Question

The heights of children 2 years old are normally distributed with a mean of 32 inches and a standard deviation of 1.5 inches. Pediatrician regularly measure the heights of toddlers to determine whether there is a problem. Pediatrician determines that there may be a problem when a child is in the top or bottom of 5% heights. What is the height of a 2 year old that could be a problem

Answer 1

Answer:

The problematic heights are those lower than 29.5325 inches and higher than higher than 34.4675 inches

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 32, \sigma = 1.5$

Pediatrician determines that there may be a problem when a child is in the top or bottom of 5% heights.

Bottom 5%

Any height lower than the value of X when Z has a pvalue of 0.05. So $Z = -1.645$

$Z = \frac{X - \mu}{\sigma}$

$-1.645 = \frac{X - 32}{1.5}$

$X - 32 = -1.645*1.5$

$X = 29.5325$

A height lower than 29.5325 inches is problematic.

Top 5%

Any height higher than the value of X when Z has a pvalue of 0.95. So $Z = 1.645$

$Z = \frac{X - \mu}{\sigma}$

$1.645 = \frac{X - 32}{1.5}$

$X - 32 = 1.645*1.5$

$X = 34.4675$

A height higher than 34.4675 inches is problematic.