Question

6.12 The average length of steel nails is 5 centime ters, with a standard deviation of 0.05 centimeters. Assuming that the lengths are normally distributed, what percentage of the nails are

(b) between 4.95 and 5.05 centimeters in length?

(c) shorter than 4.90 centimeters?

(a) longer than 5.05 centimeters?

Answer 1

Let $X=$ the length of steel nail: $N(\mu, \sigma^2),$

Given $\mu=5cm, \sigma=0.05cm.$

(a)

$$P(X>5.05)=1-P(X\leq 5.05)$$

$$=1-P(Z\leq \dfrac{5.05-5}{0.05})=1-P(Z\leq 1)$$

$$\approx 0.1587$$

$15.87\%$

(b)

$$P(4.95<X<5.05)$$

$$=P(X<5.05)-P(X\leq 4.95)$$

$$=P(Z<\dfrac{5.05-5}{0.05})-P(Z\leq \dfrac{4.95-5}{0.05})$$

$$=P(Z<1)-P(Z\leq -1)$$

$$\approx0.84134- 0.15866$$

$$\approx0.6827$$

$68.27\%$

(c)

$$P(X<4.90)=P(Z<\dfrac{4.90-5}{0.05})=P(Z<-2)$$

$$\approx 0.02275$$

$2.275\%$