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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.R.4b
Chapter 6, Problem 6.R.4b

Arm Circumferences Arm circumferences of adult men are normally distributed with a mean of 33.64 cm and a standard deviation of 4.14 cm (based on Data Set 1 “Body Data” in Appendix B). A sample of 25 men is randomly selected and the mean of the arm circumferences is obtained.
b. What is the mean of all such sample means?

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1
Identify the key information provided in the problem: the population mean (μ = 33.64 cm), the population standard deviation (σ = 4.14 cm), and the sample size (n = 25).
Recall the property of the sampling distribution of the sample mean: the mean of the sampling distribution of the sample mean is equal to the population mean (μ).
State the formula for the mean of the sampling distribution of the sample mean: μ_x̄ = μ, where μ_x̄ is the mean of the sample means and μ is the population mean.
Substitute the given population mean (μ = 33.64 cm) into the formula to determine the mean of the sample means.
Conclude that the mean of all such sample means is equal to the population mean, which is 33.64 cm.

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Key Concepts

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

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean refers to the distribution of means obtained from all possible samples of a specific size drawn from a population. According to the Central Limit Theorem, this distribution will be approximately normal if the sample size is sufficiently large, regardless of the population's distribution. The mean of this sampling distribution equals the population mean.
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Sampling Distribution of Sample Proportion

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of the sample means will approach a normal distribution as the sample size increases, typically n ≥ 30 is considered sufficient. This theorem is crucial for making inferences about population parameters based on sample statistics, as it allows for the application of normal probability methods even when the original population distribution is not normal.
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Calculating the Mean

Mean of Sample Means

The mean of all sample means, also known as the expected value of the sample mean, is equal to the population mean. In this case, since the population mean of arm circumferences is given as 33.64 cm, the mean of all sample means for samples of size 25 will also be 33.64 cm. This concept is fundamental in inferential statistics, as it underpins the reliability of sample estimates.
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Related Practice
Textbook Question

In Exercises 8 and 9, assume that women have standing eye heights that are normally distributed with a mean of 59.7 in. and a standard deviation of 2.5 in. (based on anthropometric survey data from Gordon, Churchill, et al.).

a. If an eye recognition security system is positioned at a height that is uncomfortable for women with standing eye heights less than 54 in., what percentage of women will find that height uncomfortable?

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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).


a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending.

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Textbook Question

Mensa Membership in Mensa requires a score in the top 2% on a standard intelligence test. The Wechsler IQ test is designed for a mean of 100 and a standard deviation of 15, and scores are normally distributed.


c. If 4 subjects take the Wechsler IQ test and they have a mean of 131 but the individual scores are lost, can we conclude that all 4 of them have scores of at least 131?

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Textbook Question

Birth Weights Based on Data Set 6 “Births” in Appendix B, birth weights of girls are normally distributed with a mean of 3037.1 g and a standard deviation of 706.3 g.


b. What is the value of the median?

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Textbook Question

Birth Weights Based on Data Set 6 “Births” in Appendix B, birth weights of girls are normally distributed with a mean of 3037.1 g and a standard deviation of 706.3 g.


c. What is the value of the mode?

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Textbook Question

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.


Sampling Distribution of the Sample Variance


a. Find the value of the population variance σ2.

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