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Chapter 3: Problem 36

a. A meteorologist in Denver recorded \(Y=\) the number of days of rain during a 30 -day period. Does \(Y\) have a binomial distribution? If so, are the values of both \(n\) and \(p\) given? b. A market research firm has hired operators who conduct telephone surveys. A computer is used to randomly dial a telephone number, and the operator asks the answering person whether she has time to answer some questions. Let \(Y=\) the number of calls made until the first person replies that she is willing to answer the questions. Is this a binomial experiment? Explain.

Short Answer

Expert verified
(a) Yes, \( Y \) is binomial, but values of \( n \) and \( p \) are not given. (b) No, this is a geometric distribution. Not binomial.

Step by step solution

01

Define Criteria for a Binomial Distribution

A random variable follows a binomial distribution if it satisfies the following conditions: 1) There are a fixed number of trials, 2) Each trial is independent, 3) There are only two possible outcomes for each trial, and 4) The probability of success is constant in each trial.
02

Examine Part (a): Meteorologist's Recording

For the meteorologist recording the number of rainy days, each day can be seen as a trial with two outcomes: it either rains or it doesn't. There are 30 days observed, making it a fixed number of trials. Assuming each day is independent and the probability of rain is consistent, it meets the binomial criteria. However, the values of neither \( n \) (number of trials) nor \( p \) (probability of rain) are explicitly given.
03

Examine Part (b): Telephone Surveys

In the case of telephone surveys, the number of calls made until the first willing respondent is not a fixed number, as it could potentially be infinite. This situation describes a geometric distribution rather than a binomial distribution, as it lacks a fixed number of trials.

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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 an area of mathematics focused on analyzing uncertain phenomena. It provides the foundation for understanding how likely events are to occur, helping us make predictions in everyday life.
Probability is often expressed as a number between 0 and 1, where 0 means an event will not happen, and 1 indicates certainty. For example, flipping a fair coin has a probability of 0.5 for landing on heads.
Key concepts in probability theory include:
  • Random Variables: These are variables that take on different possible values, each associated with a probability. There are discrete and continuous types of random variables. In the example of the meteorologist, the number of rainy days, denoted as \( Y \), is a discrete random variable.
  • Probability Distributions: This describes how probabilities are distributed over the values of the random variable. For example, in a binomial distribution, probabilities are allocated based on a fixed number of trials, like the 30 days observed by the meteorologist.
  • Expected Value: Also known as the mean, it provides a measure of the central tendency of a probability distribution, which can help predict the average outcome if the experiment were repeated many times.
Understanding these fundamentals helps in identifying whether situations, such as those in the meteorologist's recordings or the telephone surveys, fall into specific probability distributions.
Geometric Distribution
In probability theory, the geometric distribution is concerned with the number of trials needed for the first success in repeated Bernoulli trials. Each trial has two possible outcomes: success or failure, with a constant probability of success.
The main characteristics of a geometric distribution are:
  • Trials continue until the first success, potentially taking an infinite number of trials, unlike a fixed number for binomial distributions.
  • The probability of success is constant for each trial, which in the telephone survey case is whether a respondent agrees to participate.
  • The random variable \( Y \) represents the number of trials until the first success, fitting the description of the telephone survey example. Hence, it correctly models this distribution rather than a binomial one.
The probability mass function for a geometric distribution is given by:\[P(Y = k) = (1-p)^{k-1}p\]where \( k \) is the number of trials required for the first success, and \( p \) is the probability of success on each trial. This formula helps calculate the probability of various outcomes, such as the number of calls until the first agreement.
Statistical Experiment Design
Designing a statistical experiment involves carefully planning how to collect data in order to ensure that the results accurately represent the phenomena being studied.
Important elements of proper experiment design include:
  • Defining Variables: Clearly identify dependent and independent variables. In the meteorologist's scenario, the number of rainy days could be the dependent variable, influenced by various weather factors.
  • Setting Conditions: Ensure trials are independent and outcomes have clearly defined probabilities. This is crucial for correctly using distributions like binomial or geometric to model outcomes.
  • Sampling: Decide whether to take a fixed sample size (aligned with binomial distribution) or allow an indeterminate number (appropriate for geometric distribution), as explained in the survey example.
  • Testing Hypotheses: Use results to make informed decisions, such as determining the typical number of rainy days in a month or understanding caller willingness in surveys.
By meticulously designing experiments, data collected will yield reliable insights, allowing for accurate predictions and deeper understanding of the random variables and their distributions.

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Most popular questions from this chapter

Show that the hypergeometric probability function approaches the binomial in the limit as \(N \rightarrow \infty\) and \(p=r / N\) remains constant. That is, show that $$\lim _{N \rightarrow \infty} \frac{\left(\begin{array}{l} r \\ y \end{array}\right)\left(\begin{array}{l} N-r \\ n-y \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)}=\left(\begin{array}{l} n \\ y \end{array}\right) p^{y} q^{n-y}$$ for \(p=r / N\) constant.

Sampling for defectives from large lots of manufactured product yields a number of defectives, \(Y\), that follows a binomial probability distribution. A sampling plan consists of specifying the number of items \(n\) to be included in a sample and an acceptance number a. The lot is accepted if \(Y \leq a\) and rejected if \(Y>a\). Let \(p\) denote the proportion of defectives in the lot. For \(n=5\) and \(a=0\), calculate the probability of lot acceptance if a. \(p=0\) b. \(p=.1\) C. \(p=.3\) d. \(p=.5\) e. \(p=1.0\) A graph showing the probability of lot acceptance as a function of lot fraction defective is called the operating characteristic curve for the sample plan. Construct the operating characteristic curve for the plan \(n=5, a=0 .\) Notice that a sampling plan is an example of statistical inference. Accepting or rejecting a lot based on information contained in the sample is equivalent to concluding that the lot is either good or bad. "Good" implies that a low fraction is defective and that the lot is therefore suitable for shipment.

Asupplier of heavy construction equipment has found that new customers are normally obtained through customer requests for a sales call and that the probability of a sale of a particular piece of equipment is .3. If the supplier has three pieces of the equipment available for sale, what is the probability that it will take fewer than five customer contacts to clear the inventory?

It is known that \(5 \%\) of the members of a population have disease \(A,\) which can be discovered by a blood test. Suppose that \(N\) (a large number) people are to be tested. This can be done in two ways: 1\. Each person is tested separately, or 2\. the blood samples of \(k\) people are pooled together and analyzed. (Assume that \(N=n k\), with \(n\) an integer.) If the test is negative, all of them are healthy (that is, just this one test is needed). If the test is positive, each of the \(k\) persons must be tested separately (that is, a total of \(k+1\) tests are needed). a. For fixed \(k,\) what is the expected number of tests needed in option \(2 ?\) b. Find the \(k\) that will minimize the expected number of tests in option 2 . c. If \(k\) is selected as in part (b), on the average how many tests does option 2 save in comparison with option 1?

Five cards are dealt at random and without replacement from a standard deck of 52 cards. What is the probability that the hand contains all 4 aces if it is known that it contains at least 3 aces?
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