A particular fruit's weights are normally distributed

Question

A particular fruit's weights are normally distributed, with a mean of 476 grams and a standard deviation of 19 grams.

If you pick 11 fruit at random, what is the probability that their mean weight will be between 473 grams and 496 grams

Answer 1

Let xbar be the mean fruit weight

z score for mean = (xbar - population mean)/(standard deviation/sqrt(sample size))

trying to find P(473 < xbar < 496)

we standardize the above range to z scores by computing z-score for 473 and 496

P((473-476)/(19/sqrt(11))) < Z < (496-476)/(19/sqrt(11))) =

P(-0.524 < Z < 3.49) =

P(Z < 3.49) - P(Z<-0.524) = 0.9998 - 0.3 = 0.6668