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Chapter 6: Q1CQQ (page 226)

Bone Density Test.

In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.

Bone Density Sketch a graph showing the shape of the distribution of bone density test scores.

Short Answer

Expert verified

The graph for the shape of distribution of bone density test scores is shown below:

Step by step solution

01

Given information

The bone mineral density test scores are normally distributed with mean value of 0 and standard deviation of 1.

02

Describe the random variable

Let Z be the random variable for bone mineral density test scores.

\(\begin{aligned}{c}Z \sim N\left( {\mu ,{\sigma ^2}} \right)\\ \sim N\left( {0,{1^2}} \right)\end{aligned}\)

The shape of the curve is expected to be bell-shaped, symmetric at mean value of 0.

03

Sketch the graph

The steps to sketch the graph are:

  1. Draw horizontal axis with values extending from –3 to 3 with a gap of 1 unit each.
  2. Sketch a bell shape curve such that it increases from –3 to 0 and then decreases smoothly from 0 to 3, maintaining symmetry at 0.

The graph is shown below:

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Most popular questions from this chapter

Standard normal distribution. In Exercise 17-36, 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. In each case, Draw a graph, then find the probability of the given bone density test score. If using technology instead of Table A-2, round answers to four decimal places.

Less than 1.28

Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.

Find P99, the 99th percentile. This is the bone density score separating the bottom 99% from the top 1%.

Low Birth Weight The University of Maryland Medical Center considers “low birth weights” to be those that are less than 5.5 lb or 2495 g. Birth weights are normally distributed with a mean of 3152.0 g and a standard deviation of 693.4 g (based on Data Set 4 “Births” in Appendix B).

a. If a birth weight is randomly selected, what is the probability that it is a “low birth weight”?

b. Find the weights considered to be significantly low, using the criterion of a birth weight having a probability of 0.05 or less.

c. Compare the results from parts (a) and (b).

In Exercises 13–20, use the data in the table below for sitting adult malesand females (based on anthropometric survey data from Gordon, Churchill, et al.). Thesedata are used often in the design of different seats, including aircraft seats, train seats,theater seats, and classroom seats. (Hint: Draw a graph in each case.)

Sitting Back-to-Knee Length (Inches)

Mean

St. Dev

Distribution

Males

23.5 in

1.1 in

Normal

Females

22.7 in

1.0 in

Normal

Significance Instead of using 0.05 for identifying significant values, use the criteria that a value x is significantly high if P(x or greater) 0.025 and a value is significantly low if P(x or less) 0.025. Find the female back-to-knee length, separating significant values from those that are not significant. Using these criteria, is a female back-to-knee length of 20 in. significantly low?

Jet Ejection Seats The U.S. Air Force once used ACES-II ejection seats designed for men weighing between 140 lb and 211 lb. Given that women’s weights are normally distributed with a mean of 171.1 lb and a standard deviation of 46.1 lb (based on data from the National Health Survey), what percentage of women have weights that are within those limits? Were many women excluded with those past specifications?
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