Question

The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is between 57,000 and 63,000 miles?A) Very likely.
B) 0.4332.
C) 0.8664.
​D) 1.00.

Answer 1

Answer: C) 0.8664.

Step-by-step explanation:

Given, The tread life of tires mounted on light-duty trucks follows the normal probability distribution with $\mu=60,000\text{ miles}\ \&\ \ \sigma=4,000\text{ miles}$ .

Let $\overline{x}$ be the sample mean tire life of any 4 tires.

Now , the probability that the mean tire life of these four tires is between 57,000 and 63,000 miles will be :-

$P(57000<\overline{x}<63000)=P(\dfrac{57000-60000}{\dfrac{4000}{\sqrt{4}}}<\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{63000-60000}{\dfrac{4000}{\sqrt{4}}})\\\\=P(-1.5<z<1.5)\ \ \ [\because\ z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}} ]\\\\= P(z<1.5)-P(z<-1.5)=P(z<1.5)-(1-P(z<1.5))\\\\=0.9332-(1-0.9332)\ \ [\text{By z-table}]\\\\=0.8664$

For Very likely , the probability should lie between 0.95 and 1.

Hence, the correct answer is C) 0.8664.