Central limit theorem
Part 1: Suppose we have a distribution on x that appears to be normal (symmetrical at x-axis
aka mean [µ], area beneath curve = 1, bell shaped curve). We can take a sample of size n for
1) x̄is also normal
2) the mean of the distribution is a µ
3) the standard deviation becomes σx̄= √ n
Ex: Researchers found the height of wheat (x) follows a normal distribution, with µ = 0.9m and σ
a) What’s the probability that a single wheat plant has a height between 0.85m and 0.82m?
P(0.85 ≤ x ≤ 0.82) z1= (0.85-0.9)/0.12 = -0.42 ; z2= (0.92-0.9)/0.12 = 0.17
P(z ≤0.17) – P(z≤ -0.42) = 0.5675 – 0.3372
b) What’s the probability that the mean height of 5 plants is between 0.85m and 0.82m?
P(0.85 ≤ x̄≤ 0.82)