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Fundamentals of Statistics | 4th Edition | ISBN: 9780321838704 | Authors: Michael Sullivan,III
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Chapter 7, Problem 13RE

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QUESTION:

According to a Gallup poll, 46% of Americans 18 years old or older stated that they had read at least six books (fiction and nonfiction) within the past year. You conduct a random sample of 250 Americans 18 years old or older.

(a) Verify that the conditions for using the normal distribution to approximate the binomial distribution are met.

(b) Approximate the probability that exactly 125 read at least six books within the past year. Interpret this result.

(c) Approximate the probability that fewer than 120 read at least six books within the past year. Interpret this result.

(d) Approximate the probability that at least 140 read at least six books within the past year. Interpret this result.

(e) Approximate the probability that between 100 and 120, inclusive, read at least six books within the past year. Interpret this result.

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QUESTION:

According to a Gallup poll, 46% of Americans 18 years old or older stated that they had read at least six books (fiction and nonfiction) within the past year. You conduct a random sample of 250 Americans 18 years old or older.

(a) Verify that the conditions for using the normal distribution to approximate the binomial distribution are met.

(b) Approximate the probability that exactly 125 read at least six books within the past year. Interpret this result.

(c) Approximate the probability that fewer than 120 read at least six books within the past year. Interpret this result.

(d) Approximate the probability that at least 140 read at least six books within the past year. Interpret this result.

(e) Approximate the probability that between 100 and 120, inclusive, read at least six books within the past year. Interpret this result.

ANSWER:

Step 1 of 5

a)

The satisfies the requirements for the approximation we need \(n p(1-p) \geq 10\)

We know that

Here a random sample of 250 Americans 18 years old or older.

n = 250.

Then, \(\mathrm{np}(1-\mathrm{p})>10=250(0.46)(1-0.46)=62.1>10\).

So, the normal distribution can be used to approximate the binomial probabilities.

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