Stop the carl A car company has found that the lifetime of its disc brake pads varies from car to car according to a Normal distrilustion with mean $\mu =55000$miles and standard deviation$\sigma =4500$miles. The company installs a new brand of brake pads on an SRS of $8$cars.

(a) If the new brand has the same lifetime distribution as the previous bye of brake pad, what is the sampling distribution of the mean lifetime $\overline{x}$?

(b) The average life of the pads an these $8$cars turns out to be $\overline{x}=51800$miles. Find the probability that the sample mean lifetime is $51800$miles or less if the lifetime distribution is unchanged, What conclusion would you draw?

Given in the question that

Population mean$\left(\mu \right)=55000$

Population standard deviation$\left(\sigma \right)=4500$

Sample size$\left(n\right)=8$

We have to find that the sampling distribution of the mean lifetime $\overline{x}$.

The sample distribution of $\overline{x}$is written as :

$\overline{x}~N\left({\mu }_{\overline{X}},{\sigma }_{\overline{X}}\right)$

$\overline{x}~N\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)$

$\overline{x}~N\left(55000,\frac{4500}{\sqrt{8}}\right)$

$\overline{x}~N(55000,1590.99)$

Thus, the sampling distribution is normally distributed with mean $=55000$and standard deviation$=1590.99$

Given in the question that he average life of the pads an these $8$cars turns out to be $\overline{x}=51800$mile we have to find the probability that the sample mean lifetime is $51800$miles or less and also if the lifetime distribution is unchanged, What conclusion would you draw.

The probability that the sample mean is $51800$or less is calculated as follows:

$P\left(\overline{x}<191\right)=P\left(\frac{{\displaystyle \frac{\overline{x}-\mu }{\sigma }}}{\sqrt{n}}<\frac{{\displaystyle \frac{51800-\mu }{\sigma }}}{\sqrt{n}}\right)$

localid="1649200029249" $=P\left(Z,\frac{{\displaystyle \frac{51800-55000}{4500}}}{\sqrt{8}}\right)$

$=P(Z<-2.01)$

$=P\left(Z<-2.01\right)$

$=0.0222$

Thus, the required probability is $0.5346$

The computed probability is below$0.05$.thus, there is low chances for the occurance of the event.