Textbook Question
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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Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 5, Problem 5.CR.16c
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?
Verified step by step guidance1
Identify the problem as finding the value corresponding to the top 5% of a normal distribution. This involves finding the z-score that corresponds to the 95th percentile (since the top 5% is above this point).
Use a z-score table or statistical software to find the z-score corresponding to the 95th percentile. The z-score for the 95th percentile is approximately 1.645.
Apply the z-score formula to relate the z-score to the actual value in the distribution: \( z = \frac{x - \mu}{\sigma} \), where \( z \) is the z-score, \( x \) is the value we are solving for, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
Rearrange the formula to solve for \( x \): \( x = z \cdot \sigma + \mu \). Substitute \( z = 1.645 \), \( \mu = 44 \), and \( \sigma = 5 \) into the formula.
Perform the calculation to find \( x \), which represents the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In this context, the life spans of car batteries follow this distribution, characterized by its bell-shaped curve, where the mean is 44 months and the standard deviation is 5 months.
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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 this question, the Z-score will help determine the life expectancy threshold that corresponds to the top 5% of battery life spans.
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Percentiles
Percentiles are measures that indicate the value below which a given percentage of observations in a group of observations falls. For example, the 95th percentile indicates that 95% of the data points fall below this value. To find the shortest life expectancy in the top 5%, we need to identify the 95th percentile of the normal distribution of battery life spans.
Related Practice
Textbook Question
In a survey of U.S. adults, 81% feel they have little or no control over data collected about them by companies. You randomly select 250 U.S. adults and ask them whether they feel they have control over data collected about them by companies. Use this information in Exercises 11 and 12. (Source: Pew Research Center)
Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.
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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
A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup)
c. Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning.
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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 (b) between 8 and 11, inclusive,
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Textbook Question
Use the probability distribution in Exercise 3 to find the probability of randomly selecting a game in which DeMar DeRozan had (c) between two and four personal fouls, inclusive.
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