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Chapter 14: Problem 15

In a group of 10 college students, three are psychology majors. You choose three of the 10 students at random and ask their major. The distribution of the number of psychology majors you choose is a. binomial with \(n=10\) and \(p=0.3\). b. binomial with \(n=3\) and \(p=0.3\). c. not binomial. In a test for ESP (extrasensory perception), a subject is told that cards that the experimenter can see, but that the subject cannot see, contain either a star, a circle, a wave, or a square. As the experimenter looks at each of 4 cards in turn, the subject names the shape on the card. A subject who is just guessing has probability \(0.25\) of guessing correctly on each card. Questions \(14.16\) to \(\underline{14.18}\) use this information.

Short Answer

Expert verified
The student selection is not binomial; the ESP test is binomial with \( n=4\), \( p=0.25\).

Step by step solution

01

Understand the Distribution Context

We want to determine the type of distribution when selecting three students out of ten, where only three are psychology majors. Binomial distributions involve the number of successes in a fixed number of independent trials, each with the same probability of success.
02

Analyze Binomial Criteria

The binomial distribution requires that each trial be independent, have the same probability, and have two outcomes. In this scenario, the probability changes since students are not replaced, violating the condition of fixed probability for each trial.
03

Determine the Suitable Distribution

Given that the probability of choosing a psychology major changes as students are selected, this does not fit the binomial distribution's criteria of constant probability. Hence, this situation is not a binomial distribution.
04

ESP Test Distribution Analysis

In the ESP test, the subject guesses the shape on 4 cards. With an independent probability of 0.25 per guess and four trials, this fits the criteria for a binomial distribution: independent trials, fixed number of trials (4), and a constant probability of success (0.25).

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Key Concepts

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

Probability Distribution
Understanding probability distribution is key in statistics education. A probability distribution is a function or a rule that describes how likely different outcomes are to happen in a random experiment. In essence, it assigns a probability to each possible outcome of an experiment. There are different types of probability distributions, such as binomial, normal, and uniform. Each has its own set of rules regarding how probabilities are distributed among possible outcomes.

Probability distributions are crucial because they help us predict how an event might unfold over many repetitions. They provide a framework for understanding randomness and managing uncertainty in data. For instance, if you flip a coin, the probability distribution would show you that there's a 50% chance of landing on heads and a 50% chance of landing on tails. Understanding this concept aids in predicting outcomes more accurately.
Independent Trials
An important component of a binomial distribution is the concept of independent trials. Independent trials mean that the outcome of one trial does not affect the outcome of another. For example, if you toss a coin multiple times, each toss is an independent event. What happens on one toss doesn鈥檛 influence the next.

When working with binomial distribution problems, ensuring that trials are independent is essential. In statistics education, this helps students to differentiate between scenarios where outcomes are interconnected and those where they are separate. For a situation to meet the criteria for a binomial distribution, like the ESP test described in the exercise, each trial鈥攐r in this case, each guess of the card shape鈥攎ust not affect any other trial.
Constant Probability
Constant probability is another key requirement of a binomial distribution. It ensures that each trial in an experiment has the same chance of success throughout all trials. For example, when guessing randomly in an ESP test, if there is a constant probability of 0.25 for each guess to be right, then this meets one of the binomial distribution's criteria.

However, in scenarios where the probability of success changes from one trial to another, as seen in the exercise with choosing three psychology students, the constant probability condition is violated. This distinction is vital in statistics education to accurately model and understand real-world situations using the correct probability distribution model.

Constant probability assures consistency in predictions across repeated experiments, which is essential in conducting and analyzing reliable statistical experiments.
Statistics Education
Statistics education plays a pivotal role in helping students develop a foundational understanding of different statistical concepts, such as probability distribution, independent trials, and constant probability. Learning these concepts equips students to analyze data, make predictions, and solve problems in various professional fields.

By engaging with real-world examples, like those in the exercise, students better understand the application of these statistical principles. This not only aids in academic pursuits but also empowers students to make informed decisions based on data analysis.

Encouraging a deep comprehension of these topics helps foster critical thinking and analytical skills鈥攁 benefit for students of any discipline. With practical examples and interactive learning approaches, statistics education can demystify complex concepts, making them accessible and engaging for learners.

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Most popular questions from this chapter

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.

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 \(\pi\).

Is This Coin Balanced? While he was a prisoner of war during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. If the coin is perfectly balanced, the probability of a head is \(0.5\). Is there reason to think that Kerrich's coin was not balanced? To answer this question, find the probability that tossing a balanced coin 10,000 times would give a count of heads at least this far from 5000 (that is, at least 5067 heads or no more than 4933 heads).

Using Benford's Law. According to Benford's law (Example 12.7, page 281) the probability that the first digit of the amount of a randomly chosen invoice is a 1 or a 2 is \(0.477 .\) You examine 90 invoices from a vendor and find that 29 have first digits 1 or 2 . If Benford's law holds, the count of 1 s and \(2 \mathrm{~s}\) will have the binomial distribution with \(n=90\) and \(p=0.477\). Too few 1 s and 2 s suggests fraud. What is the approximate probability of 29 or fewer 1 s and \(2 s\) if the invoices follow Benford's law? Do you suspect that the invoice amounts are not genuine?

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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