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
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.Q.4a
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)
Verified step by step guidance1
Step 1: Understand the problem. The random variable x is normally distributed with a mean (μ) of 9.2 and a standard deviation (σ) of approximately 1.62. We are tasked with finding the probability P(x < 5.97).
Step 2: Standardize the value of x = 5.97 using the z-score formula: z = (x - μ) / σ. Substitute the given values into the formula: z = (5.97 - 9.2) / 1.62.
Step 3: Simplify the numerator (5.97 - 9.2) and then divide by the standard deviation (1.62) to calculate the z-score. This will give you the standardized value corresponding to x = 5.97.
Step 4: Use a standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to the calculated z-score. This cumulative probability represents P(x < 5.97).
Step 5: Interpret the result. The cumulative probability obtained from the z-table or software is the probability that the random variable x is less than 5.97 in the given normal distribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean (μ) and standard deviation (σ). It is symmetric around the mean, meaning that approximately 68% of the data falls within one standard deviation from the mean. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.
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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. Z-scores are essential for finding probabilities in a normal distribution, as they allow us to convert any normal random variable into a standard normal variable.
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Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes on a value less than or equal to a specific value. For a normally distributed variable, the CDF can be used to find probabilities associated with specific ranges of values, such as P(x < 5.97) in this case. The CDF is crucial for determining probabilities in statistical analysis.
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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.
Textbook Question
Find each probability using the standard normal distribution.
b. P(z < 2.23)
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
The random variable x is normally distributed with the given parameters. Find each probability.
d. μ = 18.5, σ ≈ 4.25, P(19.6 < x < 26.1)
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?