Question

a certain brand of automobile tire has a mean life - span of 34,000 miles and a standard deviation of 2,400 miles. (assume the life - spans of the tires have a bell - shaped distribution.)
for the life span of 38,000 miles, z - score is 1.67 (round to the nearest hundredth as needed.)
for the life span of 31,000 miles, z - score is - 1.25 (round to the nearest hundredth as needed.)
according to the z - scores, would the life spans of any of these tires be considered unusual?
no
yes
(a) the life spans of three randomly selected tires are 29,200 miles, 38,800 miles, and 34,000 miles. using the empirical rule, find the percentile that corresponds to each life span.
the life span 29,200 miles corresponds to a percentile of 3
the life span 38,800 miles corresponds to a percentile of
the life span 34,000 miles corresponds to a percentile of 50 (round to the nearest whole number as needed.)

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For the empirical rule in a normal distribution: about 68% of the data lies within $z = - 1$ and $z=1$, about 95% lies within $z=-2$ and $z = 2$, and about 99.7% lies within $z=-3$ and $z = 3$. Percentiles can be estimated using the properties of the normal distribution.

Step2: Calculate z - score for 29200 miles

Given $\mu = 34000$, $\sigma=2400$, and $x = 29200$. Then $z=\frac{29200 - 34000}{2400}=\frac{-4800}{2400}=-2$. According to the empirical rule, about 2.5% of the data lies below $z=-2$. So the percentile is approximately 2.5, rounded to the nearest whole number is 3.

Step3: Calculate z - score for 38800 miles

Given $x = 38800$, $\mu = 34000$, $\sigma=2400$. Then $z=\frac{38800 - 34000}{2400}=\frac{4800}{2400}=2$. According to the empirical rule, about 97.5% of the data lies below $z = 2$. So the percentile is approximately 97.5, rounded to the nearest whole number is 98.

Answer:

The life - span 38800 miles corresponds to a percentile of 98.