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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 5.4.20
Chapter 5, Problem 5.4.20

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 The zloty is the official currency of Poland. During a recent period of two years, the day-ahead prices for renewable energy in Poland (in zlotys per mega-watt hour) have a mean of 158.51 and a standard deviation of 33.424. Random samples of size 100 are drawn from this population, and the mean of each sample is determined. (Adapted from Multidisciplinary Digital Publishing Institute)

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1
Step 1: Recall the Central Limit Theorem (CLT). The CLT states that for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution. The mean of the sampling distribution will equal the population mean, and the standard deviation of the sampling distribution (known as the standard error) will be the population standard deviation divided by the square root of the sample size.
Step 2: Identify the given values from the problem. The population mean (μ) is 158.51, the population standard deviation (σ) is 33.424, and the sample size (n) is 100.
Step 3: Calculate the mean of the sampling distribution. According to the CLT, the mean of the sampling distribution of sample means is the same as the population mean. Therefore, the mean of the sampling distribution is μ = 158.51.
Step 4: Calculate the standard error of the sampling distribution. The formula for the standard error (SE) is: SE=σn. Substitute the given values: SE=33.424100. Simplify the denominator to find the standard error.
Step 5: Sketch the graph of the sampling distribution. The graph will be a normal distribution centered at the mean (158.51) with a standard deviation equal to the calculated standard error. Label the x-axis with values around the mean, spaced by increments of the standard error, and indicate the bell-shaped curve.

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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

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 sample means vary from sample to sample and is crucial for understanding the variability and reliability of estimates derived from sample data.
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Sampling Distribution of Sample Proportion

Mean and Standard Deviation of Sampling Distribution

The mean of the sampling distribution of sample means is equal to the population mean, while the standard deviation (known as the standard error) is calculated by dividing the population standard deviation by the square root of the sample size (σ/√n). These measures help quantify the expected value and variability of sample means, which are essential for making statistical inferences.
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Sampling Distribution of Sample Proportion
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.


Old Faithful In a sample of 100 eruptions of the Old Faithful geyser at Yellowstone National Park, the mean interval between eruptions was 129.58 minutes and the standard deviation was 108.54 minutes. A random sample of size 30 is selected from this population. What is the probability that the mean interval between eruptions is between 120 minutes and 140 minutes?

Textbook Question

Bags of Baby Carrots The weights of bags of baby carrots are normally distributed, with a mean of 32 ounces and a standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the most a bag of baby carrots can weigh and not need to be repackaged?

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(z < - 1.11)

Textbook Question

Conservation About 74% of the residents in a town say that they are making an effort to conserve water or electricity. One hundred ten residents are randomly selected. What is the probability that the sample proportion making an effort to conserve water or electricity is greater than 80%? Interpret your result.

Textbook Question

Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean mu=174 and standard deviation sigma=20. Find the indicated probability.


P(x > 182)

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.


As the sample size increases, the standard deviation of the distribution of sample means increases.