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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.CR.6
Chapter 5, Problem 5.CR.6

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.


To the left of z = 0.72

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Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve to the left of z = 0.72. This represents the cumulative probability for a z-score of 0.72 in a standard normal distribution.
Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0), with a mean of 0 and a standard deviation of 1. The area under the curve to the left of a given z-score represents the cumulative probability up to that z-score.
Step 3: Use a z-table or technology (such as a graphing calculator, statistical software, or an online tool) to find the cumulative probability corresponding to z = 0.72. In a z-table, locate the row for 0.7 and the column for 0.02, as these correspond to the z-score of 0.72.
Step 4: If using technology, input the z-score of 0.72 into the cumulative distribution function (CDF) for the standard normal distribution. For example, in a calculator, you might use the function normcdf(-∞, 0.72) or its equivalent.
Step 5: Interpret the result. The value obtained from the z-table or technology represents the proportion of the data under the standard normal curve to the left of z = 0.72. This is the solution to the problem.

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Key Concepts

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

Standard Normal Distribution

The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve, as it allows for the standardization of different normal distributions.
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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. In the context of the standard normal distribution, the z-score indicates the position of a value along the curve, which is essential for determining the area to the left or right of that score.
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Area Under the Curve

The area under the curve of a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be found using z-scores and standard normal tables or technology. In this case, finding the area to the left of z = 0.72 involves calculating the cumulative probability up to that z-score, which reflects the proportion of data points below that value.
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Related Practice
Textbook Question

Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (c) less than two. (d) Are any of these events unusual? Explain your reasoning.

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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 percent of the IQ scores are greater than 112?

Textbook Question

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.


Between z = 0 and z = 2.95

Textbook Question

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


a. μ = 9.2, σ ≈ 1.62, P(x < 5.97)

Textbook Question

The initial pressures for bicycle tires when first filled are normally distributed, with a mean of 70 pounds per square inch (psi) and a standard deviation of 1.2 psi.

b. A random sample of 15 tires is drawn from this population. What is the probability that the mean tire pressure of the sample is less than 69 psi?

Textbook Question

The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months.


c. What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?