A population of values has a normal distribution with mean of 51.2 and standard dev. of 12.5. You intend to draw a random sample of size of 112.

Question

Find the probability that a sample of size  is randomly selected with a mean between 51.1 and 51.8.

Answer 1

sample size (n) = 112

population mean = 51.2

population standard deviation 12.5

xbar = sample mean, want to find P(51.1 < xbar < 51.8)

x is non-standard normally distributed N(51.2,12.5)

convert xbar = 51.1 and 51.8 to standard normally distributed N(0,1) via equation z = (xbar - population mean)/standard error, where standard error = population standard deviation/square root(n) = 1.18

z is N(0,1).

convert xbar = 51.1: (51.1-51.2)/1.18 = -0.08

convert xbar = 51.8: (51.8-51.2)/1.18 = 0.508

P(51.1 < xbar < 51.8) --> P(-0.08 < z < 0.508) = 0.2269