Question

the heights of two-year-old children are normally distributed with a mean of 32 inches and a standard deviation of 1.5 inches. which of the following represent year - olds that are less than 35 inches tall? (hint: use the formula $z=\frac{x - mu}{sigma}$ to find the z - score.)

Explanation:

Step1: Identify given values

We know that the mean \(\mu = 32\) inches, the standard deviation \(\sigma = 1.5\) inches, and the value \(x = 35\) inches. The formula for the z - score is \(Z=\frac{x - \mu}{\sigma}\).

Step2: Calculate the z - score

Substitute the values into the formula: \(Z=\frac{35 - 32}{1.5}=\frac{3}{1.5} = 2\). So the z - score corresponding to \(x = 35\) is \(z = 2\). We want to find the proportion of two - year - olds with height less than 35 inches, which is the area to the left of \(z = 2\) in the standard normal distribution.

Step3: Analyze the graphs

In a normal distribution, the mean is at the center. The first graph (the blue - shaded one) seems to represent the area to the left of a certain value (in this case, the area corresponding to heights less than 35 inches). The second graph has a small blue area on the left, which would correspond to a very low z - score (very small height), not the area for heights less than 35 inches. So the graph that represents two - year - olds less than 35 inches tall is the first (blue - shaded) graph.

Answer:

The first (blue - shaded) graph