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
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 5.5.4
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
Verified step by step guidance1
Step 1: Recall the rule for using a normal distribution to approximate a binomial distribution. The approximation is valid if both np ≥ 5 and nq ≥ 5, where n is the sample size, p is the probability of success, and q is the probability of failure.
Step 2: Calculate np by multiplying the sample size n by the probability of success p. Use the formula: np = n × p.
Step 3: Calculate nq by multiplying the sample size n by the probability of failure q. Use the formula: nq = n × q.
Step 4: Check whether both conditions np ≥ 5 and nq ≥ 5 are satisfied. If both conditions are met, the normal distribution can be used as an approximation.
Step 5: Conclude whether the normal distribution is a valid approximation based on the results of the calculations in Steps 2 and 3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
A binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p). In this context, the distribution of successes can be approximated by a normal distribution under certain conditions.
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03:28
Mean & Standard Deviation of Binomial Distribution
Normal Approximation
The normal approximation to the binomial distribution is applicable when both np and nq are greater than or equal to 5. This rule of thumb ensures that the binomial distribution is sufficiently symmetric and bell-shaped, allowing for the use of normal distribution techniques to estimate probabilities and make inferences about the data.
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Conditions for Normal Approximation
To determine if a normal distribution can be used to approximate a binomial distribution, check the conditions np ≥ 5 and nq ≥ 5. In this case, n is the sample size, p is the probability of success, and q is the probability of failure. If both conditions are satisfied, the normal approximation is valid, simplifying calculations and analyses.
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06:23Using the Normal Distribution to Approximate Binomial Probabilities
Related Practice
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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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
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.
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.