Question

In a survey of first graders, their mean height was 49.5 inches with a standard deviation of 3.6 inches. Assuming the heights are normally distributed, what height represents the first quartile of these students? a. 45.00 inches b. 51.93 inches c. 47.07 inches d. 48.35 inches

Answer 1

c. 47.07 inches

Explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

2) Solution for the problem

Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:

`X \sim N(49.5,3.6)`

Where
`\mu=49.5` and
`\sigma=3.6`

We are interested on the first quartile, that means the value that accumulates 25% of the area blow.

And the best way to solve this problem is using the normal standard distribution and the z score given by:

`z=(x-\mu)/(\sigma)`

We can find a value on the normal standard distribution that accumulats 0.25 of the area on the left and then we can solve for x from the z score formula.

The value that accumulates 0.25 of the area on the left can be founded using the following excel code:

"=NORM.INV(0.25,0,1)" and gives to us
`z=-0.674`

Solving for x from the z score formula we got:

`x =\mu + z\sigma`

And replacing the values we got:

`x=49.5 + (-0.674*3.6)=47.07` inches. And that would represent the first quartile for this case.