Chapter 7.3, Problem 64E

To determine

To calculate:The probability that the mean number of flaws exceeds 2 per square yard.

Answer to Problem 64E

The required probability is 0.0001.

Explanation of Solution

Given information:

The mean, $\mu =1.6$

The standard deviation, $\sigma =1.2$

Number of materials studied = 200 square yards

Formula used:

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 standard deviation is $\frac{\sigma }{\sqrt{n}}$ .

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

Calculation:

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 2 for x, 1.6 for $\mu$ , 1.2 for $\sigma$and 200 for $n$in the above formula and simplify.

  $\begin{array}{c}z=\frac{2-1.6}{\frac{1.2}{\sqrt{200}}}\\ =\frac{0.4}{1.2}\times \sqrt{200}\\ =4.71\end{array}$

Thus, the corresponding probability is:

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

Hence, the required probability is 0.0001.