Heights of 10 year old children

Question

In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 55.6 inches, and standard deviation of 6 inches.

What is the probability that the height of a randomly chosen child is between 47.1 and 70.2 inches? Do not round until you get your your final answer, and then round to 3 decimal places.

Answer 1

Hi Bailey,

This one has a couple more steps than your previous problems, but still involves the classic z=x-mu/sigma equation. Here, we have two x-values, which means we need to get two z-values. I will call the lower bound z₁ and the upper bound z₂. Same with their corresponding x-values, those will be x₁ and x₂. Now:

x₁=47.1

mu=55.6

sigma=6

z₁=(47.1-55.6)/6

z₁= -1.42

z₂=(70.2-55.6)/6

z₂=2.43

Now, we need to return to the z-table. The probability that z₁ is less than -1.42 (and therefore the probability that x is less than 47.1) is 0.0778. Again, look at the column for -1.4 and look at the row for 0.02. This is the value you arrive at for z_1.

Now, let's talk about z₂. Same strategy. Look at the column for 2.4, look at the row for 0.03. P(Z₂<2.43)=0.9147

To get the probability between the two, simply subtract:

P=0.9925-0.0778

P=0.9147, to 3 digits

P=0.915

I encourage you to try some more problems with z-scores on your own before posting any more here. Practice makes perfect.