Question

a certain brand of automobile tire has a mean life span of 38,000 miles and a standard deviation of 2,450 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 that corresponds to each life span. for the life span of 34,000 miles, z - score is - 1.63. (round to the nearest hundredth as needed.) for the life span of 37,000 miles, z - score is - 0.41. (round to the nearest hundredth as needed.) for the life span of 31,000 miles, z - score is \square. (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 dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation.
We know that $\mu = 38000$ miles and $\sigma=2450$ miles, and for $x = 31000$ miles.

Step2: Substitute values into the formula

Substitute $x = 31000$, $\mu=38000$ and $\sigma = 2450$ into the z - score formula:
$z=\frac{31000 - 38000}{2450}$

Step3: Calculate the numerator

First, calculate the numerator: $31000-38000=-7000$

Step4: Calculate the z - score

Then, divide the numerator by the denominator: $z=\frac{-7000}{2450}\approx - 2.86$ (rounded to the nearest hundredth)

Answer:

-2.86