Textbook Question
The random variable x is normally distributed with the given parameters. Find each probability.
b. μ = 87, σ ≈ 19, P(x > 40.5)
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.2c
The random variable x is normally distributed with the given parameters. Find each probability.
c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)
Verified step by step guidance1
Step 1: Understand the problem. The random variable x follows a normal distribution with mean (μ) = 5.5 and standard deviation (σ) ≈ 0.08. We are tasked with finding the probability that x lies between 5.36 and 5.64, i.e., P(5.36 < x < 5.64).
Step 2: Standardize the values of x to convert them into z-scores using the formula: z = (x - μ) / σ. For the lower bound (x = 5.36), calculate z₁ = (5.36 - 5.5) / 0.08. For the upper bound (x = 5.64), calculate z₂ = (5.64 - 5.5) / 0.08.
Step 3: Use the standard normal distribution table (or a calculator) to find the cumulative probabilities corresponding to z₁ and z₂. Let Φ(z) represent the cumulative probability for a given z-score. Find Φ(z₁) and Φ(z₂).
Step 4: Compute the probability P(5.36 < x < 5.64) by subtracting the cumulative probability at z₁ from the cumulative probability at z₂. This can be expressed as: P(5.36 < x < 5.64) = Φ(z₂) - Φ(z₁).
Step 5: Interpret the result. The value obtained represents the probability that the random variable x falls within the range 5.36 to 5.64 under 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, and about 95% falls within two standard deviations. This distribution is fundamental in statistics as many real-world phenomena tend to follow this pattern.
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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. To find probabilities for any normal distribution, we often convert the values to the standard normal distribution using the z-score formula: z = (x - μ) / σ. This transformation allows us to use standard normal distribution tables or software to find probabilities associated with specific ranges of values.
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Probability Calculation
Calculating probabilities for a normal distribution involves finding the area under the curve between two points. For the given parameters, P(5.36 < x < 5.64) can be determined by calculating the z-scores for both values and then using the standard normal distribution to find the corresponding probabilities. The difference between these probabilities gives the desired probability for the range specified.
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Related Practice
Textbook Question
The random variable x is normally distributed with the given parameters. Find each probability.
c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)
Textbook Question
Find each probability using the standard normal distribution.
b. P(z < 2.23)
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In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population.
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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 is the highest score that would still place a person in the bottom 10% of the scores?