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Chapter 9: Problem 5

Each person in a random sample of 20 students at a particular university was asked whether he or she is registered to vote. The responses \((\mathrm{R}=\) registered, \(\mathrm{N}=\) not registered) are given here: R R N R N N R R R N R R R R R N R R R N Use these data to estimate \(\pi\), the true proportion of all students at the university who are registered to vote.

Short Answer

Expert verified
The estimated true proportion of all students at the university who are registered to vote is the same as the proportion of registered students in the sample.

Step by step solution

01

Identify Registered Students

Count the number of registered students in the sample, denoted as 'R'. You have to go through the responses one by one and count the total.
02

Calculate Proportion

Next, calculate the proportion of registered students. Accomplish this by dividing the number of registered students (from Step 1) by total number of students in the sample. This number is given as 20.
03

Estimate True Proportion

Finally, use the proportion obtained in Step 2 as an estimate of the true proportion \(\pi\) of all students at the university who are registered to vote.

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Key Concepts

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

Random Sample
A random sample is like picking a handful of jellybeans from a large jar without looking. You are trying to get a mixture that represents the whole jar.
In research, a random sample helps us get a sense of a larger group or population without having to ask everyone.
  • Each individual is chosen entirely by chance.
  • Every person has an equal chance of being selected.
By using a random sample of 20 students, the exercise aims to understand how many students are registered to vote at the entire university.
This is important because making the sample random ensures that the answers we get aren't biased by who we picked.
True Proportion
Proportion gives us a way to express a relationship between quantities. When we talk about the true proportion in this exercise, it relates to the total number of students registered to vote compared to the whole university.
Think of it as a pie chart where each slice represents part of the total.
  • The goal is to estimate the true proportion, \( \pi \), using the result from the random sample.
The true proportion will tell us what part of the entire student body is actually registered to vote.
The estimate from our sample will not always be exact, but it gives us a good idea of what's happening in the larger group.
Voting Registration
Voting registration is essential for allowing students to participate in elections. It measures engagement and civic responsibility.
In the context of the exercise, it serves as a practical example to understand how to estimate values like proportion.
  • The exercise simplifies this by using a sample of responses to calculate the proportion of registered voters.
  • This helps in understanding broader patterns of voter registration at the university.
Registering to vote is a straightforward process, yet pivotal in enabling people to express their opinions through voting.
Tracking this kind of data helps universities and communities encourage higher voter participation.

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Most popular questions from this chapter

Seventy-seven students at the University of Virginia were asked to keep a diary of a conversation with their mothers, recording any lies they told during these conversations (San Luis Obispo Telegram-Tribune, August 16 , 1995). It was reported that the mean number of lies per conversation was \(0.5\). Suppose that the standard deviation (which was not reported) was \(0.4 .\) a. Suppose that this group of 77 is a random sample from the population of students at this university. Construct a 95\% confidence interval for the mean number of lies per conversation for this population. b. The interval in Part (a) does not include 0 . Does this imply that all students lie to their mothers? Explain.

A random sample of 10 houses in a particular area, each of which is heated with natural gas, is selected, and the amount of gas (in therms) used during the month of January is determined for each house. The resulting observations are as follows: \(\begin{array}{llllllllll}103 & 156 & 118 & 89 & 125 & 147 & 122 & 109 & 138 & 99\end{array}\) a. Let \(\mu_{j}\) denote the average gas usage during January by all houses in this area. Compute a point estimate of \(\mu_{J}\) b. Suppose that 10,000 houses in this area use natural gas for heating. Let \(\tau\) denote the total amount of gas used by all of these houses during January. Estimate \(\tau\) using the given data. What statistic did you use in computing your estimate? c. Use the given data to estimate \(\pi\), the proportion of all houses that used at least 100 therms. d. Give a point estimate of the population median usage based on the given sample. Which statistic did you use?

would result in a wider large-sample cont?dence interval for \(\pi\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

The article "Consumers Show Increased Liking for Diesel Autos" (USA Today, January 29,2003 ) reported that \(27 \%\) of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area and that you want to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion to within \(.05\) with \(95 \%\) confidence? Compute the required sample size first using 27 as a preliminary estimate of \(\pi\) and then using the conservative value of .5. How do the two sample sizes compare? What sample size would you recommend for this study?

The Chronicle of Higher Education (January 13 , 1993) reported that \(72.1 \%\) of those responding to a national survey of college freshmen were attending the college of their first choice. Suppose that \(n=500\) students responded to the survey (the actual sample size was much larger). a. Using the sample size \(n=500\), calculate a \(99 \%\) confidence interval for the proportion of college students who are attending their first choice of college. b. Compute and interpret a \(95 \%\) confidence interval for the proportion of students who are not attending their first choice of college. c. The actual sample size for this survey was much larger than 500 . Would a confidence interval based on the actual sample size have been narrower or wider than the one computed in Part (a)?
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