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 - 1.31 and draw a sketch of the region.
sketch the region. choose the correct graph below.
o a.
1.31
o b.
-1.31
1.31
o c.
-1.31
o d.
-1.31

Explanation:

Step1: Analyze the normal distribution property

In a standard normal distribution (mean = 0, standard deviation = 1), we are looking for the region where \(z<- 1.31\). The normal distribution is symmetric about the mean \(z = 0\). The \(z\)-score is a measure of how many standard deviations an element is from the mean. A negative \(z\)-score (\(z=-1.31\)) means the value is 1.31 standard deviations below the mean.

Step2: Examine the graphs

• Graph A: The region is on the right - hand side (positive \(z\)-scores), which is incorrect as we want \(z < - 1.31\) (negative \(z\)-scores).
• Graph B: It shows a region that is symmetric around the mean (covers both positive and negative \(z\)-scores), which is not what we want (\(z < - 1.31\) is only on the left - hand side).
• Graph C: The region is on the left - hand side (negative \(z\)-scores) and is the correct representation of \(z < - 1.31\).
• Graph D: The shaded region is not the correct tail. The left - hand tail for \(z < - 1.31\) should be a larger portion (since the normal curve is continuous and the total area under the curve is 1).

Answer:

C.