Question

Suppose X, the amount of money a student at a university spent on books in 2017, was normally distributed with mean $550 and standard deviation $250 (that is, µ = 550 and σ = 250). Compute the probability that a randomly selected student at this university spent between $520 and $580 on books in 2017 (that is, compute P(520 ≤ X ≤ 580)). (Show work)

Answer 1

Mean of money spent by students on books in 2017 (µ) = 550

The standard deviation (σ) = 250

To find P(520 ≤ X ≤ 580), we need to find the Z-scores (Z) corresponding to 520 and 580. Using these values we will find the probability values using a standard normal distribution table that represents the area to the left of the Z-score.

For 520,

Z₁ = (520 - µ) / σ

= (520 - 550) / 250

= -0.12

The probability to the left of Z₁ (P(X<=520)) = 45.224%

For 580,

Z₂ = (580 - µ) / σ

= (580 - 550) / 250

= 0.12

The probability to the left of Z₂ (P(X<=580)) = 54.776%

Hence using these values,

P(520 ≤ X ≤ 580) = P(X<=580) - P(X<=520)

= 54.776 - 45.224

= 9.552%

Thus the probability that a randomly selected student at this university spent between $520 and $580 on books in 2017 = 9.552%