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The problem in the image asks to compute a 90% confidence interval for the proportion of X-ray machines that malfunction based on a sample. Let's work through the solution step by step.

### Problem Details:
- Sample size \( n = 175 \)
- Number of machines that malfunction \( x = 49 \)
- Confidence level = 90%

### Step-by-step Solution:

1. **Sample Proportion ( \( \hat{p} \) )**:
   \[
   \hat{p} = \frac{x}{n} = \frac{49}{175} = 0.28
   \]

2. **Standard Error (SE)** for the proportion:
   \[
   SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.28(1 - 0.28)}{175}} = \sqrt{\frac{0.28 \times 0.72}{175}} = \sqrt{0.001152} \approx 0.03394
   \]

3. **Z-score for a 90% confidence interval**: 
   The critical z-value for a 90% confidence interval is approximately 1.645.

4. **Margin of Error (ME)**:
   \[
   ME = z \times SE = 1.645 \times 0.03394 \approx 0.0558
   \]

5. **Confidence Interval**:
   The confidence interval is calculated as:
   \[
   \hat{p} \pm ME = 0.28 \pm 0.0558
   \]
   - **Lower limit**: \( 0.28 - 0.0558 = 0.2242 \)
   - **Upper limit**: \( 0.28 + 0.0558 = 0.3358 \)

### Conclusion:
- **Lower limit**: 0.22 (rounded to two decimal places)
- **Upper limit**: 0.34 (rounded to two decimal places)

Would you like further clarification or details on any part of the process? 

Here are 5 related questions to explore:

1. How is the margin of error influenced by the sample size?
2. What is the Z-score for a 95% confidence interval, and how does it affect the interval width?
3. Can we apply the same method if the sample size was much smaller?
4. What are the assumptions underlying the use of this confidence interval formula?
5. How does increasing the confidence level affect the width of the confidence interval?

**Tip**: Always round the sample proportion and intermediate steps to at least three decimal places to maintain accuracy in the final results.

A researcher wishes to estimate the proportion of X-ray machines that malfunction. A random sample of 175 machines is taken, and 49 of the machines in the sample malfunction. Based upon this, compute a 90% confidence interval for the proportion of all X-ray machines that malfunction. Then find the lower limit and upper limit of the 90% confidence interval.

This problem demonstrates how to compute a 90% confidence interval for the proportion of malfunctioning X-ray machines. Using a sample size of 175 machines with 49 malfunctions, it explains the use of sample proportion, standard error, margin of error, and Z-scores to calculate the confidence interval.

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