Tire lifetimes: The lifetime of a certain type of automobile tire (in thousands of miles) is normally distributed with mean $\mu=41$ and standard deviation $\sigma=4$. Use the $\pi 1-84$ Plus calculator to answer the following. $\infty$ (a) Find the 13th percentile of the tire lifetimes. (b) Find the 65 th percentile of the tire lifetimes. (c) Find the first quartile of the tire liferimes. (d) The tire company wants to guarantee that its tires will last at least a certain number of miles. What number of miles (in thousands) should the company guarantee so that only $4 \%$ of the tires violate the guarantee? Round the answers to at least two decimal places. Part 1 of 4 The 13 th percentile of the tire lifetimes is $\square$ thousand miles. Part 2 of 4 The 65 th percentile of the tire lifetimes is $\square$ thousand miles. Part 3 of 4 The first quartile of the tire lifetimes is $\square$ thousand miles. $\square$ Part 4 of 4 The company should guarantee $\square$ thousand miles.

Step 1 ：The 13th percentile corresponds to a z-score of -1.13. Using the z-score formula \(X = Z*\sigma + \mu\), we get \(X = -1.13*4 + 41\)

Step 2 ：Calculating the above expression, we get \(X = 36.48\)

Step 3 ：\(\boxed{36.48}\) thousand miles is the 13th percentile of the tire lifetimes

Step 4 ：The 65th percentile corresponds to a z-score of 0.39. Using the z-score formula \(X = Z*\sigma + \mu\), we get \(X = 0.39*4 + 41\)

Step 5 ：Calculating the above expression, we get \(X = 42.56\)

Step 6 ：\(\boxed{42.56}\) thousand miles is the 65th percentile of the tire lifetimes

Step 7 ：The first quartile corresponds to the 25th percentile, which has a z-score of -0.67. Using the z-score formula \(X = Z*\sigma + \mu\), we get \(X = -0.67*4 + 41\)

Step 8 ：Calculating the above expression, we get \(X = 38.32\)

Step 9 ：\(\boxed{38.32}\) thousand miles is the first quartile of the tire lifetimes

Step 10 ：The 4th percentile corresponds to a z-score of -1.75. Using the z-score formula \(X = Z*\sigma + \mu\), we get \(X = -1.75*4 + 41\)

Step 11 ：Calculating the above expression, we get \(X = 34\)

Step 12 ：\(\boxed{34}\) thousand miles is the number of miles the company should guarantee so that only 4% of the tires violate the guarantee