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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 5.4.7
Chapter 5, Problem 5.4.7

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.


A sampling distribution is normal only when the population is normal.

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1
Understand the concept of a sampling distribution: A sampling distribution is the probability distribution of a statistic (e.g., the sample mean) obtained from a large number of samples drawn from a population.
Recall the Central Limit Theorem (CLT): The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large (typically n ≥ 30), regardless of the population's distribution.
Identify the condition when the sampling distribution is normal: If the population itself is normal, then the sampling distribution of the sample mean will also be normal, regardless of the sample size.
Evaluate the given statement: The statement 'A sampling distribution is normal only when the population is normal' is false because the sampling distribution can also be approximately normal for non-normal populations if the sample size is large enough, as per the CLT.
Rewrite the statement as true: 'A sampling distribution is normal if the population is normal, or it is approximately normal for large sample sizes due to the Central Limit Theorem.'

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Key Concepts

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

Sampling Distribution

A sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It describes how the statistic varies from sample to sample and is crucial for understanding the behavior of estimators in statistics.
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Central Limit Theorem

The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, typically when the sample size is 30 or more. This theorem is fundamental in inferential statistics, allowing for normal approximation in hypothesis testing.
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Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. Many statistical methods assume normality, and understanding this concept is essential for interpreting results and making inferences about populations based on sample data.
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Related Practice
Textbook Question

In Exercises 39 and 40, determine whether the finite correction factor should be used. If so, use it in your calculations when you find the probability.


Parking Infractions In a sample of 1000 fines issued by the City of Toronto for parking infractions in September of 2020, the mean fine was \$49.83 and the standard deviation was \$52.15. A random sample of size 60 is selected from this population. What is the probability that the mean fine is less than \$40?

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Textbook Question

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

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Textbook Question

Graphical Analysis In Exercises 11–16, determine whether the graph could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal distribution, estimate the mean and standard deviation.

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Textbook Question

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.


If the sample size is at least 30, then you can use z-scores to determine the probability that a sample mean falls in a given interval of the sampling distribution.

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Textbook Question

Graphical Analysis In Exercises 11–16, determine whether the graph could represent a variable with a normal distribution. Explain your reasoning. If the graph appears to represent a normal distribution, estimate the mean and standard deviation.

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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.


0.94

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