Question

Consider two sample means distributions corresponding to the same x distribution. The first sample mean distribution is based on samples of size n=100 and the second is based on sample of size n=225.Which sample mean distribution has the smaller standard error? Explain.

What percentage of the area lies under the standard normal curve
(a) to the left of the µ?
(b) between µ –   σ and µ + σ?
(c) between µ - 3 σ and µ + 3 σ?  

Answer 1

Z = (X - µ) / σ

a) P(to the left of µ) = P(Z < 0) = 0.5 = 50%

b) P(between µ-σ and µ+σ) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826 = 68.26%

c) P(between µ-3σ and µ+3σ) = P(-3 < Z < 3) = P(Z < 3) - P(Z < -3) = 0.9987 - 0.0013 = 0.9974 = 99.74%