Carpal tunnel syndrome is a painful wrist condition that can be treated with surgery or less invasively with wrist splints. In a study of 180 patients with the condition, half had surgery and half used wrist splints. In the surgery group, 70 patients showed improvement after three months while 42 of those who used wrist splints improved. Is surgery more effective than the use of wrist splints for improving symptoms of the condition?

Calculate a 95% confidence interval for the difference (surgery - wrist splints) in improvement rates.

Question 2 options:

0.467 to 0.778

0.177 to 0.445

0.233 to 0.542

-0.177 to 0.445

-0.445 to 0.177

To determine if surgery is more effective than the use of wrist splints for improving symptoms of carpal tunnel syndrome, we can calculate a 95% confidence interval (CI) for the difference in improvement rates between the two groups.

### Step 1: Define the problem
- Let \( p_1 \) be the proportion of patients who improved in the surgery group.
- Let \( p_2 \) be the proportion of patients who improved in the wrist splints group.
- The goal is to calculate the 95% confidence interval for the difference in proportions: \( p_1 - p_2 \).

### Step 2: Calculate the proportions
- Surgery group: 
  \[
  p_1 = \frac{70}{90} = \frac{7}{9} \approx 0.7778
  \]
- Wrist splints group: 
  \[
  p_2 = \frac{42}{90} = \frac{42}{90} \approx 0.4667
  \]

### Step 3: Calculate the difference in proportions
\[
\hat{p_1} - \hat{p_2} = 0.7778 - 0.4667 = 0.3111
\]

### Step 4: Calculate the standard error (SE) of the difference in proportions
The standard error for the difference in proportions is given by:
\[
SE = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}
\]
where \( n_1 = n_2 = 90 \).

Substitute the values:
\[
SE = \sqrt{\frac{0.7778 \times 0.2222}{90} + \frac{0.4667 \times 0.5333}{90}} \approx \sqrt{\frac{0.1728}{90} + \frac{0.2489}{90}} \approx \sqrt{0.00192 + 0.00277} \approx \sqrt{0.00469} \approx 0.0685
\]

### Step 5: Calculate the 95% confidence interval
The 95% confidence interval is given by:
\[
(\hat{p_1} - \hat{p_2}) \pm Z_{\alpha/2} \times SE
\]
where \( Z_{\alpha/2} \) is the critical value for a 95% confidence interval, which is approximately 1.96.

\[
CI = 0.3111 \pm 1.96 \times 0.0685
\]

\[
CI = 0.3111 \pm 0.1343
\]

\[
CI = (0.1768, 0.4454)
\]

### Step 6: Choose the correct option
The confidence interval for the difference in improvement rates is approximately (0.177, 0.445).

The correct answer is **0.177 to 0.445**.

---

Would you like more details or have any questions? 

Here are 5 related questions you might consider:

1. How does the size of the confidence interval reflect the precision of the estimate?
2. What assumptions are necessary for the confidence interval calculation?
3. How would the confidence interval change if the sample size increased?
4. What is the interpretation of a 95% confidence interval in this context?
5. How would you conduct a hypothesis test to compare the two treatments?

**Tip:** When interpreting confidence intervals, remember that if the interval does not include zero, it suggests a statistically significant difference between the two groups.

Learn how to calculate a 95% confidence interval for the difference in improvement rates between surgery and wrist splints for carpal tunnel syndrome. Includes step-by-step calculation using proportions and standard error formulae.

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