Section 6.6Exercises

CLARIFYING THE CONCEPTS

Question 6.394

1. Provide an example of why we would need to use the normal approximation to the binomial distribution. (p. 385)

6.6.1

For certain values of and , it may be inconvenient to calculate probabilities for the binomial distribution. For example, if and , it may be tedious to calculate , which, in the absence of technology, would involve 43 applications of the binomial probability formula.

Question 6.395

2. What are the requirements for using the normal approximation to the binomial distribution? (p. 386)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 3–8 Example 44 Requirements for a
normal approximation to
the binomial distribution
Exercises 9–24 Example 45 The normal
approximation to the
binomial distribution

For Exercises 3–8, determine whether the requirements are met for using the normal approximation to the binomial probability distribution.

Question 6.396

3. is a binomial random variable with and

6.6.3

Met

Question 6.397

4. is a binomial random variable with and

Question 6.398

5. is a binomial random variable with and

6.6.5

Unmet

Question 6.399

6. is a binomial random variable with and

Question 6.400

7. is a binomial random variable with and

6.6.7

Unmet

Question 6.401

8. is a binomial random variable with and

For Exercises 9–16, let be a binomial random variable with and . Use the normal approximation to find the following probabilities:

Question 6.402

9.

6.6.9

0.1272

Question 6.403

10.

Question 6.404

11.

6.6.11

0.4364

Question 6.405

12.

Question 6.406

13.

6.6.13

0.4364

Question 6.407

14.

Question 6.408

15.

6.6.15

0.3616

Question 6.409

16.

For Exercises 17–24, let be a binomial random variable with and . Use the normal approximation to find the following probabilities:

Question 6.410

17.

6.6.17

0.7764

Question 6.411

18.

Question 6.412

19.

6.6.19

0.6772

Question 6.413

20.

Question 6.414

21.

6.6.21

0.0853

Question 6.415

22.

Question 6.416

23.

6.6.23

0.0992

Question 6.417

24.

APPLYING THE CONCEPTS

Question 6.418

25. e-Cigarettes. A 2014 study17 found that 6.5% of adolescents have tried e-cigarettes. (Though e-cigarettes have been marketed as cessation helpers, the study found that use of e-cigarettes does not discourage, and may encourage, conventional cigarette use among adolescents.) For a sample of 1000 U.S. adolescents, approximate the following probabilities:

  1. More than 65 have tried e-cigarettes.
  2. At least 65 have tried e-cigarettes.
  3. Less than 65 have tried e-cigarettes.
  4. Between 60 and 70 (inclusive) have tried e-cigarettes.

6.6.25

(a) 0.4761 (b) 0.5239 (c) 0.4761 (d) 0.5223

Question 6.419

26. Make Mine Medium. fivethirtyeight.com reported that 31% of those surveyed like their steaks cooked medium.18 For a sample of 200 people, approximate the following probabilities:

  1. At least 70 people like their steaks cooked medium.
  2. More than 69 people like their steaks cooked medium.
  3. At most 69 people like their steaks cooked medium.
  4. Between 70 and 75 (inclusive) like their steaks cooked medium.

Question 6.420

27. Hurricane Response. A survey found that 19% of respondents in New Orleans rated the overall response by government and volunteer agencies to major hurricanes in the past three years as good or excellent, whereas 57% of those living in other areas did so.19 Suppose that we have a sample of 100 people living in New Orleans and 100 people living in other areas. Approximate the following probabilities:

390

  1. At least 30 of the respondents living in New Orleans rated the response as good or excellent.
  2. At least 30 of the respondents living in other areas rated the response as good or excellent.
  3. Fewer than 20 of the respondents living in New Orleans rated the response as good or excellent.
  4. Fewer than 20 of the respondents living in other areas rated the response as good or excellent.

6.6.27

(a) 0.0037 (b) 1 (c) 0.5517 (d) 0

Question 6.421

28. Disease Outbreak. A survey found that only 9% of Americans were “very confident” that the U.S. government is prepared to handle a major outbreak of an infectious disease.20 Suppose that we have a sample of 100 Americans. Approximate the following probabilities:

  1. Exactly 9 Americans are very confident.
  2. At least 9 Americans are very confident.
  3. More than 9 Americans are very confident.
  4. At most, 9 Americans are very confident.
  5. Fewer than 9 Americans are very confident.

Use the Normal Approximation to the Binomial Distributions applet for Exercise 29.

Question 6.422

29. Select . The rectangles represent the binomial probabilities and the area under the curve represents the normal probabilities.

  1. For and , is there a tight fit between the rectangles and the curve?
  2. What does this mean for whether the normal approximation should be used for a binomial distribution with and ?
  3. Verify whether the conditions are met for applying the normal approximation.

6.6.29

(a) No (b) The normal distribution is not a good approximation to the binomial distribution, so not appropriate. (c) , so the conditions are not met.