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Sampling at school. For a sociology class project you are asked to conduct a survey on 20 students at your school. You decide to stand outside of your dorm's cafeteria and conduct the survey on a random sample of 20 students leaving the cafeteria after dinner one evening. Your dorm is comprised of \(45 \%\) males and \(55 \%\) females. (a) Which probability model is most appropriate for calculating the probability that the \(4^{\text {th }}\) person you survey is the \(2^{\text {nd }}\) female? Explain. (b) Compute the probability from part (a). (c) The three possible scenarios that lead to \(4^{\text {th }}\) person you survey being the \(2^{\text {nd }}\) female are $$ \\{M, M, F, F\\},\\{M, F, M, F\\},\\{F, M, M, F\\} $$ One common feature among these scenarios is that the last trial is always female. In the first three trials there are 2 males and 1 female. Use the binomial coefficient to confirm that there are 3 ways of ordering 2 males and 1 female. (d) Use the findings presented in part (c) to explain why the formula for the coefficient for the negative binomial is \(\left(\begin{array}{c}n-1 \\\ k=1\end{array}\right)\) while the formula for the binomial coefficient is \(\left(\begin{array}{l}n \\ k\end{array}\right)\).

Step by step solution

01

Identify the Probability Model

In part (a), we need to recognize that we are looking for the probability that a specific event (finding the second female) happens on the fourth trial, with a known outcome probability for each gender. This suggests the use of a negative binomial distribution because we want the second female occurrence on the fourth trial.

02

Calculate the Probability with Negative Binomial

For part (b), the formula for the negative binomial probability is given by \(P(X=n) = \binom{n-1}{r-1} p^r (1-p)^{n-r}\) where \(p\) is the probability of success (female encountered, 0.55), \(r\) is the number of successes we want (2 females), and \(n\) is the trial in which the \(r^{th}\) success occurs (4th trial). Thus, the probability is calculated as \(P(X=4) = \binom{3}{1} (0.55)^2 (0.45)^{2}\). Calculating this gives \(3 \times 0.3025 \times 0.2025 = 0.1831\).

03

Determine Binomial Coefficient for Scenario Counting

In part (c), we confirm that there are 3 ways to arrange 2 males (M) and 1 female (F) in 3 trials, which precede the always-female fourth trial. The binomial coefficient for 2 males and 1 female in 3 trials is \(\binom{3}{2} = 3\). This corresponds to the sequences \{M, M, F\}, \{M, F, M\}, and \{F, M, M\}.

04

Compare Negative Binomial and Binomial Coefficients

Part (d) requires us to explain the difference in the coefficients' formulas. The negative binomial coefficient \(\binom{n-1}{k-1}\) reflects the number of sequences for achieving \(k-1\) successes in \(n-1\) trials before obtaining the \(k^{th}\) success on the nth trial. In contrast, the binomial coefficient \(\binom{n}{k}\) calculates the number of ways to have \(k\) successes in \(n\) trials without specifying any order of the final trial.

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Key Concepts

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

Probability Model

A probability model is a mathematical representation that describes the possible outcomes of a random process and assigns probabilities to these outcomes. Probability models are used extensively in statistics to predict and analyze real-world random events. In this exercise's context, the negative binomial distribution is employed as the probability model. The negative binomial distribution is ideal when we are interested in determining the probability of a certain number of successes following a set number of trials. Here, we want to calculate the probability that the fourth surveyed student is the second female. By considering gender as our variable, and with known probabilities for each, it fits the use of the negative binomial probability model, requiring two successful findings (females) by the fourth trial.

Binomial Coefficient

The binomial coefficient, often represented as \(inom{n}{k}\), is a fundamental concept in combinatorics. It counts the number of ways to choose \(k\) successes (or items) from \(n\) trials (or selections), without regard to the order. In simpler words, it tells us how many combinations of successes can occur in a series of trials. In the solving step, we analyze sequences like \{M, M, F\}, \{M, F, M\}, and \{F, M, M\} that precede the final female survey participant to illustrate the use of the binomial coefficient. Calculating this for 2 males and 1 female in 3 trials, \(inom{3}{2} = 3\) verifies there are three combinations, which matches our sequence count.

Survey Sampling Methods

Survey sampling methods refer to the techniques used to select individuals or groups from a population to be surveyed. These methods ensure that the selected sample accurately represents the larger population. In this scenario, the student conducts the survey outside a dorm's cafeteria, employing random sampling on students exiting the venue. This is an example of convenience sampling because it's based on the ease of access to the survey participants. Although not as accurate as stratified or systematic sampling, convenience sampling is often used in educational settings for quick and accessible data collection.

Gender Distribution in Sampling

Gender distribution in sampling is the consideration of gender proportion within a sample, which can be critical in studies involving social attributes. In the context of the student's survey, the dormitory's overall population consists of 45% males and 55% females. Understanding this distribution helps in interpreting the data and aligning survey findings with the demographic characteristics of the population. It's important to comprehend that in a simple random sample, these proportions might not exactly reflect in a small sample size like 20, but they orient the expectations for gender representation in sampling outcomes. Correct sampling ensures that the gender distribution in the sample aligns with the population's actual dynamics.