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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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 5.1.54
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)
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z lies between -1.54 and 1.54. This involves using the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
Step 2: Recall that the standard normal distribution table (z-table) provides the cumulative probability P(Z ≤ z) for a given z-value. For this problem, you will need to find the cumulative probabilities for z = 1.54 and z = -1.54.
Step 3: Use the symmetry property of the standard normal distribution. Since the distribution is symmetric about z = 0, the cumulative probability for z = -1.54 is equal to 1 minus the cumulative probability for z = 1.54.
Step 4: Calculate the probability P(-1.54 < z < 1.54) by subtracting the cumulative probability for z = -1.54 from the cumulative probability for z = 1.54. Mathematically, this is expressed as P(-1.54 < z < 1.54) = P(Z ≤ 1.54) - P(Z ≤ -1.54).
Step 5: If using technology (e.g., a calculator or statistical software), input the z-values directly to find the cumulative probabilities and subtract them as described in Step 4. Alternatively, use a z-table to look up the cumulative probabilities and perform the subtraction.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it essential for calculating probabilities related to normally distributed data.
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Z-scores
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 allow for the comparison of scores from different normal distributions and are crucial for finding probabilities in the standard normal distribution.
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Probability Calculation
Probability calculation in the context of the standard normal distribution involves finding the area under the curve between two z-scores. This area represents the probability of a random variable falling within that range. Tools such as z-tables or statistical software can be used to determine these probabilities efficiently, especially for non-standard values.
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Related Practice
Textbook Question
Graphical Analysis In Exercises 17–22, find the indicated z-score(s) shown in the graph.
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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
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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)
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Textbook Question
Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.
Renewable Energy During a recent period of two years, the day-ahead prices for renewable energy in Germany (in euros per mega-watt hour) have a mean of 31.58 and a standard deviation of 12.293. Random samples of size 75 are drawn from this population, and the mean of each sample is determined.
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Textbook Question
In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.
P(x ≤ 150)
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