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Chapter 5: Problem 49

Briefly explain the following. a. A binomial experiment b. A trial c. A binomial random variable

Short Answer

Expert verified
a. A binomial experiment is a statistical experiment that has properties like n repeated trials, each trial has two outcomes (success or failure), the probability of success is same in every trial and trials are independent. An example is coin flipping. b. A trial is a single operation which is performed to get the outcome or the result. Examples are rolling a die or flipping a coin. c. A binomial random variable is the number of successes in a binomial experiment. Its values can be any number from 0 to n, inclusive. For example, the quantity of 'heads' in 10 coin flips.

Step by step solution

01

Explanation of a Binomial Experiment

A binomial experiment is a statistical experiment that has the following properties: The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; the outcome on one trial does not affect the outcome on other trials. Example could be flipping a coin. Each flip is an independent event, the outcome of one flip does not affect the outcomes of other flips.
02

Explanation of a Trial

A trial in statistics is referred to the act, process, or operation of trying test performance or an experiment. In simpler words, it is a single operation which is performed to get the outcome or the result. For instance, rolling a die, flipping a coin, are all trials as we perform these activities to get an outcome.
03

Explanation of a Binomial Random Variable

A binomial random variable is the number of successes in a binomial experiment. The values of the random variable can be any whole number between 0 and n, inclusive. A binomial random variable can be thought of as the number of successes in n trials. For example, if we flip a coin (where heads is considered 'success') 10 times, the number of heads is a binomial random variable.

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Most popular questions from this chapter

Which of the following are binomial experiments? Explain why. a. Drawing 3 balls with replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls b. Drawing 3 balls without replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls c. Selecting a few households from New York City and observing whether or not they own stocks when it is known that \(28 \%\) of all households in New York City own stocks

Let \(x\) be the number of heads obtained in two tosses of a coin. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|lll} \hline x & 0 & 1 & 2 \\ \hline P(x) & .25 & .50 & .25 \\ \hline \end{array} $$ Calculate the mean and standard deviation of \(x\). Give a brief interpretation of the value of the mean.

A large proportion of small businesses in the United States fail during the first few years of operation. On average, \(1.6\) businesses file for bankruptcy per day in a particular large city. a. Using the Poisson formula, find the probability that exactly 3 businesses will file for bankruptcy on a given day in this city. b. Using the Poisson probabilities table, find the probability that the number of businesses that will file for bankruptcy on a given day in this city is \(\begin{array}{lll}\text { i. } 2 \text { to } 3 & \text { ii. more than } 3 & \text { iii. less than } 3\end{array}\)

Which of the following are binomial experiments? Explain why. a. Rolling a die many times and observing the number of spots b. Rolling a die many times and observing whether the number obtained is even or odd c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when \(54 \%\) of all voters are known to be in favor of this proposition.

Let \(x\) be the number of cars that a randomly selected auto mechanic repairs on a given day. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|ccccc} \hline x & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .05 & .22 & .40 & 23 & .10 \\ \hline \end{array} $$ Find the mean and standard deviation of \(x\), Give a brief interpretation of the value of the mean.
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