Bad carpet The number of flaws per square yard is a type of carpet material that varies with mean $1.6$flaws per square yard and standard deviation $1.2$flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector studies $200$square yards of the material records the number of flaws found in each square yard, and calculates $x$, the mean number of flaws per square yard inspected. Find the probability that the mean number of flaws exceeds $2$ per square yard. Follow the four-step process.

It is given in the question that, the mean,$\mu =1.6$

the standard deviation, $\sigma =1.2$

the number of materials studied = $200$square yards.

Find the probability that the mean number of flaws exceeds $2$per square yard.

The central limit theorem states that if the sample size of a sampling distribution is $30$or more, then the sample mean is approximately normal whose mean is $\mu$, and the standard deviation is $\frac{\sigma }{\sqrt{n}}$.

The z-value of a distribution can be found by dividing the difference between the population mean and sample mean by the standard deviation that is, $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$.

The sample size of 200 square yards is at least 30; so apply the central limit theorem.

Find the z value by using the formula $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$.

Substitute $2forx,1.6for\mu ,1.2for\sigma and200forn$in the above formula and simplify.

$z=\frac{2-1.6}{\frac{1.2}{\sqrt{200}}}$

$=\frac{0.4}{1.2}\times \sqrt{200}$

$=4.71$

Thus, the corresponding probability is:

$P(\overline{x}>2)=P(z>4.71)=P(Z<-4.71)=0.0001$

Hence, the required probability is$0.0001$