Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.5.17

Chapter 5, Problem 5.5.17

Approximating a Binomial Distribution In Exercises 17 and 18, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Bachelor’s Degrees Twenty-two percent of adults over 18 years of age have a bachelor’s degree. You randomly select 20 adults over 18 years of age and ask whether they have a bachelor’s degree.

Verified step by step guidance

Step 1: Determine if the normal approximation to the binomial distribution can be used. The rule of thumb is that the normal approximation is appropriate if both np ≥ 5 and n(1-p) ≥ 5, where n is the number of trials and p is the probability of success.

Step 2: Calculate np, where n = 20 (number of trials) and p = 0.22 (probability of success). Use the formula np = n × p.

Step 3: Calculate n(1-p), where n = 20 and p = 0.22. Use the formula n(1-p) = n × (1-p).

Step 4: Check the conditions np ≥ 5 and n(1-p) ≥ 5. If both conditions are satisfied, the normal approximation can be used. If not, explain why the approximation is not valid.

Step 5: If the normal approximation is valid, calculate the mean (μ) and standard deviation (σ) of the binomial distribution. Use the formulas μ = n × p and σ = √(n × p × (1-p)).

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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: n (the number of trials) and p (the probability of success on each trial). In this context, the success is defined as an adult having a bachelor's degree.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution can be used when certain conditions are met, specifically when both np and n(1-p) are greater than or equal to 5. This allows for the use of the normal distribution to estimate probabilities and calculate the mean and standard deviation, simplifying analysis for larger sample sizes.

Mean and Standard Deviation of a Binomial Distribution

The mean (μ) of a binomial distribution is calculated as μ = np, while the standard deviation (σ) is given by σ = √(np(1-p)). These formulas provide essential measures of central tendency and variability, which are crucial for understanding the distribution of successes in the context of the given problem.

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Mean & Standard Deviation of Binomial Distribution

Related Practice

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=1.365

Textbook Question

"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.

You randomly select 300 U.S. women ages 18 to 29 and ask them whether they participate in at least one sport. Of the 72 who say yes, 50% say they participate in volleyball. How likely is this result? Do you think this sample is a good one? Explain your reasoning."

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=18, p=0.90, q=0.10

Textbook Question

Paint Cans A machine is set to fill paint cans with a mean of 128 ounces and a standard deviation of 0.2 ounce. A random sample of 40 cans has a mean of 127.9 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain.

Textbook Question

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the z-score that has 78.5% of the distribution’s area to its left.

Textbook Question

Bags of Baby Carrots The weights of bags of baby carrots are normally distributed, with a mean of 32 ounces and a standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the most a bag of baby carrots can weigh and not need to be repackaged?