TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2019W/3.6

Tire company

A tire manufacturer believes that the tread life of its snow tires can be distributed by a Normal model with a mean (expectation) of 32 000 miles and a standard deviation of 2 500 miles.

(a) If you buy a set of these tires, would it be reasonable for you to hope that they will last 40 000 miles? Explain your answer.

(b) Approximately what fraction of these tires can be expected to last less than 30 000 miles?

(c) Approximately what fraction of these tires can be expected to last between 30 000 and 35 000 miles?

(d) Calculate the interquartile range of this distribution. Recall, the interquartile range is the difference between upper and lower quartile, i.e.

$IQR=x_{0.75}-x_{0.25}$.

(e) In a marketing strategy, a local tire dealer wants to offer a refund to any customer whose tires fail to last a certain number of miles. However, the dealer does not want to take too big risk. If the dealer is willing to give refunds to no more than one of every 25 customers, for what milage can he guarantee these tires to last?

Note: Table of standard Normal distribution should be used for all computations.

Lösungsvorschlag von Gittenburg

--Gittenburg 11:42, 29. Okt. 2019 (CET)

(a):

> 1 - pnorm(40000, 32000, 2500)
[1] 0.0006871379

(b):

> pnorm(30000, 32000, 2500)
[1] 0.2118554

(c):

> pnorm(35000, 32000, 2500) - pnorm(30000, 32000, 2500)
[1] 0.6730749

(d):

> qnorm(0.75, 32000, 2500) - qnorm(0.25, 32000, 2500)
[1] 3372.449

(e)

> qnorm(1/25, 32000, 2500)
[1] 27623.28