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Chapter 4: Problem 43

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status. What is the probability that at most five of the freshmen reply 鈥測es鈥�?

Short Answer

Expert verified
The probability that at most five freshmen reply 鈥測es鈥� is approximately 0.116167.

Step by step solution

01

Define the Binomial Distribution

The problem can be modeled using a binomial distribution, where each freshman has a probability of 0.713 of responding "yes." Let the random variable \(X\) represent the number of students who reply "yes." Thus, \(X\) follows a binomial distribution: \(X \sim B(n=8, p=0.713)\), where \(n=8\) is the number of trials and \(p=0.713\) is the probability of success on each trial.
02

Calculate "At Most Five" Probability

We need to calculate \(P(X \leq 5)\). This probability can be broken into the sum of probabilities for \(X = 0, 1, 2, 3, 4,\) and \(5\). The formula for the probability of \(k\) successes in a binomial distribution is \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\).
03

Calculate Each Probability

Calculate each probability using the binomial probability formula:\[P(X = 0) = \binom{8}{0} 0.713^0 (1-0.713)^8, \P(X = 1) = \binom{8}{1} 0.713^1 (1-0.713)^7, \P(X = 2) = \binom{8}{2} 0.713^2 (1-0.713)^6, \P(X = 3) = \binom{8}{3} 0.713^3 (1-0.713)^5, \P(X = 4) = \binom{8}{4} 0.713^4 (1-0.713)^4, \P(X = 5) = \binom{8}{5} 0.713^5 (1-0.713)^3. \]
04

Sum the Probabilities

Add up all the probabilities calculated in Step 3 to find \(P(X \leq 5)\):\[P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).\]
05

Calculate the Sum

Using the binomial formula calculations from Step 3, calculate the sum:\[P(X \leq 5) = 0.000006 + 0.000136 + 0.001415 + 0.008240 + 0.030922 + 0.075448 = 0.116167.\] Thus, the probability that at most five respond "yes" is approximately 0.116167.

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Key Concepts

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

Probability
Probability measures how likely an event is to occur. For instance, when dealing with a binomial distribution, we often refer to the likelihood of a specific number of successes out of a set number of trials. In the context of our problem, we are examining the probability that a certain number of freshmen will respond 'yes' to a survey. By calculating probabilities for individual outcomes and summing them, we can determine comprehensive probabilities for a range of outcomes. This is particularly useful in real-world applications, where decisions are frequently based on probabilistic outcomes.
Random Variable
A random variable is a numerical representation of the outcome of a random event. With our exercise, the random variable, denoted as \(X\), represents how many freshmen believe that same-sex couples should have the right to legal marital status. In this situation, \(X\) could be any value from 0 to 8 because we are selecting 8 freshmen. Each potential value for \(X\) corresponds to a specific probability of occurring. Random variables are fundamental in statistics as they allow us to quantify randomness and uncertainty.
Binomial Probability Formula
The Binomial Probability Formula calculates the likelihood of exactly \(k\) successes in \(n\) independent trials, where each trial has the same probability of success, \(p\). It is expressed as: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\] Here, \(\binom{n}{k}\) represents combinations, or how many ways we can choose \(k\) successes from \(n\) trials. This formula is pivotal in determining specific probabilities within a binomial distribution, like finding the likelihood of receiving a certain number of 'yes' responses from our selected freshmen.
Statistical Analysis
Statistical analysis involves collecting and scrutinizing data to extract meaningful patterns and conclusions. In our scenario, we use statistical methods to assess survey responses from first-time, full-time freshmen. By applying the binomial distribution and relevant formulas, we can analyze the probability distribution of these survey responses. This allows us to draw conclusions about the broader population. Statistical analysis translates abstract data into actionable insights, enabling decision-makers to make informed choices based on empirical evidence.

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Most popular questions from this chapter

A group of Martial Arts students is planning on participating in an upcoming demonstration. Six are students of Tae Kwon Do; seven are students of Shotokan Karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in that first demonstration. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many Shotokan Karate students do we expect to be in that first demonstration?

Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {0.15} \\ \hline 2 & {0.35} \\ \hline 3 & {0.40} \\ \hline 4 & {0.10} \\ \hline\end{array}$$ What is the probability the baker will sell exactly one batch? \(P(x=1)=\)_________

Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many instructors do you expect on the committee who are not technically proficient? e. Find the probability that at least five on the committee are not technically proficient. f. Find the probability that at most three on the committee are not technically proficient.

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. What is the probability that the San Jose Sharks win six games in that upcoming month? a. 0.1476 b. 0.2336 c. 0.7664 d. 0.8903

Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. Construct a probability distribution table for the data.
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