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Section 4

The Normal Approximation to the Binomial Distribution

Explain why a normal distribution can be used as an approximation to a binomial distribution.

What conditions must be met to use the normal distribution to approximate the binomial distribution?

Why is a correction for continuity necessary?

When is the normal distribution not a good approximation for the binomial distribution?

Use the normal approximation to the binomial to find the probabilities for the specific value(s) of X.a. n = 30, p = 0.5, X = 18b. n = 50, p = 0.8, X = 44c. n = 100, p = 0.1, X = 12

Use the normal approximation to find the probabilities for the specific value(s) of X.a. $n=10, p=0.5, X \geq 7$b. $n=20, p=0.7, X \leq 12$c. $n=50, p=0.6, X \leq 40$

Check each binomial distribution to see whether it can be approximated by a normal distribution (i.e., are $n p \geq 5$ and $n q \geq 5 ?$ ).a. $n=20, p=0.5$b. $n=10, p=0.6$c. $n=40, p=0.9$

Check each binomial distribution to see whether it can be approximated by a normal distribution (i.e., are $n p \geq 5$ and $n q \geq 5 ?$ ).a. $n=50, p=0.2$b. $n=30, p=0.8$c. $n=20, p=0.85$

In a recent year, about 22% of Americans 18 years and older are single. What is the probability that in a random sample of 200 Americans 18 or older more than 30 are single?

In a recent year, the rate of U.S. home ownership was 65.9%. Choose a random sample of 120 households across the United States. What is the probability that 65 to 85 (inclusive) of them live in homes that they own?

A mail order company has an 8% success rate. If it mails advertisements to 600 people, find the probability of getting fewer than 40 sales.

Seventy-six percent of small business owners do not have a college degree. If a random sample of 60 small business owners is selected, find the probability that exactly 48 will not have a college degree.

According to the World Almanac, 72% of households own smartphones. If a random sample of 180 households is selected, what is the probability that more than 115 but fewer than 125 have a smartphone?

Twenty-two percent of work injuries are back injuries. If 400 work-injured people are selected at random, find the probability that 92 or fewer have back injuries.

College students often make up a substantial portion of the population of college cities and towns. State College, Pennsylvania, ranks first with 71.1% of its population made up of college students. What is the probability that in a random sample of 150 people from State College, more than 50 are not college students?

About 12.5% of restaurant bills are incorrect. If 200 bills are selected at random, find the probability that at least 22 will contain an error. Is this likely or unlikely to occur?

The top web browser in 2015 was Chrome with 51.74% of the market. In a random sample of 250 people, what is the probability that fewer than 110 did not use Chrome?

The percentage of female Americans 25 years old and older who have completed 4 years of college or more is 26.1. In a random sample of 200 American women who are at least 25, what is the probability that at most 50 have completed 4 years of college or more?

According to the U.S. Census, 67.5% of the U.S. population were born in their state of residence. In a random sample of 200 Americans, what is the probability that fewer than 125 were born in their state of residence?

Women comprise 80.3% of all elementary school teachers. In a random sample of 300 elementary teachers, what is the probability that less than three-fourths are women?

The mayor of a small town estimates that 35% of the residents in the town favor the construction of a municipal parking lot. If there are 350 people at a town meeting, find the probability that at least 100 favor construction of the parking lot. Based on your answer, is it likely that 100 or more people would favor the parking lot?

Recall that for use of a normal distribution as an approximation to the binomial distribution, the conditions $n p \geq 5$ and $n q \geq 5$ must be met. For each given probability, compute the minimum sample size needed for use of the normal approximation.a. p = 0.1b. p = 0.3 c. p = 0.5d. p = 0.8e. p = 0.9