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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.Q.4b
Chapter 5, Problem 5.Q.4b

The random variable x is normally distributed with the given parameters. Find each probability.
b. μ = 87, σ ≈ 19, P(x > 40.5)

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Step 1: Understand the problem. The random variable x is normally distributed with a mean (μ) of 87 and a standard deviation (σ) of approximately 19. We are tasked with finding the probability P(x > 40.5).
Step 2: Standardize the value of x = 40.5 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (40.5 - 87) / 19.
Step 3: Simplify the z-score calculation to find the standardized value. This will give you the z-score corresponding to x = 40.5.
Step 4: Use a standard normal distribution table or a statistical software/tool to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(x ≤ 40.5).
Step 5: Since the problem asks for P(x > 40.5), use the complement rule: P(x > 40.5) = 1 - P(x ≤ 40.5). Subtract the cumulative probability from 1 to find the desired probability.

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Key Concepts

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

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean, and about 95% falls within two standard deviations. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.
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Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. Z-scores allow for the comparison of scores from different normal distributions and are essential for finding probabilities associated with specific values in a normal distribution.
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Probability Calculation

Probability calculation in the context of a normal distribution involves determining the likelihood of a random variable falling within a certain range. This is often done using Z-scores and standard normal distribution tables or software. For the given problem, calculating P(x > 40.5) requires finding the Z-score for 40.5 and then using the standard normal distribution to find the corresponding probability.
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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

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 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

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.

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 a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What is the highest score that would still place a person in the bottom 10% of the scores?