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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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 5.1.58
Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features.
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Start by drawing a horizontal axis, which will represent the values of the variable. Label the axis with appropriate values centered around the mean (450). Include values at equal intervals, such as 350, 400, 450, 500, and 550, which are spaced by the standard deviation (50).
Draw a vertical axis perpendicular to the horizontal axis. This axis represents the probability density or frequency. Label it appropriately, though specific values are not necessary for this problem.
Sketch the bell-shaped curve of the normal distribution. The peak of the curve should be at the mean (450), and the curve should be symmetric about this point. The height of the curve decreases as you move away from the mean in both directions.
Mark the points on the horizontal axis that are one, two, and three standard deviations away from the mean. These points are 400, 350, and 300 to the left of the mean, and 500, 550, and 600 to the right of the mean. These intervals represent the empirical rule: approximately 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three.
Describe the features of the curve: it is symmetric about the mean, unimodal (one peak), and asymptotic (the tails approach but never touch the horizontal axis). The total area under the curve equals 1, representing the entire probability 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, symmetric around the mean. It is defined by two parameters: the mean (average) and the standard deviation (spread). In this case, a mean of 450 indicates the center of the distribution, while a standard deviation of 50 indicates how spread out the values are around the mean.
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Mean and Standard Deviation
The mean is the average value of a dataset, serving as the central point of the normal curve. The standard deviation measures the dispersion of data points from the mean; a larger standard deviation results in a wider curve. In this scenario, the mean of 450 and a standard deviation of 50 suggest that most data points will fall within the range of 350 to 550, encompassing approximately 68% of the data within one standard deviation.
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Features of the Normal Curve
The normal curve has several key features: it is symmetric about the mean, with the highest point at the mean, and it approaches the horizontal axis but never touches it. The area under the curve represents the total probability, which equals 1. Additionally, the empirical rule states that about 68% of data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean.
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Related Practice
Textbook Question
Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.
P(- 0.89 < z < 0)
Textbook Question
In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.
n=20, p=0.65, q=0.35
Textbook Question
In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.
P(x≥109)
a. P(x>109.5)
b. P(x<108.5)
c. P(x<109.5)
d. P(x>108.5)
Textbook Question
Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.
Find the z-score that has 2.275% of the distribution’s area to its left.
Textbook Question
In Exercises 21–24, a control chart is shown. Each chart has horizontal lines drawn at the mean mu, at mu ±2sigma, and at mu±3sigma. Determine whether the process shown is in control or out of control. Explain.
An engine part has been designed to have a diameter of 55 millimeters. The standard deviation of the process is 0.001 millimeter.