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Chapter 4: Problem 44

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 least two of the freshmen reply 鈥測es鈥�?

Short Answer

Expert verified
The probability is approximately 99.9%.

Step by step solution

01

Identify Random Variable and Distribution

Define the random variable and the appropriate probability distribution. Let the random variable \( X \) represent the number of students who believe that same-sex couples should have the right to legal marital status when selecting eight students randomly. Since each student's response could be considered as a Bernoulli trial with two outcomes (yes or no), and there are a fixed number of trials, \( X \) follows a Binomial Distribution: \( X \sim \text{Binomial}(n=8, p=0.713) \), where \( n = 8 \) is the number of trials (students) and \( p = 0.713 \) is the probability of each student saying "yes".
02

Express the Desired Probability

We need to find the probability that at least two students reply "yes." Mathematically, this is defined as \( P(X \geq 2) \). Since the binomial distribution covers the full range from 0 to 8, \( P(X \geq 2) = 1 - P(X < 2) = 1 - (P(X = 0) + P(X = 1)) \).
03

Calculate \( P(X = 0) \)

Using the probability mass function of a Binomial distribution, \( P(X = 0) = \binom{8}{0} \cdot (0.713)^0 \cdot (1-0.713)^8 = 1 \cdot 1 \cdot (0.287)^8 \). Calculate \( (0.287)^8 \), which is approximately \( 0.0000115 \).
04

Calculate \( P(X = 1) \)

Similarly, for one student saying "yes," \( P(X = 1) = \binom{8}{1} \cdot (0.713)^1 \cdot (0.287)^7 = 8 \cdot 0.713 \cdot (0.287)^7 \). Calculate \( (0.287)^7 \), which is approximately \( 0.000173 \). Thus, \( P(X = 1) = 8 \cdot 0.713 \cdot 0.000173 \approx 0.00099 \).
05

Compute \( P(X \geq 2) \)

Add \( P(X = 0) \) and \( P(X = 1) \), \( 0.0000115 + 0.00099 \approx 0.0010015 \). Calculate \( P(X \geq 2) = 1 - 0.0010015 = 0.9989985 \).
06

Interpret the Result

The probability that at least two of the eight randomly chosen freshmen agree that same-sex couples should have the right to legal marital status is approximately 0.999 or 99.9%.

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Key Concepts

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

Probability
Probability is a fundamental concept in statistics that quantifies how likely an event is to occur. Think of it as a way to measure the chance of something happening.
It ranges from 0 to 1, where 0 means the event will not happen, and 1 means the event is certain to happen.
In our context, probability is used to determine how likely it is that a certain number of students will agree with a statement.
In our exercise, we are dealing with a particular event: the number of students who believe that same-sex couples should have the right to legal marital status. The probability of a single student responding "yes" is 0.713.
The problem becomes an exercise in calculating cumulative probabilities, especially when looking for events like "at least two students say yes."
We achieve this by calculating a complementary probability and using the fact that the total probability of all possible outcomes is always 1.
  • To calculate the probability of at least two students agreeing, we found the probabilities for 0 and 1, then subtracted their sum from 1.
Random Variable
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. It's a core concept in statistical analysis, allowing us to model and analyze situations where outcomes are not deterministic.
In the language of statistics, a random variable enables us to translate real-world uncertainties into mathematical terms.
In this exercise, we define a random variable, denoted as \( X \). This random variable represents the number of students out of eight who respond "yes" to a particular survey question.
As such, \( X \) can take on any whole number value from 0 to 8, reflecting the range of possible outcomes from the random selection of students.
Defining a clear random variable is crucial for setting up a probability distribution model. Set this up correctly, and it leads directly to using appropriate tools and formulas to calculate the probabilities of different scenarios associated with \( X \).
In this problem, \( X \) fits the framework of a binomial distribution because each student's answer is an independent event with the same probability of success ("yes" response).
  • This structure allows us to use binomial distribution formulas to find probabilities.
Bernoulli Trials
A Bernoulli trial is a random experiment where there are only two possible outcomes: success and failure.
Named after the Swiss mathematician Jacob Bernoulli, each trial is independent, and the probability of success remains constant.
A series of Bernoulli trials can be thought of as a process that generates a binomial distribution, where each trial contributes to the overall outcomes.
In the context of our problem, each student surveyed represents a single Bernoulli trial. The student's response is either "yes" (success) or "no" (failure).
With eight students, we conduct eight Bernoulli trials, each having the same probability of "success" 鈥� 0.713.
The concept of Bernoulli trials helps us to frame problems where multiple trials are involved, each having the binary outcome structure.
Such a framework allows the utilization of binomial probability formulas to calculate overall success rates in repeated trials.
  • This aligns perfectly with scenarios in our exercise, such as determining the probability of multiple students sharing the same opinion.

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Most popular questions from this chapter

Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl. 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 adults in America do you expect to survey until you find one who will watch the Super Bowl? e. Find the probability that you must ask seven people. f. Find the probability that you must ask three or four people.

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} & {x^{\star} P(x)} \\ \hline 1 & {0.35} \\ \hline 2 & {0.20} \\ \hline 3 & {0.15} \\ \hline 4 & {} \\\ \hline 5 & {0.10} \\ \hline 6 & {0.05} \\ \hline\end{array}$$ Define the random variable \(X\).

Use the following information to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities. 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 of the 12 students do we expect to attend the festivities? e. Find the probability that at most four students will attend. f. Find the probability that more than two students will attend.

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day. Which type of distribution can the Poisson model be used to approximate? When would you do this?

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non- business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample. In words, define the random variable \(X.\)
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