Answer to Question #338992 in Statistics and Probability for Fati

Question

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

Answer 1

Suppose that $X$ is a random variable that has a normal distribution with parameters $\mu=32$ and $\sigma=1.5$. Consider the probability: $P(X\geq\alpha)=\frac{1}{1.5\sqrt{2\pi}}\int_{\alpha}^{+\infty}e^{-\frac12\left(\frac{x-32}{1.5}\right)^2}dx$. By substitutions we receive that $P(X\geq\alpha)=0.05$ for $\alpha\approx34.47$(it is rounded to two decimal places). Thus, heights that are greater than $\alpha\approx34.47$ are problematic. Consider $P(X\leq\beta)=\frac{1}{1.5\sqrt{2\pi}}\int_{-\infty}^{\beta}e^{-\frac12\left(\frac{x-32}{1.5}\right)^2}dx$. By substitutions we obtain that heights that are lower than $\beta\approx29.53$ (it is rounded to two decimal places) are problematic.

Answer: heights that are lower than $29.53$ and higher than $34.47$ are problematic. Values are rounded to two decimal places.