Step 1 :Given that the sample size (n) is 1571 and the number of successes (phones that broke before the warranty expired) is 90, we can calculate the sample proportion (\(\hat{p}\)) as the number of successes divided by the sample size, which is \(\hat{p} = \frac{90}{1571} = 0.0573\).
Step 2 :The z-score (Z) corresponding to the desired confidence level of 90% is 1.645.
Step 3 :We can calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.0573(1-0.0573)}{1571}} = 0.0059\).
Step 4 :Finally, we can calculate the confidence interval using the formula \(\hat{p} \pm Z_{\alpha/2} \times SE\). The lower limit of the confidence interval is \(0.0573 - 1.645 \times 0.0059 = 0.0476\) and the upper limit of the confidence interval is \(0.0573 + 1.645 \times 0.0059 = 0.0669\).
Step 5 :Final Answer: With 90% confidence the proportion of all smart phones that break before the warranty expires is between 0.0476 and 0.0669. So, the final answer is \(\boxed{[0.0476, 0.0669]}\).