Q144E Critical-part failures in NASCAR... [FREE SOLUTION]

91影视

Statistics For Business And Economics

James T. McClave, P. George Benson, Terry Sincich

$Math Studyset 91影视 Explanations$ Math

13th Edition 路 888 pages

ISBN: 9780134506593

Chapter 4: Q144E (page 277)

Critical-part failures in NASCAR vehicles. In NASCAR races such as the Daytona 500, 43 drivers start the race; however, about 10% of the cars do not finish due to the failure of critical parts. University of Portland professors conducted a study of critical-part failures from 36 NASCAR races (The Sport Journal, Winter 2007). The researchers discovered that the time (in hours) until the first critical-part failure is exponentially distributed with a mean of .10 hours.
a. Find the probability that the time until the first critical part failure is 1 hour or more.
b. Find the probability that the time until the first critical part failure is less than 30 minutes.

Short Answer

Expert verified

a. The probability that the time until the first critical part failure is more than an hour is 0.000045.

b. The probability that the time until the first critical part failure is less than 30 minutes is 0.9933

Step by step solution

Given Information

The time until the first critical part failure is exponentially distributed with a mean of 0.1hour

Let x be the time until the first critical part failure.

The p.d.f of the exponential distribution is given by

$f (x) = \frac{1}{胃} e^{- \frac{1}{胃} x}; x > 0$

(a) Compute the probabilities for given conditions

The probability that the time until the first critical part failure is more than an hour is computed as

$\begin{array}{rcl} p (x > 1) & = & 10 {鈭�}_{1}^{鈭�} e^{- 10 x} d x \\ = & {\frac{10 脳 e^{- 10 x}}{- 10}]}_{1}^{鈭�} \\ = & - e^{- 10 鈭�} + e^{- 10} \\ = & 0.000045 \end{array}$

Thus, the probability is 0.000045

(b) Compute the probability

The probability that the time until the first critical part failure is less than 30 minutes is computed as

Therefore,

$\begin{array}{rcl} p (x < \frac{1}{2}) & = & 10 {鈭�}_{0}^{\frac{1}{2}} e^{- 10 x} d x \\ = & {\frac{10 脳 e^{- 10 x}}{- 10}]}_{0}^{\frac{1}{2}} \\ = & - e^{- 5} + e^{- 0} \\ = & 1 - 0.0067 \\ = & 0.9933 \end{array}$