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Chapter 5: Q12 (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.

Investigating Dates In a survey sponsored by TGI Friday鈥檚, 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 鈥測es鈥� or 鈥渘o.鈥�

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: 鈥測es鈥� or 鈥渘o.鈥�

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 鈥渋f they investigate dates on social media before meeting them.鈥� The response is either 鈥測es鈥� or 鈥渘o.鈥�

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: 鈥測es鈥� or 鈥渘o.鈥�

  • 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

Using the same SAT questions described in Exercise 2, is 20 a significantly high number of correct answers for someone making random guesses?

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.

Surveying Senators The Senate members of the 113th Congress include 80 males and 20 females. Forty different senators are randomly selected without replacement, and the gender of each selected senator is recorded.

In Exercises 21鈥�25, refer to the accompanyingtable, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from 鈥淧revalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,鈥� by Ohayon et al., Neurology, Vol. 78, No. 20).

Use the range rule of thumb to determine whether 4 is a significantly high number of sleepwalkersin a group of 5 adults.

x

P(x)

0

0.172

1

0.363

2

0.306

3

0.129

4

0.027

5

0.002

In Exercises 7鈥�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.

A sociologist randomly selects single adults for different groups of three, and the random variable xis the number in the group who say that the most fun way to flirt is in person(based on a Microsoft Instant Messaging survey).

x

P(x)

0

0.091

1

0.334

2

0.408

3

0.166

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.

M&Ms Data Set 27 鈥淢&M Weights鈥� in Appendix B includes data from 100 M&M candies, and 19 of them are green. Mars, Inc. claims that 16% of its plain M&M candies are green. For the following, assume that the claim of 16% is true, and assume that a sample consists of 100 M&Ms.

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 19 green M&Ms significantly high?

b. Find the probability of exactly 19 green M&Ms.

c. Find the probability of 19 or more green M&Ms.

d. Which probability is relevant for determining whether the result of 19 green M&Ms is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 19 green M&Ms significantly high?

e. What do the results suggest about the 16% claim by Mars, Inc.?
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