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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

In Exercises 6鈥�10, use the following: Five American Airlines flights are randomly selected, and the table in the margin lists the probabilities for the number that arrive on time (based on data from the Department of Transportation). Assume that five flights are randomly selected.

Find the mean of the number of flights among five that arrive on time.

x

P(x)

0

0+

1

0.006

2

0.051

3

0.205

4

0.409

5

0.328

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�

Acceptance Sampling. Exercises 35 and 36 involve the method of acceptance sampling, whereby a shipment of a large number of items is accepted based on test results from a sample of the items.

Aspirin The MedAssist Pharmaceutical Company receives large shipments of aspirin tablets and uses this acceptance sampling plan: Randomly select and test 40 tablets, then accept the whole batch if there is only one or none that doesn鈥檛 meet the required specifications. If one shipment of 5000 aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?

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.

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning line positions on voting ballots. Among 41 different ballots, Democrats were assigned the top line 40 times. Assume that Democrats and Republicans are assigned the top line using a method of random selection so that they are equally likely to get that top line.

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 40 top lines for Democrats significantly high?

b. Find the probability of exactly 40 top lines for Democrats

c. Find the probability of 40 or more top lines for Democrats.

d. Which probability is relevant for determining whether 40 top lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 top lines for Democrats significantly high?

e. What do the results suggest about how the clerk met the requirement of assigning the line positions using a random method?

In Exercises 21鈥�24, assume that when adults with smartphones are randomly selected, 54% use them in meetings or classes (based on data from an LG Smartphone survey).

If 20 adult smartphone users are randomly selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.
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