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The problem asks to compute a 90% confidence interval for the proportion of X-ray machines that malfunction, based on a random sample of 175 machines, out of which 49 malfunctioned.

We can solve this problem using the formula for the confidence interval for a population proportion \( p \):

\[
CI = \hat{p} \pm Z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

Where:
- \( \hat{p} \) is the sample proportion of machines that malfunction.
- \( Z_{\alpha/2} \) is the Z-score corresponding to the confidence level (for a 90% confidence interval, \( Z_{\alpha/2} \) is approximately 1.645).
- \( n \) is the sample size (175 in this case).

Let's calculate step-by-step.

1. Sample proportion \( \hat{p} \) is calculated as:
\[
\hat{p} = \frac{49}{175}
\]

2. The confidence interval formula involves the standard error (SE):
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\]

3. Finally, use the Z-score for 90% confidence (1.645) to calculate the margin of error.

Let me compute that for you.

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. 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.

Learn how to compute a 90% confidence interval for the proportion of malfunctioning X-ray machines, based on a sample of 175 machines where 49 malfunctioned. This involves using the sample proportion, Z-score for 90% confidence, and standard error.

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