Question

assume that a randomly - selected subject is given a bone density test. those test scores are normally distributed with a mean of 0 and a standard deviation of 1. find the probability that a given score is less than - 1.68 and draw a sketch of the region.
sketch the region. choose the correct graph below.

Explanation:

Step1: Identify the distribution

The test - scores follow a standard normal distribution \(Z\sim N(0,1)\). We want to find \(P(Z < - 1.68)\).

Step2: Use the standard - normal table

We look up the value of \(z=-1.68\) in the standard - normal table (the cumulative distribution function of the standard normal distribution \(\varPhi(z)\)). The standard - normal table gives the area to the left of \(z\) under the standard - normal curve. Looking up \(z =-1.68\) in the table, we find that \(P(Z < - 1.68)=0.0465\).

Step3: Sketch the region

The standard - normal curve is symmetric about \(z = 0\). The region of interest is the area to the left of \(z=-1.68\) under the standard - normal curve. This corresponds to option C (since option C shows the shaded region to the left of \(z=-1.68\)).

Answer:

The probability is \(0.0465\). The correct graph for the sketch of the region is C.