Question

A machine produces bolts which are N(4,0.09) where measurements are in cm. Bolts are measured accurately and any bolt smaller than 3.5cm or longer than 4.4cm is rejected. out of 500 bolts how many would be accepted?

Answer 1

$$P(3.5<X<4.4)=P(X<4.4)-P(X\leq 3.5)$$

$$=P(Z<\dfrac{4.4-\mu}{\sigma})-P(Z\leq\dfrac{3.5-\mu}{\sigma})$$

$$=P(Z<\dfrac{4.4-4}{0.3})-P(Z\leq\dfrac{3.5-4}{0.3})$$

$$\approx P(Z<1.333333)-P(Z\leq -1.666667)$$

$$\approx 0.9087888-0.0477904\approx0.8609984$$

$$0.8609984(500)\approx430$$

430 bolts out of 500 bolts would be accepted.