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Chapter 14: Problem 22

Binomial Setting? In each of the following situations, is it reasonable to use a binomial distribution for the random variable \(X\) ? Give reasons for your answer in each case. a. An auto manufacturer chooses one car from each hour's production for a detailed quality inspection. One variable recorded is the count \(X\) of finish defects (dimples, ripples, etc.) in the car's paint. b. The pool of potential jurors for a murder case contains 100 persons chosen at random from the adult population of a large city. Each person in the pool is asked whether he or she opposes the death penalty; \(X\) is the number who say Yes. c. Joe buys a ticket in his state's Pick 3 lottery game every week; \(X\) is the number of times in a year that he wins a prize.

Short Answer

Expert verified
b. Yes, binomial setting. c. Yes, binomial setting. a. No, not a binary outcome.

Step by step solution

01

Understanding the Binomial Setting

A binomial distribution is ideal when an experiment meets four specific criteria: (1) there is a fixed number of trials, (2) each trial is binary (two possible outcomes, e.g., success/failure), (3) the probability of success remains constant across trials, and (4) the trials are independent.
02

Analyze Situation (a)

In situation (a), each hour's production results in one car being selected, but the count of finish defects is not binary; it can have more than two possible counts. Thus, it does not fit the binomial setting.
03

Analyze Situation (b)

Here, the pool consists of a fixed number of 100 jurors. The response to whether they oppose the death penalty is binary ('Yes' or 'No'), the probability can be considered constant (assuming random sampling from a large population), and each response is independent. Hence, situation (b) fits the binomial setting.
04

Analyze Situation (c)

Joe buys a lottery ticket every week for a year. This constitutes a fixed number of trials (52 weeks), with each trial having two possible outcomes (win/lose). The probability of winning can be assumed constant, and each trial is independent of others. Therefore, situation (c) also fits the binomial setting.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is a mathematical framework used to analyze random phenomena. In simple terms, it helps us predict the likelihood of different outcomes in various situations.
In the case of a binomial distribution, probability theory is especially useful. It helps us determine the chance of a certain number of successes in a fixed number of binary trials. Each trial can either result in a success or a failure, making this a classic application of probability theory.
To determine whether a situation can be modeled using a binomial distribution, we can use probability theory's principles. These include checking for independence, a consistent probability of success, and a fixed number of trials. Whenever these conditions are met, probability theory can accurately model and predict results using a binomial distribution.
Statistical Analysis
Statistical analysis involves using mathematical methods to make sense of numerical data. When dealing with random occurrences, it helps identify patterns and trends. Binomial distribution, in particular, is a tool within statistical analysis that's used for this purpose.
In statistical analysis, a binomial distribution can simplify complex random variables into easier-to-manage figures. This distribution is used to analyze how often a particular result occurs in a given number of trials.
  • Situation (b) in the exercise aptly uses statistical analysis when assessing jurors鈥� responses. The scenario meets the criteria for a binomial setting, allowing the analysis to focus on the pattern of 'Yes' responses.
  • Statistical analysis in situation (c) looks at Joe winning the lottery. The fixed number of weeks Joe buys tickets makes it possible to use statistical tools to calculate his wins.
Statistical analysis, supported by the rules of a binomial distribution, provides a clear path to interpretation and decision-making based on data.
Random Variables
A random variable is a numerical value assigned to the outcome of a random event. It quantifies the uncertainty and randomness inherent in the event. In a binomial distribution, each trial can result in one of two outcomes, making the scenario perfect for random variables.
In situation (a) from the exercise, the variable counts finish defects, which could range from zero to multiple defects. However, since the outcome isn't limited to just two possible results, it does not fit the binomial model, which requires only binary outcomes.
On the other hand, random variables in situations (b) and (c) meet binomial criteria. In (b), the random variable is the number of 'Yes' answers from jurors鈥攂eing binary (Yes/No) ensures compatibility with a binomial setting. Similarly, in (c), Joe's lottery wins are also represented as a random variable with binary outcomes, winning or not winning, aligning well with the binomial distribution. Understanding random variables is crucial as they play a fundamental role in statistical analysis and probability assessments.

