Textbook Question
In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.
0.1
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.R.64
In Exercises 63–68, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.
P(x ≤ 36)
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Step 1: Interpret the binomial probability in words. The problem asks for the probability of obtaining 36 or fewer successes in a binomial experiment.
Step 2: Recall that a binomial distribution can be approximated by a normal distribution when the sample size is large and both np and n(1-p) are greater than or equal to 5. Verify these conditions for the given problem.
Step 3: Identify the mean (μ) and standard deviation (σ) of the binomial distribution. Use the formulas μ = n * p and σ = sqrt(n * p * (1 - p)), where n is the number of trials and p is the probability of success.
Step 4: Apply the continuity correction. Since the binomial probability is P(x ≤ 36), convert it to the normal distribution probability P(x ≤ 36.5) to account for the discrete-to-continuous adjustment.
Step 5: Standardize the value 36.5 using the z-score formula z = (x - μ) / σ, where x is the value of interest, μ is the mean, and σ is the standard deviation. Then, use the standard normal distribution table or a calculator to find the corresponding probability.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Probability
Binomial probability refers to the likelihood of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is calculated using the binomial formula, which incorporates the number of trials, the number of successes, and the probability of success on each trial. This concept is essential for understanding discrete outcomes in scenarios like coin flips or quality control.
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Calculating Probabilities in a Binomial Distribution
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is significant in statistics because many phenomena tend to follow this distribution due to the Central Limit Theorem, which states that the sum of a large number of independent random variables will approximate a normal distribution, regardless of the original distribution.
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Continuity Correction
Continuity correction is a technique used when approximating a discrete probability distribution, like the binomial distribution, with a continuous distribution, such as the normal distribution. This correction involves adjusting the discrete value by 0.5 in either direction to better align the probabilities. For example, when calculating P(x ≤ 36) in a binomial context, one would use P(x ≤ 36.5) in the normal approximation to account for the discrete nature of the binomial variable.
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Related Practice
Textbook Question
In Exercises 55–60, find the indicated probabilities and interpret the results.
Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (a) less than 5500 MMT CO2 eq. Compare your answers with those in Exercise 33.
Textbook Question
In Exercises 55–60, find the indicated probabilities and interpret the results.
The mean annual salary for physical therapists in the United States is about \$87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (a) less than \(84,000? Assume sigma = \)10,500.
Textbook Question
An auto parts seller finds that 1 in every 200 parts sold is defective. Use the geometric distribution to find the probability that (c) none of the first 20 parts sold are defective.
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Textbook Question
Find the positive z-score for which 94% of the distribution’s area lies between -z and z.
Textbook Question
In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.
P(x < 55)