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Chapter 14: Problem 1

List the characteristics of a multinomial experiment.

Short Answer

Expert verified
Answer: The key characteristics of a multinomial experiment are: 1. Mutually exclusive outcomes - Outcomes cannot occur simultaneously. 2. Fixed number of trials - The number of trials performed is predetermined. 3. Categorical outcomes - Outcomes are described by categories or labels rather than numerical values. 4. Constant probability - The probability of each outcome remains the same throughout the experiment. 5. Independent trials - The outcome of one trial does not influence or affect the outcome of others.

Step by step solution

01

1. Mutually Exclusive Outcomes

In a multinomial experiment, the outcomes are mutually exclusive, meaning that it is impossible for two or more outcomes to occur simultaneously. When one outcome occurs, all other outcomes can't happen. For example, when you roll a standard six-sided die, the outcomes are mutually exclusive 鈥� if you roll a 3, you can't also roll a 4, 5, or any other number at the same time.
02

2. Fixed Number of Trials

When conducting a multinomial experiment, the number of trials performed must be fixed in advance. Each trial is independent of the others and has the same likelihood of each outcome occurring. For example, flipping a coin 10 times involves 10 fixed trials.
03

3. Categorical Outcomes

The outcomes of a multinomial experiment are categorical rather than numerical, meaning they are described by their category or label rather than by a numerical value. An example of this is the result of a political survey where respondents are classified as: "Democrat", "Republican", or "Independent". The outcome variables are categorical rather than numerical.
04

4. Constant Probability

In a multinomial experiment, the probability of each outcome should remain constant from trial to trial. This means that the likelihood of each outcome occurring remains the same throughout the entire experiment. For example, in a fair six-sided die roll, the probability of rolling any given number should be \(\frac{1}{6}\) for every roll, regardless of any previous rolls.
05

5. Independent Trials

The trials of a multinomial experiment should be independent. This means that the outcome of one trial does not influence or affect the outcome of any other trials in the experiment. In other words, the outcomes of the trials are unrelated, regardless of the order in which they occur.

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Most popular questions from this chapter

Random samples of 200 observations were selected from each of three populations, and each observation was classified according to whether it fell into one of three mutually exclusive categories: You want to know whether the data provide sufficient evidence to indicate that the proportions of observations in the three categories depend on the population from which they were drawn. a. Give the value of \(X^{2}\) for the test. b. Give the rejection region for the test for \(\alpha=.01\). c. State your conclusions. d. Find the approximate \(p\) -value for the test and interpret its value.

You can use a goodness-of-fit test to determine whether all of the criteria for a binomial experiment have actually been met in a given application. Suppose that an experiment consisting of four trials was repeated 100 times. The number of repetitions on which a given number of successes was obtained is recorded in the table: Estimate \(p\) (assuming that the experiment was binomial), obtain estimates of the expected cell frequencies, and test for goodness of fit. To determine the appropriate number of degrees of freedom for \(X^{2}\), note that \(p\) was estimated by a linear combination of the observed frequencies.

Use the Chi-Square Probabilities applet to find the value of \(\chi^{2}\) with the following area \(\alpha\) to its right: a. \(\alpha=.05, d f=15\) b. \(\alpha=.01, d f=11\)

Suppose you are interested in following two independent traits in snap peas- seed texture \((\mathrm{S}=\) smooth, \(\mathrm{s}=\) wrinkled \()\) and seed color \((\mathrm{Y}=\) yellow, \(\mathrm{y}=\) green \()-\) in a second-generation cross of heterozygous parents. Mendelian theory states that the number of peas classified as smooth and yellow, wrinkled and yellow, smooth and green, and wrinkled and green should be in the ratio \(9: 3: 3: 1 .\) Suppose that 100 randomly selected snap peas have \(56,\) \(19,17,\) and 8 in these respective categories. Do these data indicate that the 9: 3: 3: 1 model is correct? Test using \(\alpha=.01\)

Suppose that a consumer survey summarizes the responses of \(n=307\) people in a contingency table that contains three rows and five columns. How many degrees of freedom are associated with the chi-square test statistic?
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