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Most popular questions from this chapter

College A dmissions. A small liheral arts college in Ohio would like to have an entering class of 500 st udents next year. Past experience shows that about \(40 \%\) of the students admitted will decide to attend. The college is planning to admit 1250 students. Suppose that students make their decisions independently and that the probability is \(0.40\) that a randomly chosen student will accept the offer of admission. a. What are the mean and standard deviation of the number of students who accept the admissions offer from this college? b. Using the Normal approximation, what is the probability that the college gets more students than it wants? Check that you can safely use the approximation. c. Use software or an online binomial calculator to compute the exact probability that the college gets more students than it wants. How good is the approximation in part (b)? d. To decrease the probability of getting more students than are wanted, does the college need to increase or decrease the number of students it admits? Using software or an online binomial calculator, what is the largest number of students that the college can admit if administrators want the exact probability of getting more students than they want to be no larger than \(5 \%\) ?

Estimating \(\pi\) from Random Numbers. Kenyon College student Eric Newman used basic geometry to evaluate software random number generators as part of a summer research project. He generated 2000 independent random points \((X, Y)\) in the unit square. (That is, \(X\) and \(Y\) are independent random numbers between 0 and 1 , each having the density function illustrated in Eigure \(12.5\) (page 284). The probability that \((X, Y)\) falls in any region within the unit square is the area of the region.) 13 a. Sketch the unit square, the region of possible values for the point \((X, Y)\). b. The set of points \((X, Y)\) where \(X^{2}+Y^{2}<1\) describes a circle of radius 1. Add this circle to your sketch in part (a) and label the intersection of the two regions \(A\). c. Let \(T\) be the total number of the 2000 points that fall into the region \(A\). T follows a binomial distribution. Identify \(n\) and \(p\). (Hint: Recall that the area of a circle is \(\pi r^{2}\).) d. What are the mean and standard deviation of T? e. Explain how Eric used a random number generator and the facts given here to estimate \(\pi\).

Hot Spot. Hot Spot is a California lottery game. Players pick 1 to 10 Spots (sets of numbers, each from 1 to 80 ) that they want to play per draw. For example, if you select a 4 Spot, you play four numbers. The lottery draws 20 numbers, each from 1 to 80 . Your prize is based on how many of the numbers you picked match one of those selected by the lottery. The odds of winning depend on the number of Spots you choose to play. For example, the overall odds of winning some prize in 4 Spot is approximately \(0.256\). You decide to play the 4 Spot game and buy 5 tickets. Let \(X\) be the number of tickets that win some prize. a. \(X\) has a binomial distribution. What are \(n\) and \(p\) ? b. What are the possible values that \(X\) can take? c. Find the probability of each value of \(X\). Draw a probability histogram for the distribution of \(X\). (See Figure \(14.2\) on page \(\underline{331}\) for an example of a probability histogram.) d. What are the mean and standard deviation of this distribution? Mark the location of the mean on your histogram.

Response Rates for Random Digit Dialing. When an opinion poll uses random digit dialing to select respondents for polls, the response rate is approximately \(10 \%\) for people contacted by cell phone. You watch a pollster dial cell numbers that have been selected in this manner. \(X\) is the number of calls dialed before the pollster obtains a second response to the poll.

Larry reads that half of all super jumbo eggs contain double yolks. So he always buys super jumbo eggs and uses two whenever he cooks. If eggs do or don't contain two yolks independently of each other, the number of eggs with double yolks when Larry uses two chosen at random has the distribution a. binomial with \(n=2\) and \(p=1 / 2\). b. binomial with \(n=2\) and \(p=1 / 3\). c. binomial with \(n=3\) and \(p=1 / 2\).
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