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.
To the left of z= -1.28 and to the right of z= 1.28
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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 5.1.20
Using and Interpreting Concepts
Finding Area In Exercises 17–22, find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area.
Verified step by step guidance1
Step 1: Understand the problem. The shaded region under the standard normal curve represents the area between z = -0.9 and z = 0. This area corresponds to the probability of a standard normal random variable falling within this range.
Step 2: Recall that the standard normal curve is symmetric about z = 0, and the total area under the curve is 1. The area under the curve between two z-values can be found using the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the standard normal table or technology (such as a graphing calculator or statistical software) to find the cumulative probability for z = -0.9 and z = 0. The cumulative probability for a z-value represents the area under the curve to the left of that z-value.
Step 4: To find the area of the shaded region, subtract the cumulative probability at z = -0.9 from the cumulative probability at z = 0. This gives the area between these two z-values.
Step 5: If using technology, input the z-values into the software or calculator to directly compute the area. Alternatively, use the standard normal table to look up the cumulative probabilities and perform the subtraction manually.
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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 and areas under the curve.
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Area Under the Curve
In statistics, the area under the curve (AUC) of a probability distribution represents the probability of a random variable falling within a certain range. For the standard normal distribution, this area can be found using Z-scores and standard normal tables or technology. The shaded region in the graph indicates the probability associated with the Z-scores between -0.9 and 0.
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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 crucial for standardizing scores across different distributions, allowing for comparison and the calculation of probabilities using the standard normal distribution.
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Related Practice
Textbook Question
Which Is More Likely? Assume that the fertility rates in Exercise 32 are normally distributed. Are you more likely to randomly select a state with a fertility rate of less than 65 or to randomly select a sample of 15 states in which the mean of the state fertility rates is less than 65? Explain.
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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=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.
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Textbook Question
Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.
0.6736
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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 = 45, sigma =15, n = 100
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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 > 65)
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