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Chapter 4: Q49E (page 239)

The next presidential election. A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters are randomly selected from the population of all eligible voters.

a. Analyze the potential outcomes for the sample using the binomial random variable.

b. Use a binomial probability table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential election.

c. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election.

Short Answer

Expert verified
  1. A binomial random variable can have any number between 0 as well as n.
  2. PX猢�12=0.772
  3. P16猢�X猢�20=0.714

Step by step solution

01

Given information

70% of all eligible voters will vote in the next presidential election

n=20

02

(a)Finding potential outcomes

The experiment comprises of n similar as well as independent trials, with n predetermined. Every trial can have one of two results: success (S) as well as failures (F), with the likelihood of succeeding (p) being equal from trial to trial. A binomial random variable can have any number between 0 as well as n.

03

(b) Finding the probability that more than 12 of the eligible voters sampled will vote in the next presidential election

PX猢�12=PX=12+PX=13+...+PX=20=鈭�i=122020ipi1-p20-i=0.772

Thus, the required probability is 0.772.

04

(c) Finding the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election

P16猢�X猢�20=PX=16+PX=17+...+PX=20=鈭�i=162020ipi1-p20-i=0.714

Thus, the required probability is 0.714.

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Most popular questions from this chapter

Tax returns audited by the IRS. According to the Internal Revenue Service (IRS), the chances of your tax return being audited are about 1 in 100 if your income is less than \(1 million and 9 in 100 if your income is \)1 million or more (IRS Enforcement and Services Statistics).

  1. What is the probability that a taxpayer with income less than \(1 million will be audited by the IRS with income \)1 million or more?
  2. If five taxpayers with incomes under \(1 million are randomly selected, what is the probability that exactly one will be audited? That more than one will be audited?
  3. Repeat part b, assuming that five taxpayers with incomes of \)1 million or more are randomly selected.
  4. If two taxpayers with incomes under \(1 million are randomly selected, and two with incomes more than \)1 million are randomly selected, what is the probability that none of these taxpayers will be audited by the IRS?
  5. What assumptions did you have to make in order to answer these questions using the methodology presented in this section?

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USGA golf ball specifications. According to the US. Golf Association (USGA), "The diameter of the [golf] ball must not be less than 1.680 inches" (USGA, 2016). The USGA periodically checks the specifications of golf balls by randomly sampling balls from pro shops around the country. Two dozen of each kind are sampled, and if more than three do not meet requirements, that kind of ball is removed from the USGA's approved-ball list.

  1. What assumptions must be made and what information must be known in order to use the binomial probability distribution to calculate the probability that the USGA will remove a particular kind of golf ball from its approved-ball list?
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  3. Refer to part b. If x has a binomial distribution, then so does the number, y, of balls in the sample that meet the USGA's minimum diameter. [Note: x + y = 24] Describe the distribution of y. In particular, what are p, q and n? Also, find E(y) and the standard deviation of y.
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