The height of 3-year-old boys is approximately normally distributed. Duncan and Shane are 3-year-old boys.
Duncan is 32.0 inches tall and is at the 32nd percentile of the distribution. Shane is 34.0 inches tall and is at
the 62nd percentile of the distribution. Which of the following is closest to the mean of the height distribution?
Use the z-table to obtain z-scores that correspond to the percentiles.
The 32nd percentile corresponds to a z-score of -0.47. Shane's height is 0.47 standard deviations below the mean.
The 62nd percentile corresponds to a z-score of 0.31. Duncan's height is 0.31 standard deviations above the mean.
Using the equations for z-score:
Duncan
$$ z= \frac{x-\mu}{\sigma} $$
$$ -0.47= \frac{32-\mu}{\sigma} $$
$$ -0.47\sigma = 32-\mu $$
Shane
$$ z= \frac{x-\mu}{\sigma} $$
$$ 0.31= \frac{34-\mu}{\sigma} $$
$$ 0.31\sigma=34-\mu $$
We can solve for the two variables by treating the problem like a system of equations. Performing elimination through subtracting the equations:
$$ -0.47\sigma - 0.31\sigma= 32-\mu - (34-\mu) $$
$$ -0.78\sigma = -2 $$
$$ \sigma = 2.564 $$
We can substitute this back into either equation to obtain \(\mu\).
$$ -0.47(2.56)=32-\mu $$
$$ \mu = 32 + 0.47(2.564) $$
$$ \mu = \boxed{33.21} $$