Question

a certain brand of automobile tire has a mean life span of 36,000 miles and a standard deviation of 2,200 miles. (assume the life spans of the tires have a bell - shaped distribution )
(a) the life spans of three randomly selected tires are 34,000 miles, 37,000 miles, and 31,000 miles. find the z - score that corresponds to each life span.
for the life span of 34,000 miles, z - score is - 0.91 (round to the nearest hundredth as needed.)
for the life span of 37,000 miles, z - score is 0.45 (round to the nearest hundredth as needed.)
for the life span of 31,000 miles, z - score is (round to the nearest hundredth as needed.)

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value from the data set, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 36000$ and $\sigma=2200$.

Step2: Calculate z - score for $x = 31000$

Substitute $x = 31000$, $\mu = 36000$, and $\sigma = 2200$ into the formula: $z=\frac{31000 - 36000}{2200}=\frac{- 5000}{2200}\approx - 2.27$

Answer:

$-2.27$