Probability Stats

Question

A particular fruit's weights are normally distributed, with a mean of 472 grams and a standard deviation of 18 grams.

If you pick 15 fruit at random, what is the probability that their mean weight will be between 470 grams and 477 grams

Answer 1

xbar = mean weight is normally distributed with mean 472 and SD 18

want to find P(470 < xbar < 477)

for each value of inequality (xbar) compute z stat:

z = (xbar - mean)/(SD/sqrt(15), where z is distributed as mean 0 and standard deviation 1

for xbar = 470, z = (470 - 472/(18/sqrt(15)) = -0.43

for xbar = 477, z = (477- 472)/(18/sqrt(15)) = 1.08

so P(470 < xbar < 477) = P(-0.43 < z < 1.08) = P(z < 1.08) - P(z < -0.43) = 0.8599 - 0.3336 = 0.5263