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Chapter 4: Problem 49

Use the following information to answer the next six 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 select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. On average \((\mu),\) how many freshmen would you expect to have to ask until you found one who replies "yes?"

Short Answer

Expert verified
You'd expect to ask about 1.403 freshmen on average.

Step by step solution

01

Understanding the Problem

We are dealing with a probability scenario where we are selecting freshmen until one replies 'yes' regarding their belief that same-sex couples should have the right to legal marital status. This fits the geometric probability model.
02

Identifying the Parameter of Interest

We need to find the expected number of trials (freshmen to ask) required to get the first 'success' (a 'yes' answer), which is represented by the mean \(\mu\).
03

Mean of Geometric Distribution

For a geometric distribution, the mean (expected value) is given by the formula \(\mu = \frac{1}{p}\), where \(p\) is the probability of success on each trial.
04

Substitute the Probability into the Formula

Here, the probability \(p\) of a freshman responding 'yes' is 71.3%, or 0.713. Substitute \(p = 0.713\) into the formula:\[ \mu = \frac{1}{0.713} \]
05

Calculate the Expected Value

Perform the calculation: \[ \mu = \frac{1}{0.713} \approx 1.403 \].This means, on average, you would expect to ask slightly more than 1 person until you get a 'yes' response.

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Key Concepts

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

Expected Value
The concept of expected value is a cornerstone in probability and statistics, particularly when dealing with probability models like the geometric distribution. It provides a theoretical mean or average outcome for a random process. In simple terms, expected value tells you what you can "expect" to happen in the long run if you were to repeat the process many times. For geometric distributions, the expected value can be understood as the average number of trials needed to achieve the first success in a series of independent experiments. This is calculated using the formula \( \mu = \frac{1}{p} \), where \( \mu \) represents the expected value, and \( p \) is the probability of success on each trial.By substituting the probability of success into the formula, we understand how likely you are to encounter the desired outcome. This helps predict the long-term average outcome of the random process.
Probability of Success
Probability of success is a key component in calculating the expected value for a geometric distribution. It represents the chance of a successful outcome occurring in a single trial. In our context, a 'success' is when a randomly selected freshman responds 'yes' to the question about same-sex marriage rights.This probability is often expressed as a number between 0 and 1, or as a percentage. For the example being discussed, the probability \( p \) is 71.3% or 0.713. This means there's a 71.3% chance of getting a 'yes' from any given student on your first try.Understanding this probability allows for predicting outcomes effectively. It makes it clear how often, on average, you expect successes to happen and how this affects the expected number of trials or attempts needed.
Geometric Probability Model
The geometric probability model is used in situations where the same experiment is repeated until the first success occurs. It is an important tool when dealing with questions that involve trials until a particular event occurs, such as finding a freshman who says 'yes' to a question.A few central characteristics of a geometric distribution include:
  • Each trial is independent of the others.
  • The probability of success (\( p \)) is constant for each trial.
  • The random variable \( X \) represents the number of trials needed to get the first success.
The model's principal formula for the expected value \( \mu = \frac{1}{p} \) enables calculation of how many trials are needed, on average, to achieve the first successful outcome, given a constant probability of success. This model is particularly useful for scenarios in which you are interested in knowing how long it will take, on average, to wait for a success.

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Most popular questions from this chapter

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. On average, for every 25 patients calling in, how many do you expect to have the flu?

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time. Define the random variable \(X\).

Approximately 8% of students at a local high school participate in after- school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number that participated in after-school sports all four years of high school. 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 seniors are expected to have participated in after-school sports all four years of high school? e. Based on numerical values, would you be surprised if none of the seniors participated in after school sports all four years of high school? Justify your answer numerically. f. Based on numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically.

Find the expected value from the expected value table. $$\begin{array}{|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} \\ \hline 2 & {0.1} & {2(0.1)=0.2} \\ \hline 4 & {0.3} & {4(0.3)=1.2} \\ \hline 6 & {0.4} & {6(0.4)=2.4} \\ \hline 8 & {0.2} & {8(0.2)=1.6} \\ \hline\end{array}$$

Find the standard deviation. $$\begin{array}{|c|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} & {(x-\mu)^{2} P(x)} \\ \hline 2 & {0.1} & {2(0.1)=0.2} & {(2-5.4)^{2}(0.1)=1.156} \\\ \hline 4 & {0.3} & {4(0.3)=1.2} & {(4-5.4)^{2}(0.3)=0.588} \\ \hline 6 & {0.4} & {6(0.4)=2.4} & {(6-5.4)^{2}(0.4)=0.144} \\ \hline 8 & {0.2} & {8(0.2)=1.6} & {(8-5.4)^{2}(0.2)=1.352} \\ \hline \end{array}$$
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