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

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

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 = (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 = (X - \mu)/(\sigma)`

`-1.645 = (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 = (X - \mu)/(\sigma)`

`1.645 = (X - 32)/(1.5)`

`X - 32 = 1.645*1.5`

`X = 34.4675`

A height higher than 34.4675 inches is problematic.