Answer to Question #254224 in Statistics and Probability for kayjay

Question

the length of 5000 corn cobs produced from a local wholesaler are measured and recorded. the lengths X, are found to be normally distributed with a mean of 9cm and a variance of 0.25cm². Find the probability that a corn cob selected at random from the wholesaler has a length between 8.5cm and 9.25cm.

Answer 1

$X\sim N(\mu, \sigma^2)$

Given $\mu=9\ cm , \sigma^2=0.25\ cm^2.$

$$P(8.5<X<9.25)=P(X<9.25)-P(X\leq8.5)$$

$$=P(Z<\dfrac{9.25-9}{0.5})-P(Z\leq\dfrac{8.5-9}{0.5})$$

$$=P(Z<0.5)-P(Z\leq-1)$$

$$\approx0.691462-0.158655\approx0.532807$$

The probability that a corn cob selected at random from the wholesaler has a length between 8.5cm and 9.25cm is $0.5328.$