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 5.3.18

Chapter 5, Problem 5.3.18

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

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Identify the given information: The graph shows a standard normal distribution with a shaded area of 0.5987 to the left of the unknown z-score. The mean of the standard normal distribution is 0, and the standard deviation is 1.

Understand the relationship between the area under the curve and the z-score: The area to the left of a z-score in the standard normal distribution corresponds to the cumulative probability for that z-score.

Use a z-table or statistical software: Look up the cumulative probability of 0.5987 in a z-table (or use statistical software) to find the z-score that corresponds to this cumulative area.

Interpret the z-table: Locate the value 0.5987 in the body of the z-table. The row and column headers corresponding to this value will give the z-score. If using software, input the cumulative probability to directly obtain the z-score.

Verify the result: Ensure that the z-score found matches the given area (0.5987) by checking the cumulative probability for the calculated z-score in the z-table or software.

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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 a particular value compares to the overall distribution, especially in the context of normal distributions.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). Many statistical methods assume normality, making it crucial for interpreting z-scores and areas under the curve.

Area Under the Curve

In the context of a normal distribution, the area under the curve represents the probability of a random variable 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 calculate and interpret these areas is vital for statistical analysis and hypothesis testing.

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Related Practice

Textbook Question

Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results.

Asthma Prevalence by State The mean percent of asthma prevalence of the 50 U.S. states is 9.51%. A random sample of 30 states is selected. What is the probability that the mean percent of asthma prevalence for the sample is greater than 10%? Assume sigma=1.17%

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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.

Between z=0 and z=2.86

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Textbook Question

Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=36, find the probability of a sample mean being less than 12,750 or greater than 12,753 when mu=12750 and 1.7.

105

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Textbook Question

Graphical Analysis In Exercises 9 and 10, the graph of a population distribution is shown with its mean and standard deviation. Random samples of size 100 are drawn from the population. Determine which of the figures labeled (a)–(c) would most closely resemble the sampling distribution of sample means. Explain your reasoning.

The waiting time (in seconds) to turn left at an intersection

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Textbook Question

Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean mu=174 and standard deviation sigma=20. Find the indicated probability.

P(172 < x <192)

132

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Textbook Question

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(- 1.54 < z < 1.54)

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