91Ó°ÊÓ

It is given in the question that, the mean,$\mathrm{μ}=1.6$

the standard deviation, $\mathrm{σ}=1.2$

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 $\mathrm{μ}$, and the standard deviation is $\frac{\mathrm{σ}}{\sqrt{n}}$.

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, $z=\frac{x−\mathrm{μ}}{\frac{\mathrm{σ}}{\sqrt{n}}}$.

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 $z=\frac{x−\mathrm{μ}}{\frac{\mathrm{σ}}{\sqrt{n}}}$.

Substitute $2forx,1.6for\mathrm{μ},1.2for\mathrm{σ}and200forn$in the above formula and simplify.

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

$=\frac{0.4}{1.2}×\sqrt{200}$

$=4.71$

Thus, the corresponding probability is:

$P(\stackrel{¯}{x}>2)=P(z>4.71)=P(Z<−4.71)=0.0001$

Hence, the required probability is$0.0001$