A normal population has a mean µ = 39 and standard deviation σ = 9. What proportion of the population is between 24 and 31?

Question

A normal population has a mean µ = 39 and standard deviation σ = 9. What proportion of the population is between 24 and 31?

Answer 1

Need to compute P(24 < x < 31), where x is a normally distributed variable with mean 39 and sd = 9

we standardize limits of interval into z-statistic, by subtracting mean and dividing by standard deviation:

P((24 - 39)/9 < (x-39)/9 < (31-39)/9) = P(-1.67 < z < -0.88)

We compute that probability using the standardize normal probability table.

Because the probabilities in that table are cumulative to the left:

P(-1.67 < z < -0.88) = P(z < -0.88) - P(z<-1.67) = 0.1894 - 0.0475 = 0.1419