Question

Please provide an aswer and reference(s) to the question below from a classmate. Thank you in advance!

Class,

I am having a problem with the following problem:

A certain brand of automobile tire has a mean life span of 39,000 miles and a standard deviation of 2,250 miles.​ (Assume the life spans of the tires have a​ bell-shaped distribution.)

For the life span of 34,000 miles, z-score is = 34,000 x 39,000 = -2.22

2,250

For the life span of 34,000 miles, z-score is = 38,000 x 39,000 = -0.44

2,250

For the life span of 34,000 miles, z-score is = 31,000 x 39,000 = -3.56

2,250

The following is the part I am having trouble with:

The life spans of three randomly selected tires are 34,500 miles, 43,500 miles, and 39,000

miles. Using the empirical​ rule, find the percentile that corresponds to each life span:

1. The life span 34,500 miles corresponds to the ___th percentile?

2. The life span 43,500 miles corresponds to the ___th percentile?

3. The life span 39,000 miles corresponds to the ___th percentile?

I have found in the text in Chapter 2, p.88 where it talks about Empirical Rules and Bell-Shaped Distribution. I did find that number 3 is "50"th percentile, as it is also the mean value in this problem set. Numbers 1 and 2 I am having an issue calculating. Any help from my battle buddies would be outstanding. Thank you in advance!

Reference:

Larson, R. & Farber, B. (2015). Elementary Statistics: picturing the world. 6th edition.

Answer 1

The Figure below will help

The percentile is the percentage of values that lie below a certain specified value.

The emperical rule or the 68-95-99.7 rule states that

(1) About 68% of the values fall within 1 standard deviation of the mean i.e from

Therefore 34% lies to the left i.e from to () and 34% to the right i.e from to ()

(2) About 95% of the values fall within 2 standard deviation of the mean i.e from

Therefore 47.5% lies to the left i.e from to and 47.5% to the right i.e from to

(3) About 99.7% of the values fall within 3 standard deviation of the mean i.e from

Therefore 49.85% lies to the left i.e from to and 49.85% to the right i.e from to

Please remember, the bell curve is such that the mean \mu, lies in the center and 50% of the total data, lies to the left of the mean and 50% lies to the right of the mean.

_________________________________________________________________________

1) The lifespan of 34,500 miles = (34500 - 39000)/2250 = -2 standard deviations from the mean.

We need to find the area to the left of this. We know that 0 to mu has 50% of the data, and 47.5% lies from to . Therefore 50% - 47.5% = 2.5th Percentile

2) The lifespan of 43,500 miles = (43500 - 39000)/2250 = +2 standard deviations from the mean.

We need to find the area to the left of this. We know that 0 to mu has 50% of the data, and 47.5% lies from to . Therefore 50% + 47.5% = 97.5th Percentile

3) The lifespan of 39,000 miles = (39000 - 39000)/2250 = 0 standard deviations from the mean, which means it is the central data and hence the 50th Percentile.