The Central Limit Theorem for Sums (Optional): example 7.5

Let X = one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.

ΣX = the sum or total of 80 values. Because μ_X = 90, σ_X = 15, and n = 80, $\Sigma X$ ~ N[(80)(90),
($\sqrt{\text{80}}$)(15)]

• mean of the sums = (n)(μ_X) = (80)(90) = 7200
• standard deviation of the sums = $\text{(}\sqrt{n}\text{)(}{\sigma }_X\text{) = (}\sqrt{\text{80}}\text{)}$(15)
• sum of 80 values = Σx = 7500
  

a. Find P(Σx > 7500)

P(Σx > 7500) = 0.0127

b. Find Σx where z = 1.5.

Σx = (n)(μ_X) + (z)$\left(\sqrt{n}\right)$(σ_Χ) = (80)(90) + (1.5)($\sqrt{80}$)(15) = 7401.2