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Chapter 5: Q12 (page 209)

Identifying Binomial Distributions. In Exercises 5–12, 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.

Investigating Dates In a survey sponsored by TGI Friday’s, 1000 different adult respondents were randomly selected without replacement, and each was asked if they investigate dates on social media before meeting them. Responses consist of “yes” or “no.”

Short Answer

Expert verified

The given situation can be approximated using the binomial distribution as the outcome of the response to the question has precisely twopossible outcomes: “yes” or “no.”

Although the selections were made without replacement, the given sample follows the 5% rule as the sample size is no more than 5% of the population size. Thus, the selections can be considered independent.

All the remaining assumptions are satisfied.

Step by step solution

01

Given information

A sample of 1000 adults is surveyed and is made to answer the question “if they investigate dates on social media before meeting them.” The response is either “yes” or “no.”

02

Assumptions of binomial distribution

The following assumptions of the binomial distribution should be satisfied:

  • The procedure should have a fixed number of trials.
  • The trials should be independent.
  • Each trial should have outcomes that are of exactly two kinds: success and failure.
  • The probability of success should be the same for all the trials.
03

Examination of assumptions of binomial distribution

  • First assumption:

The number of trials is fixed and holds a value equal to 1000.

  • Second assumption:

It is given that 1000adults are selected for a survey without replacement.

Thus, they cannot be considered independent unless they fulfill the 5% rule of cumbersome calculations, which says that the sample size should be no more than 5% of the population size.

It is given that the population of adults is considered.

Therefore, it can be safely said that a sample of 1000 adults is no more than 5% of the population of all adults.

Since the sample size is less than 5% of the population size, the selections can be considered independent.

  • Third assumption:

The outcomes of the event whose probability is to be estimated must be of exactly two types.One of the outcomes is regarded as a success, while the other is considered a failure.

Here, the response to the question has exactly twopossible outcomes: “yes” or “no.”

  • Fourth assumption:

Since all the 1000 adults have answered the same question, the probability of success for all trials is the same.

Since all the assumptions are met, the given situation can be modeled using the binomial distribution.

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Most popular questions from this chapter

Identifying Binomial Distributions. In Exercises 5–12, 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.

Clinical Trial of YSORT The YSORT method of gender selection, developed by the Genetics & IVF Institute, was designed to increase the likelihood that a baby will be a boy. When 291 couples use the YSORT method and give birth to 291 babies, the genders of the babies are recorded.

Critical Thinking: Did Mendel’s results from plant hybridization experiments contradict his theory? Gregor Mendel conducted original experiments to study the genetic traits of pea plants. In 1865 he wrote “Experiments in Plant Hybridization,” which was published in Proceedings of the Natural History Society. Mendel presented a theory that when there are two inheritable traits, one of them will be dominant and the other will be recessive. Each parent contributes one gene to an offspring and, depending on the combination of genes, that offspring could inherit the dominant trait or the recessive trait. Mendel conducted an experiment using pea plants. The pods of pea plants can be green or yellow. When one pea carrying a dominant green gene and a recessive yellow gene is crossed with another pea carrying the same green>yellow genes, the offspring can inherit any one of these four combinations of genes: (1) green/green; (2) green/yellow; (3) yellow/green; (4) yellow/yellow. Because green is dominant and yellow is recessive, the offspring pod will be green if either of the two inherited genes is green. The offspring can have a yellow pod only if it inherits the yellow gene from each of the two parents. Given these conditions, we expect that 3/4 of the o§spring peas should have green pods; that is, P(green pod) = 3/4. When Mendel conducted his famous hybridization experiments using parent pea plants with the green/yellow combination of genes, he obtained 580 offspring. According to Mendel’s theory, 3/4 of the offspring should have green pods, but the actual number of plants with green pods was 428. So the proportion of offspring with green pods to the total number of offspring is 428/580 = 0.738. Mendel expected a proportion of 3/4 or 0.75, but his actual result is a proportion of 0.738.

a. Assuming that P(green pod) = 3/4, find the probability that among 580 offspring, the number of peas with green pods is exactly 428.

b. Assuming that P(green pod) = 3/4, find the probability that among 580 offspring, the number of peas with green pods is 428 or fewer.

c. Which of the two preceding probabilities should be used for determining whether 428 is a significantly low number of peas with green pods?

d. Use probabilities to determine whether 428 peas with green pods is a significantly low number. (Hint: See “Identifying Significant Results with Probabilities” in Section 5-1.)

Are the values found in Exercise 2 statistics or parameters? Why?

Significance with Range Rule of Thumb. In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).

Hybrids Assume that offspring peas are randomly selected in groups of 10.

a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 10.

b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

c. Is the result of 9 peas with green pods a result that is significantly high? Why or why not?

In Exercises 7–14, determine whether a probability

distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

When betting on the pass line in the dice game of craps at the Mohegan Sun casino in Connecticut, the table lists the probabilities for the number of bets that must be placed in order to have a win.

x

P(x)

1

0.493

2

0.25

3

0.127

4

0.064
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