Question

120 hardness measurements are made on a large slab of steel with an average of Rockwell C value of 39 and a standard deviation of 4.0. What percent of measurements should fall between 35 and 45?

Answer 1

Answer: 77.45 %

Step-by-step explanation:

We assume that the measurements are normally distributed.

Given : Mean : $\mu=39$

Standard deviation : $\sigma=4.0$

Let x be the randomly selected measurement.

Now we calculate z score for the normal distribution as :-

$z=\dfrac{x-\mu}{\sigma}$

For x = 35, we have

$z=\dfrac{35-39}{4}=-1$

For x = 45, we have

$z=\dfrac{45-39}{4}=1.5$

Now, the p-value = $P(35<x<45)=P(-1<z<1.5)$

$=P(z<1.5)-P(z<-1)\\\\=0.9331927-0.1586553=0.7745374\approx0.7745$

In percent , $0.7745\times100=77.45\%$

Hence, the percent of measurements should fall between 35 and 45 = 77.45 %