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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 5.4.26
Chapter 5, Problem 5.4.26

Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


SAT Italian Subject Test The scores on the SAT Italian Subject Test for the 2018–2020 graduating classes are normally distributed, with a mean of 628 and a standard deviation of 110. Random samples of size 25 are drawn from this population, and the mean of each sample is determined.

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Step 1: 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, regardless of the population's distribution. In this case, the population is already normally distributed, so the sampling distribution of the sample mean will also be normal.
Step 2: Identify the population mean (μ) and population standard deviation (σ) from the problem. Here, the population mean is μ = 628, and the population standard deviation is σ = 110.
Step 3: Calculate the mean of the sampling distribution of the sample mean. According to the CLT, the mean of the sampling distribution is equal to the population mean. Therefore, the mean of the sampling distribution is μₓ̄ = μ = 628.
Step 4: Calculate the standard deviation of the sampling distribution of the sample mean, also known as the standard error (SE). The formula for the standard error is: σₓ̄=σn, where n is the sample size. Substitute σ = 110 and n = 25 into the formula to find the standard error.
Step 5: Sketch the graph of the sampling distribution. Since the sampling distribution is normal, draw a bell-shaped curve centered at the mean μₓ̄ = 628. Label the x-axis with values representing the mean and standard deviations (e.g., μₓ̄ ± σₓ̄, μₓ̄ ± 2σₓ̄, etc.).

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Key Concepts

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

Central Limit Theorem (CLT)

The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n ≥ 30). This theorem is fundamental in statistics as it allows for the use of normal probability techniques to make inferences about population parameters based on sample statistics.
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Calculating the Mean

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. It is characterized by its mean, which equals the population mean, and its standard deviation, known as the standard error, which is the population standard deviation divided by the square root of the sample size. Understanding this concept is crucial for estimating population parameters and conducting hypothesis tests.
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Sampling Distribution of Sample Proportion

Mean and Standard Deviation of Sampling Distribution

For a given population with mean (μ) and standard deviation (σ), the mean of the sampling distribution of the sample mean is equal to μ, while the standard deviation of the sampling distribution (standard error) is calculated as σ/√n, where n is the sample size. In the context of the SAT Italian Subject Test, this means that for samples of size 25, the mean will remain 628, and the standard deviation will be 110/√25 = 22.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

Finding Probability In Exercises 41–46, find the probability of z occurring in the shaded region of the standard normal distribution. If convenient, use technology to find the probability.


Textbook Question

Explain how to transform a given x-value of a normally distributed variable x into a z-score.

Textbook Question

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=24, p=0.85, q=0.15

Textbook Question

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.


P(- 1.54 < z < 1.54)

Textbook Question

Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.

Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.


P(x ≤ 150)