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Ch. 5 - Normal Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 5 - Normal Probability DistributionsProblem 5.4.5
Chapter 5, Problem 5.4.5

True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


As the sample size increases, the mean of the distribution of sample means increases.

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1
Understand the concept of the sampling distribution of the sample mean: The sampling distribution of the sample mean is the probability distribution of all possible sample means from a population. Its mean is equal to the population mean (denoted as μ).
Recall the property of the mean of the sampling distribution: The mean of the sampling distribution of the sample mean does not depend on the sample size. It is always equal to the population mean (μ).
Analyze the statement: The statement claims that as the sample size increases, the mean of the distribution of sample means increases. This is incorrect because the mean of the sampling distribution remains constant and equal to μ, regardless of the sample size.
Rewrite the statement as a true statement: 'As the sample size increases, the variability (standard deviation) of the distribution of sample means decreases, but the mean of the distribution of sample means remains constant and equal to the population mean.'
Conclude: The original statement is false, and the corrected true statement has been provided in the previous step.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem states that as the sample size increases, the distribution of the sample means approaches a normal distribution, regardless of the shape of the population distribution. This theorem is fundamental in statistics as it allows for the use of normal probability techniques for inference, even when the original data is not normally distributed.
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Calculating the Mean

Sample Mean

The sample mean is the average of a set of values taken from a larger population. It serves as an estimate of the population mean and is calculated by summing all sample values and dividing by the number of observations. Understanding how the sample mean behaves as sample size increases is crucial for making inferences about the population.
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Sampling Distribution of Sample Proportion

Law of Large Numbers

The Law of Large Numbers states that as the size of a sample increases, the sample mean will converge to the expected value (population mean). This principle underlines the importance of larger sample sizes in achieving more accurate and reliable estimates, reinforcing the idea that variability decreases with larger samples.
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Related Practice
Textbook Question

Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.

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Textbook Question

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.


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Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


P91

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Textbook Question

"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.

You randomly select 250 U.S. men ages 18 to 29 and ask them whether they participate in at least one sport. You find that 80% say no. How likely is this result? Do you think this sample is a good one? Explain your reasoning."

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Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


P1.5

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Textbook Question

Given the mean of a normal distribution, how can you find the median?

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