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Chapter 5: Q11 (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.

Credit Card Survey In an AARP Bulletin survey, 1019 different adults were randomly selected without replacement. Respondents were asked if they have one or more credit cards, and responses were recorded as 鈥測es鈥� and 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 鈥淚f you have one or more credit cards,鈥� has precisely two possible 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, and, thus, the selections can be considered as independent.

All other assumptions are met by the procedure.

Step by step solution

01

Given information

A sample of 1019 adults is surveyed and is made to answer the question 鈥渋f they have one or more credit cards.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 1019.

  • Second assumption:

It is given that 1019 adults were 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 1019 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.

  • Fourth assumption:

Since all the 1019 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

Significance with Range Rule of Thumb. In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

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a. Find the mean and standard deviation for the numbers of girls in groups of 16 births.

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