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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.R.51b
Chapter 5, Problem 5.R.51b

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.


The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

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Step 1: Understand the problem. We are given a population of values {1, 2, 0, 3} representing the goals scored by four defenders. We need to consider all possible samples of size 2 (with replacement) from this population, calculate the mean of each sample, and then analyze the sampling distribution of sample means.
Step 2: List all possible samples of size 2 with replacement. Since there are 4 elements in the population and sampling is done with replacement, the total number of samples is 4 × 4 = 16. The samples are: (1,1), (1,2), (1,0), (1,3), (2,1), (2,2), (2,0), (2,3), (0,1), (0,2), (0,0), (0,3), (3,1), (3,2), (3,0), (3,3).
Step 3: Calculate the mean of each sample. For each sample, compute the mean using the formula: μ=x+y2, where x and y are the two values in the sample. For example, for the sample (1,1), the mean is (1+1)/2 = 1. Repeat this for all 16 samples.
Step 4: Find the mean and standard deviation of the sampling distribution of sample means. The mean of the sampling distribution is the average of all sample means. The standard deviation of the sampling distribution is calculated using the formula: ∑(x-μ)2n, where x is each sample mean, μ is the mean of the sampling distribution, and n is the number of samples.
Step 5: Compare the mean and standard deviation of the sampling distribution with the population mean and standard deviation. The population mean is calculated as ∑xN, where x represents the population values and N is the population size. The population standard deviation is calculated using the formula: ∑(x-μ)2N. Compare these values to the mean and standard deviation of the sampling distribution to observe the relationship.

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Key Concepts

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

Sampling with Replacement

Sampling with replacement means that after selecting an item from a population, it is returned to the population before the next selection. This allows for the same item to be chosen multiple times in different samples. In the context of the question, it means that when forming samples of size 2 from the defenders' goals, each goal can be selected more than once, leading to a larger number of possible samples.
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Sampling Distribution of Sample Proportion

Mean of a Sample

The mean of a sample is calculated by summing all the values in the sample and dividing by the number of values. It provides a measure of central tendency for the sample. In this exercise, after listing all possible samples of size 2, you will compute the mean for each sample to understand the average goals scored in those samples.
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Sampling Distribution of Sample Proportion

Sampling Distribution of Sample Means

The sampling distribution of sample means is the distribution of the means of all possible samples of a given size from a population. It is important because it allows us to understand how sample means vary and how they relate to the population mean. The mean and standard deviation of this distribution can be compared to the population's mean and standard deviation to assess the accuracy and reliability of the sample estimates.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for Level 1 actuaries in the United States is about \$72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (b) more than \(68,000? Assume sigma = \)11,000.

Textbook Question

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

Textbook Question

In Exercises 61 and 62, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.


A survey of U.S. adults ages 33 to 40 earning more than \$150,000 per year found that 94% are content with how their lives have turned out so far. You randomly select 20 U.S. adults ages 33 to 40 earning more than \$150,000 and ask if they are content with their lives so far.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.


The mean annual salary for physical therapists in the United States is about \$87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (b) more than \(85,000? Assume sigma = \)10,500.

Textbook Question

In Exercises 69 and 70, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.


A survey of U.S. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (b) exactly 50.