Skip to main content
Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.1.49c
Chapter 6, Problem 6.1.49c

Significance For bone density scores that are normally distributed with a mean of 0 and a standard deviation of 1, find the percentage of scores that are


c. not significant (or less than 2 standard deviations away from the mean).

Verified step by step guidance
1
Recognize that the problem involves a standard normal distribution, which has a mean (μ) of 0 and a standard deviation (σ) of 1.
Understand that 'not significant' refers to scores that are within 2 standard deviations from the mean, i.e., between -2 and +2 on the z-score scale.
Use the cumulative distribution function (CDF) of the standard normal distribution to find the probability of a z-score being less than +2. This is represented as P(Z < 2).
Similarly, use the CDF to find the probability of a z-score being less than -2. This is represented as P(Z < -2).
Subtract P(Z < -2) from P(Z < 2) to find the percentage of scores that are within 2 standard deviations of the mean: P(-2 < Z < 2) = P(Z < 2) - P(Z < -2). Multiply the result by 100 to express it as a percentage.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the mean is 0 and the standard deviation is 1, which allows for the application of the empirical rule.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this question, scores that are less than 2 standard deviations from the mean (between -2 and 2) are considered not significant.
Recommended video:
Guided course
08:45
Calculating Standard Deviation

Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations. This rule helps in determining the percentage of scores that are not significant, as it indicates that roughly 95% of scores will lie within 2 standard deviations from the mean.
Recommended video:
5:14
Probability of Mutually Exclusive Events
Related Practice
Textbook Question

Continuity Correction In testing the assumption that the probability of a baby boy is 0.512, a geneticist obtains a random sample of 1000 births and finds that 502 of them are boys. Using the continuity correction, describe the area under the graph of a normal distribution corresponding to the following. (For example, the area corresponding to “the probability of at least 502 boys” is this: the area to the right of 501.5.)


c. The probability of more than 502 boys

154
views
Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30


d. Find the variance.

97
views
Textbook Question

Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g


d. What is the value of the variance?

173
views
Textbook Question

Sleepwalking Assume that 29.2% of people have sleepwalked (based on “Prevalence and Comorbidity of Nocturnal Wandering in the U.S. Adult General Population, by Ohayon et al., Neurology, Vol. 78, No. 20). Assume that in a random sample of 1480 adults, 455 have sleepwalked.


c. What does the result suggest about the rate of 29.2%?

99
views
Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Doorway Height The Boeing 757-200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 68.6 in. and a standard deviation of 2.8 in. (based on Data Set 1 “Body Data” in Appendix B).


d. When considering the comfort and safety of passengers, why are women ignored in this case?

98
views
Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.


d. Is the new capacity of 20 passengers safe?

111
views