13. Measuring bone density Individuals with low bone
density have a high risk of broken bones (fractures).
Physicians who are concerned about low bone density
(osteoporosis) in patients can refer them for special-
ized testing. Currently, the most common method for
testing bone density is dual-energy X-ray absorptiom-
etry (DEXA). A patient who undergoes a DEXA test
usually gets bone density results in grams per square
centimeter (g/cm²) and in standardized units.
7.
Judy, who is 25 years old, has her bone density
measured using DEXA. Her results indicate a bone
density in the hip of 948 g/cm² and a standardized
score of z = -1.45. In the reference population of
25-year-olet women like Judy, the mean bone density
in the hip is 956 g/cm².6
(b) Use the information provided to calculate the standard
deviation of bone density in the reference population.

The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The z-score measures how many standard deviations the measure is above or below the mean.

Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

For this problem, the parameters are given as follows:

[tex]X = 948, z = -1.45, \mu = 956[/tex]

Hence we solve for [tex]\sigma[/tex] to find the z-score as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.45 = \frac{948 - 956}{\sigma}[/tex]

[tex]-1.45\sigma = -8[/tex]

[tex]\sigma = \frac{8}{1.45}[/tex]

[tex]\sigma = 5.52[/tex]

The standard deviation of bone density in the reference population is of 5.52 g/cm².

More can be learned about the normal distribution at https://brainly.com/question/24808124

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