Answer to Question #348953 in Statistics and Probability for math234

Question

The diameters of washers are normally distributed with mean 50mm and variance

4mm. A washer is considered non-defective if its diameter lies between 46mm and

53mm. If one washer is randomly selected, what is the probability that:

4.2.1 its diameter less than 45mm? (6)

4.2.2 itis defective?

Answer 1

Let $X=$ diameters of washer: $X\sim N(\mu, \sigma^2).$

Given $\mu=50mm, \sigma^2=4mm^2$

4.2.1

$$P(X<45)=P(Z<\dfrac{45-50}{\sqrt{4}})$$

$$=P(Z<-2.5)\approx0.006210$$

4.2.2

$$P(46<X<53)=P(Z<\dfrac{53-50}{\sqrt{4}})$$

$$-P(Z\le\dfrac{46-50}{\sqrt{4}})=P(Z<1.5)-P(Z\le-2)$$

$$\approx0.9331928-0.0227501$$

$$\approx0.910443$$

$$P(defective)=1-0.910443=0.089557$$