A normally distributed population has mean 1214 and standard deviation 122

Question

A normally distributed population has mean 1214 and standard deviation 122. (a) Find the probability that a single randomly selected element of the population is between 1100 and 1300. (b) Find the mean and standard deviation of the mean of a randomly chosen sample of a sample of size n = 25. (c) Find the probability that the mean of a random sample of size 25 is between 1100 and 1300.

Answer 1

Let's start by listing variables we know:

μ = 1214

σ = 122

a. P(1100 < x < 1300)?

z = (1100 - 1214)/122 = -0.93

z = (1300 - 1214)/122 = 0.70

P(-0.93 < z < 0.70) = 0.7580 - 0.1762 = 0.5818 = 58.18%

b. When n = 25, the sample mean and standard deviation are as follows:

μ_x = μ = 1214

σ_x = σ/√n = 122/√25 = 24.4

c. P(1100 < x < 1300) for n = 25?

z = (1100 - 1214)/24.4 = -4.67

z = (1300 - 1214)/24.4 = 3.52

P(-4.67 < z < 3.52) = 1 = 100%