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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.38
Chapter 5, Problem 5.R.38

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.


 0.1

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Identify the problem: You are tasked with finding the z-score that corresponds to a cumulative area (or percentile) of 0.1. This means that 10% of the data lies to the left of the z-score in a standard normal distribution.
Understand the relationship: The cumulative area represents the probability that a randomly selected value from the standard normal distribution is less than or equal to the z-score. The standard normal distribution has a mean of 0 and a standard deviation of 1.
Use the Standard Normal Table or technology: Locate the cumulative area of 0.1 in the Standard Normal Table. The table typically provides cumulative probabilities for z-scores. Alternatively, use statistical software or a calculator with an inverse normal function (e.g., invNorm in a TI calculator).
Interpret the result: The z-score corresponding to a cumulative area of 0.1 will be negative because the area is less than 0.5, indicating it is on the left side of the mean in the standard normal distribution.
Verify your result: Double-check the z-score by plugging it back into the cumulative probability formula or using technology to ensure the cumulative area is approximately 0.1.

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Key Concepts

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

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 indicate how many standard deviations an element is from the mean, allowing for comparison across different datasets.
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Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the bell-shaped curve and is used to determine probabilities and percentiles for normally distributed data. The area under the curve corresponds to probabilities, making it essential for finding z-scores.
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Cumulative Area

Cumulative area refers to the total area under the curve of a probability distribution up to a certain point. In the context of the standard normal distribution, it represents the probability that a randomly selected score will be less than or equal to a specific z-score. This concept is crucial for interpreting z-scores in terms of percentiles and probabilities.
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Related Practice
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 (a) less than \(75,000? Assume sigma = \)11,000.

Textbook Question

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


Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (a) less than 5500 MMT CO2 eq. Compare your answers with those in Exercise 33.

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 (a) less than \(84,000? Assume sigma = \)10,500.

Textbook Question

An auto parts seller finds that 1 in every 200 parts sold is defective. Use the geometric distribution to find the probability that (c) none of the first 20 parts sold are defective.

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Textbook Question

Find the positive z-score for which 94% of the distribution’s area lies between -z and z.

Textbook Question

In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.


P(x ≤ 36)