Step 1: Verify the Central Limit Theorem
Find the sample proportion and its complement:
\[\hat{p} = \frac{x}{n} = \frac{45}{500} = 0.09 \qquad \hat{q} = 1 - \hat{p} = 0.91\]The Central Limit Theorem applies. We can continue.
Step 2: Find the Critical Value
For a 99% confidence level, the remaining 1% is in the tails, with 0.5% in each tail. As we saw in Lesson 18.1 Critical Values, the critical value for a 99% confidence level is $z_c = \pm 2.58$.
Step 3: Find the Margin of Error
\[\begin{align*} E &= z_c\sqrt{\frac{\hat{p}\hat{q}}{n}} \\ &= 2.58\sqrt{\frac{(0.09)(0.91)}{500}} \\ &= 2.58\sqrt{\frac{0.0819}{500}} \\ &= 2.58 \times 0.01280 \\ &\approx 0.033 \end{align*}\]Step 4: Find the Confidence Interval
\[\hat{p} + E = 0.09 + 0.033 = 0.123\] \[\hat{p} - E = 0.09 - 0.033 = 0.057\]The confidence interval is \((0.057,\ 0.123)\).
Step 5: Interpret the Confidence Interval
We are 99% confident that the true proportion of defective items produced by this factory is between 0.057 and 0.123 (between 5.7% and 12.3%).