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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.51a
Chapter 5, Problem 5.R.51a

In Exercises 51 and 52, a population and sample size are given. (a) Find 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: Calculate the mean of the population. The population consists of the values {1, 2, 0, 3}. Use the formula for the population mean: μ = ∑xiN, where N is the total number of values in the population.
Step 2: Calculate the population standard deviation. Use the formula: ∑(xi - μ)2N, where μ is the population mean calculated in Step 1.
Step 3: Identify all possible samples of size 2 from the population {1, 2, 0, 3}. Since the sample size is 2, list all combinations of two values from the population. For example, one sample could be {1, 2}, another could be {1, 0}, and so on.
Step 4: For each sample, calculate the sample mean using the formula: ∑xin, where n is the sample size (in this case, 2).
Step 5: For each sample, calculate the sample standard deviation using the formula: ∑(xi - x̄)2n - 1, where x̄ is the sample mean calculated in Step 4.

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Key Concepts

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

Population Mean

The population mean is the average of all values in a population. It is calculated by summing all the values and dividing by the total number of values. In this case, to find the mean of the goals scored by the defenders, you would add the goals (1 + 2 + 0 + 3) and divide by 4, the total number of defenders.
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Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean. To calculate it, you find the variance (the average of the squared differences from the mean) and then take the square root of that variance. This helps in understanding the consistency of the defenders' goal-scoring performance.
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Calculating Standard Deviation

Sample Size

Sample size refers to the number of observations or data points selected from a population for analysis. In this question, a sample size of 2 means that you will select 2 defenders' goal scores to analyze. The choice of sample size can affect the accuracy and reliability of statistical estimates, such as the sample mean and standard deviation.
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Related Practice
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 (c) greater than 60.

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

The random variable x is normally distributed with the given parameters. Find each probability.

b. μ = 87, σ ≈ 19, P(x > 40.5)

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 test scores for the Law School Admission Test (LSAT) in a recent year are normally distributed, with a mean of 151.88 and a standard deviation of 9.95. Random samples of size 40 are drawn from this population, and the mean of each sample is determined.

Textbook Question

The random variable x is normally distributed with the given parameters. Find each probability.


c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

Textbook Question

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


The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.