Question

In the country of United States of Heightlandia, the height measurements of ten-year-old children areapproximately normally distributed with a mean of 54.8 inches, and standard deviation of 4.9 inches.What is the probability that the height of a randomly chosen child is between 51.05 and 54.75 inches? Do notround until you get your your final answer, and then round to 3 decimal places.

Answer 1

m = 54.8 inches

std = 4.9 inches

We can standarize the variable as:

(X - m)/std

So, the probability of a height randomly choosen is between 51.05 and 54.75 inches is:

P[ 51.05 < X < 54.75]

Standarizing those values:

(51.05 - m)/std = (51.05 - 54.8)/4.9 = -75/98...

(54.75 - m)/std = (51.05 - 54.8)/4.9 = -1/98...

So, the new probability is:

P[ -75/98 < Z < -1/98] = Phi(-1/98) - Phi(-75/98)

Where Z is the new standarized variable.

We can find these Phi values from tables of the standarized normal distributions:

Phi(-1/98) = 0.77796

Phi(-75/98) = 0.49593

So the probability is:

P[ -75/98 < Z < -1/98] = Phi(-1/98) - Phi(-75/98) = 0.77796 - 0.49593

P[ 51.05 < X < 54.75] = 0.28203 = 0.282

And that is the answer