Answer to Question #284450 in Statistics and Probability for Mohamed

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)

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

$$\approx 0.02275$$

$2.275\%$