Question

A manufacturer knows that their items have a normally distributed length, with a mean of 18.8 inches, and standard deviation of 1.5 inches. If 10 items are chosen at random, what is the probability that their mean length is less than 17.7 inches?

Answer 1

Here given information is as follows:

Mean(u) = 18.8 inches, Standard deviation (0) = 1.5inches

10 items are chosen at random hence n = 10.

Here we have to find, Pr(X<17.7)

$Pr(X< 17.7) = Pr\left (\frac{X-\mu}{\sigma} < \frac{17.7-18.8}{1.5} \right )$$Where,\frac{X-\mu}{\sigma} =Z\sim N(0,1)$

$\therefore Pr(X< 17.7) = Pr\left (Z < \frac{17.7-18.8}{1.5} \right )= Pr\left (Z < -0.7333\right )$By using z-table values we get,

$\therefore Pr(X< 17.7) = Pr\left (Z < -0.7333\right ) =0.2317$

Hence, Probability that items chosen with mean length less than 17.7 inches is 0.2317