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. draw a graph and find the bone density test scores that can be used as cutoff values separating the lowest 3% and highest 3%, indicating levels that are too low or too high, respectively. sketch the region containing the lowest 3% and highest 3%. choose the correct graph below. a. b. c. d. the bone density scores are (use a comma to separate answers as needed. round to two decimal places as needed.)

Explanation:

Step1: Find the z - score for the lower tail

We want to find \(z_{\alpha}\) such that \(P(Z < z_{\alpha})=0.03\). Using the standard normal table (or a calculator with a normal - distribution function, e.g., in Excel: NORM.S.INV(0.03)), we get \(z_{\alpha}\approx - 1.88\).

Step2: Find the z - score for the upper tail

Due to the symmetry of the standard normal distribution \(N(0,1)\), if \(P(Z < z_{\alpha}) = 0.03\), then \(P(Z>z_{1 - \alpha})=0.03\) and \(z_{1 - \alpha}\approx1.88\) (since \(P(Z>z)=1 - P(Z < z)\) and for a symmetric distribution \(z_{1 - \alpha}=-z_{\alpha}\) when \(P(Z < z_{\alpha})=\alpha\) and \(P(Z>z_{1 - \alpha})=\alpha\)).

Answer:

\(-1.88,1.88\)