be used in the hypothesis test.

Answers will vary slightly due to rounding. Select the closest answer, ignoring any negative sign.

Question 5 options:

3.35

2.51

4.89

0.05

2.16

To determine whether surgery is more effective than wrist splints for improving symptoms of carpal tunnel syndrome, we can perform a hypothesis test for the difference between two proportions.

### Step 1: Define the hypotheses
- **Null Hypothesis (\(H_0\))**: The proportion of improvement is the same for both groups (surgery and wrist splints).
- **Alternative Hypothesis (\(H_1\))**: The proportion of improvement is different between the two groups.

### Step 2: Calculate the sample proportions
- **Proportion of improvement in the surgery group (\(p_1\))**:
  \[
  p_1 = \frac{62}{88} \approx 0.7045
  \]
- **Proportion of improvement in the wrist splints group (\(p_2\))**:
  \[
  p_2 = \frac{39}{76} \approx 0.5132
  \]

### Step 3: Calculate the pooled proportion (\(\hat{p}\))
Since we're comparing the two proportions, we first calculate the pooled proportion:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{62 + 39}{88 + 76} = \frac{101}{164} \approx 0.6159
\]

### Step 4: Calculate the standard error (\(SE\))
The standard error for the difference between two proportions is calculated as:
\[
SE = \sqrt{\hat{p} \left( 1 - \hat{p} \right) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}
\]
Substituting the values:
\[
SE = \sqrt{0.6159 \times 0.3841 \left(\frac{1}{88} + \frac{1}{76}\right)} \approx \sqrt{0.6159 \times 0.3841 \times 0.0264} \approx \sqrt{0.00625} \approx 0.0790
\]

### Step 5: Calculate the z test statistic
The z test statistic is calculated as:
\[
z = \frac{p_1 - p_2}{SE} = \frac{0.7045 - 0.5132}{0.0790} \approx \frac{0.1913}{0.0790} \approx 2.42
\]

### Step 6: Match the closest value
The closest value to 2.42 in the given options is **2.51**.

### Final Answer
The z test statistic value closest to our calculation is **2.51**.

Would you like more details on this solution, or have any questions?

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**Related Questions:**
1. How would you interpret the z test statistic in the context of this problem?
2. What would be the p-value associated with the z test statistic?
3. If the significance level was 0.05, would you reject the null hypothesis?
4. What are the implications of a Type I error in this context?
5. How would you calculate the confidence interval for the difference in proportions?

**Tip:** When conducting hypothesis tests, always verify that the conditions for the test (like sample size and normality) are met to ensure the results are valid.

Calculate the z test statistic for comparing the effectiveness of surgery and wrist splints in improving symptoms of carpal tunnel syndrome. Understand the hypothesis testing process, including sample proportions and standard error calculation. Suitable for advanced undergraduate students and researchers.

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