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

Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.3.21

Chapter 5, Problem 5.3.21

Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.

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Identify the problem: The graph shows a standard normal distribution (bell curve) with two shaded areas, each representing 0.05 of the total area under the curve. The task is to find the z-scores corresponding to these areas.

Understand the symmetry of the standard normal distribution: The curve is symmetric about z = 0. The two z-scores will be equidistant from the mean (z = 0), one negative and one positive.

Determine the cumulative area to the left of the positive z-score: Since the total area under the curve is 1, the cumulative area to the left of the positive z-score is 1 - 0.05 = 0.95.

Use a z-table or statistical software to find the z-score: Look up the cumulative area of 0.95 in the z-table to find the corresponding z-score. This will give the positive z-score. The negative z-score will be the same value but with a negative sign.

Verify the solution: Check that the cumulative area to the left of the negative z-score is 0.05 and the cumulative area to the left of the positive z-score is 0.95. This ensures the z-scores are correct.

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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 represents the number of standard deviations a data point is from the mean of a distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. Z-scores are essential for understanding how far a value lies from the average, allowing for comparisons across different datasets.

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 a bell-shaped curve, and all z-scores correspond to specific areas under this curve. This distribution is crucial for calculating probabilities and z-scores in statistics.

Area Under the Curve

In the context of the normal distribution, the area under the curve represents the probability of a value falling within a certain range. The total area under the curve equals 1, and specific areas correspond to probabilities associated with z-scores. Understanding how to interpret these areas is vital for finding z-scores related to given probabilities.

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Related Practice

Textbook Question

Finding Area

In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z=0.33

Textbook Question

In Exercises 1–4, a population has a mean mu and a standard deviation sigma. Find the mean and standard deviation of the sampling distribution of sample means with sample size n.

Mu = 150, sigma =25, n = 49

Textbook Question

Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.

Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P91

Textbook Question

True or False? In Exercises 5–8, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

As the sample size increases, the mean of the distribution of sample means increases.

Textbook Question

"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.

You randomly select 250 U.S. men ages 18 to 29 and ask them whether they participate in at least one sport. You find that 80% say no. How likely is this result? Do you think this sample is a good one? Explain your reasoning."