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 - 0.57 and draw a sketch of the region. sketch the region. choose the correct graph below.

Explanation:

Step1: Use the standard normal distribution table

We know that for a standard normal distribution \(Z\sim N(0,1)\), we want to find \(P(Z < - 0.57)\).
By the property of the standard normal distribution, \(P(Z < -z)=1 - P(Z < z)\) when \(z>0\). Also, we can directly use the standard - normal table (or a calculator with a normal - distribution function).
Looking up the value of \(\varPhi(0.57)\) in the standard normal table (where \(\varPhi(z)=P(Z < z)\)).
From the standard normal table, \(\varPhi(0.57)=0.7157\).

Step2: Calculate \(P(Z < - 0.57)\)

Since \(P(Z < -0.57)=1 - P(Z < 0.57)\) (because of the symmetry of the standard normal distribution \(P(Z < -z)+P(Z < z)=1\) for \(z > 0\)).
Substitute \(P(Z < 0.57) = 0.7157\) into the formula.
\(P(Z < -0.57)=1 - 0.7157=0.2843\)

For the sketch:
The standard normal distribution is symmetric about \(z = 0\). The region \(P(Z < -0.57)\) is the area to the left of \(z=-0.57\) under the standard - normal curve.

Answer:

The probability that a given score is less than \(-0.57\) is \(0.2843\). The correct graph is the one (assuming in the options) that has the shaded region to the left of \(z =-0.57\) (which is likely option D if we assume a standard set of normal - distribution graphs where for \(P(Z < -0.57)\) the left - tail is shaded).