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Chapter 5: Q8 (page 209)

Identifying Binomial Distributions. In Exercises 5鈥�12, determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are not binomial, identify at least one requirement that is not satisfied.

LOL In a U.S. Cellular survey of 500 smartphone users, subjects are asked if they find abbreviations (such as LOL or BFF) annoying, and each response was recorded as 鈥測es,鈥� 鈥渘o,鈥� or 鈥渘ot sure.鈥�

Short Answer

Expert verified

The given situation cannot be approximated using the binomial distribution as the question asked has more than two possible outcomes (鈥測es,鈥濃�渘o,鈥漚nd 鈥渘ot sure鈥�).

Step by step solution

01

Given information

The question asked in the survey is, 鈥淲hether the subjects find abbreviations annoying or not?鈥�.

Out of 500 smartphone users who were surveyed, the response to a question had more than two possible outcomes: 鈥測es,鈥� 鈥渘o,鈥� and 鈥渘ot sure.鈥�

02

Assumptions of binomial distribution

The following assumptions of the binomial distribution should be satisfied:

  • Finite and independent trials
  • Two possible outcomes (success and failure) for each trial
  • The probability of success should be the same for each of the trials
03

Violation of an assumption of binomial distribution

Out of the given set of assumptions required for a procedure to follow the binomial distribution,one such assumption is that the outcomes of the event whose probability is to be computed should be of exactly two kinds:

  • One outcome is considered as the success;
  • The remaining outcome is considered afailure.

Here, the answer to the question of whether the subject finds abbreviations annoying or not has more than two possible outcomes (鈥測es,鈥濃�渘o,鈥� and 鈥渘ot sure鈥�).

Therefore, the given situation cannot be modeled using the binomial distribution.

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Most popular questions from this chapter

In Exercises 15鈥�20, refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children.

Using Probabilities for Significant Events

a. Find the probability of getting exactly 6 girls in 8 births.

b. Find the probability of getting 6 or more girls in 8 births.

c. Which probability is relevant for determining whether 6 is a significantly high number of girls in 8 births: the result from part (a) or part (b)?

d. Is 6 a significantly high number of girls in 8 births? Why or why not?

Number of girls x

P(x)

0

0.004

1

0.031

2

0.109

3

0.219

4

0.273

5

0.219

6

0.109

7

0.031

8

0.004

Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

a. For a randomly selected subject, find the probability of a bone density test score less than 1.54.

b. For a randomly selected subject, find the probability of a bone density test score greater than -1.54.

c. For a randomly selected subject, find the probability of a bone density test score between -1.33 and 2.33.

d. Find \({Q_1}\), the bone density test score separating the bottom 25% from the top 75%.

e. If the mean bone density test score is found for 9 randomly selected subjects, find the probability that the mean is greater than 0.50.

In Exercises 25鈥�28, find the probabilities and answer the questions.

See You Later Based on a Harris Interactive poll, 20% of adults believe in reincarnation. Assume that six adults are randomly selected, and find the indicated probability.

a. What is the probability that exactly five of the selected adults believe in reincarnation?

b. What is the probability that all of the selected adults believe in reincarnation?

c. What is the probability that at least five of the selected adults believe in reincarnation?

d. If six adults are randomly selected, is five a significantly high number who believe in reincarnation?

Ultimate Binomial Exercises! Exercises 37鈥�40 involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities.

Hybrids One of Mendel鈥檚 famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel鈥檚 claim of 25% is true, and assume that a sample consists of 580 offspring peas. a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 152 yellow peas either significantly low or significantly high?

b. Find the probability of exactly 152 yellow peas.

c. Find the probability of 152 or more yellow peas.

d. Which probability is relevant for determining whether 152 peas is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 152 yellow peas significantly high?

e. What do the results suggest about Mendel鈥檚 claim of 25%?

Expected Value in Roulette When playing roulette at the Venetian casino in Las Vegas, a gambler is trying to decide whether to bet \(5 on the number 27 or to bet \)5 that the outcome is any one of these five possibilities: 0, 00, 1, 2, 3. From Example 6, we know that the expected value of the \(5 bet for a single number is -26垄. For the \)5 bet that the outcome is 0, 00, 1, 2, or 3, there is a probability of 5/38 of making a net profit of \(30 and a 33/38 probability of losing \)5.

a. Find the expected value for the \(5 bet that the outcome is 0, 00, 1, 2, or 3.

b. Which bet is better: a \)5 bet on the number 27 or a $5 bet that the outcome is any one of the numbers 0, 00, 1, 2, or 3? Why?
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