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Chapter 11: Problem 8

Describe the four characteristics of a multinomial experiment.

Short Answer

Expert verified
The four characteristics of a multinomial experiment are: 1) A fixed number of trials, 2) Discrete outcomes, 3) Constant probabilities of outcomes, and 4) The counting of the frequency of outcomes.

Step by step solution

01

Characteristic 1: Fixed Number of Trials

A multinomial experiment consists of a fixed number of independent trials. The idea that they are independent means that the outcome of one trial doesn't affect any other trial. This is a defining characteristic of a multinomial experiment.
02

Characteristic 2: Discrete Outcomes

Each trial of the experiment can result in one of three or more possible outcomes. This can be referred to as categorical or discrete outcomes, meaning outcomes that fall into various categories.
03

Characteristic 3: Probability of Outcomes

The probabilities of the possible outcomes remain constant throughout each trial. This means that the probability of each outcome doesn't change from experiment to experiment. It stays the same throughout all trials.
04

Characteristic 4: Counting Frequency of Outcomes

The result of the multinomial experiment is generally expressed as a count of the number of times each outcome occurred. The goal often is to understand the distribution of outcomes, for example, how many times a specific outcome appears.

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Key Concepts

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

Fixed Number of Trials
In a multinomial experiment, you work with a predetermined number of independent trials. This means you decide, in advance, how many times you will perform the experiment. For instance, if you are rolling a die 20 times, that number鈥�20鈥攊s your fixed number of trials.

The trials are independent. This important characteristic implies that the outcome of one trial does not affect the outcome of the others. For example, rolling a die on your second attempt is not influenced by what happened on the first attempt. Such consistency is crucial for calculating and predicting probabilities accurately in statistics.
Discrete Outcomes
Every trial in a multinomial experiment results in one of several possible discrete outcomes. These are outcomes that can be categorized or classified distinctly. Whether you are tossing a coin or selecting a flavor of ice cream, each option falls into its unique category.

Discrete outcomes are categorized because they fit neatly into distinct slots or labels. For instance, when you toss a coin, you only get heads or tails. However, in a more complex experiment like rolling a die, you have six discrete outcomes: one, two, three, four, five, or six. Recognizing these discrete outcomes is essential for accurately determining probabilities.
Probability of Outcomes
An essential trait of multinomial experiments is that the probability of each outcome remains constant from trial to trial. This means if you flip a fair coin, the probability for heads remains at 50% each time you flip it.

Consistency in probability ensures that the experiment's randomness is well defined, without being skewed by variable factors in-between trials. It allows us to utilize probability rules and predict outcomes under controlled, static conditions. Whether you are dealing with dice, cards, or color selections, knowing the probability ahead of time helps in setting up any multinomial analysis.
Counting Frequency of Outcomes
The outcome of a multinomial experiment is typically summarized by counting how often each result occurs. This counting is crucial because it forms the base for analyzing results, making predictions, and drawing conclusions.

For example, if you roll a die 100 times and find that the number "3" appears 20 times, you have counted the frequency of one of your outcomes. Such counts are often presented in a frequency distribution, offering a clear visual representation of how results spread across different outcomes. Having a method to tally these occurrences is key to understanding your experiment's underlying patterns.

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Most popular questions from this chapter

A FOX News/Opinion Dynamics poll asked a question about gun control of random samples of 900 people each during May 2009 and March 2000 . The question asked was, "Which of the following do you think is more likely to decrease gun violence: better enforcement of existing gun laws or more laws and restrictions on obtaining guns?" The numbers in the following table are approximately the same as reported in the poll, which was reported to the nearest percent. \begin{tabular}{lcccc} \hline & Better Enforcement & More Laws and Restrictions & Both & Unsure \\ \hline May 2009 & 425 & 308 & 93 & 74 \\ March 2000 & 372 & 330 & 122 & 76 \\ \hline Source: http://www.pollingreport.com/guns.htm. \end{tabular} Test at the \(5 \%\) significance level whether the distributions of responses from May 2009 and March 2000 are significantly different.

Chance Corporation produces beauty products. Two years ago the quality control department at the company conducted a survey of users of one of the company's products. The survey revealed that \(53 \%\) of the users said the product was excellent, \(31 \%\) said it was satisfactory, \(7 \%\) said it was unsatisfactory, and \(9 \%\) had no opinion. Assume that these percentages were true for the population of all users of this product at that time. After this survey was conducted, the company redesigned this product. A recent survey of 800 users of the redesigned product conducted by the quality control department at the company showed that 495 of the users think the product is excellent, 255 think it is satisfactory, 35 think it is unsatisfactory, and 15 have no opinion. Is the percentage distribution of the opinions of users of the redesigned product different from the percentage distribution of users of this product before it was redesigned? Use \(\alpha=.025\).

The following are the prices (in dollars) of the same brand of camcorder found at eight stores in Los Angeles. \(\begin{array}{llllllll}568 & 628 & 602 & 642 & 550 & 688 & 615 & 604\end{array}\) A. Using the formula from Chapter 3 , find the sample variance, \(s^{2}\), for these data. b. Make the \(95 \%\) confidence intervals for the population variance and standard deviation. Assume that the prices of this camcorder at all stores in Los Angeles follow a normal distribution. c. Test at the \(5 \%\) significance level whether the population variance is different from 750 square dollars.

A sample of 21 observations selected from a normally distributed population produced a sample variance of \(1.97\). a. Write the null and alternative hypotheses to test whether the population variance is greater than \(1.75\). b. Using \(\alpha=.025\), find the critical value of \(\chi^{2}\). Show the rejection and nonrejection regions on a chi-square distribution curve. c. Find the value of the test statistic \(x^{2}\). d. Using the \(2.5 \%\) significance level, will you reject the null hypothesis stated in part a?

The following table gives the distributions of grades for three professors for a few randomly selected classes that each of them taught during the last 2 years. \begin{tabular}{l|lccc} \hline & & \multicolumn{3}{c} { Professor } \\ \cline { 3 - 5 } & & Miller & Smith & Moore \\ \hline \multirow{4}{*} { Grade } & A & 18 & 36 & 20 \\ & B & 25 & 44 & 15 \\ & C & 85 & 73 & 82 \\ & D \& F & 17 & 12 & 8 \\ \hline \end{tabular} Using the \(2.5 \%\) significance level, test the null hypothesis that the grade distributions are homogeneous for these three professors.
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