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Chapter 8: Problem 45

Election polling Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. We want to estimate the proportion p of all registered voters in the city who plan to vote for Chavez with 95% confidence and a margin of error no greater than 0.03. How large a random sample do we need? Show your work.

Short Answer

Expert verified
A sample size of 1068 is needed.

Step by step solution

01

Understanding the Problem

We need to estimate the required sample size to achieve a margin of error of no greater than 0.03 at a 95% confidence level for estimating the proportion of voters supporting Chavez.
02

Setting Up the Formula

To find the required sample size, we use the formula for margin of error: \( E = z \sqrt{\frac{p(1-p)}{n}} \). Here, \( E \) is the margin of error, \( z \) is the critical value of the standard normal distribution, \( p \) is the estimated proportion, and \( n \) is the sample size.
03

Choosing Values for Calculation

For a 95% confidence level, the critical value \( z \) is approximately 1.96. Since we are estimating the proportion \( p \) and have no prior estimate, we use \( p = 0.5 \) for the maximum variability, which maximizes \( p(1-p) \). The margin of error \( E \) is 0.03.
04

Solving for Sample Size

Substitute the known values into the margin of error formula and solve for \( n \):\[ 0.03 = 1.96 \sqrt{\frac{0.5(1-0.5)}{n}} \]Squaring both sides to eliminate the square root gives:\[ 0.0009 = \frac{1.96^2 \times 0.25}{n} \]\[ n = \frac{1.96^2 \times 0.25}{0.0009} \].
05

Calculating the Result

Calculate \( n \) using the values:\[ n = \frac{1.960^2 \times 0.25}{0.0009} \approx \frac{0.9604 \times 0.25}{0.0009} \approx 1067.11 \]. Since we cannot survey a fraction of a person, round up to the next whole number, giving a sample size of 1068.

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Key Concepts

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

Margin of Error
The margin of error is a crucial part of any statistical survey or poll. It tells you the range within which you can expect the true value of the population parameter to lie.
  • In the exercise, we see a margin of error of 0.03. This means we are 95% confident that the actual proportion of all registered voters supporting Chavez is within 3% of our sample's proportion.
  • Why 3%? Because that's the maximum allowable distance between the sample estimate and the true population proportion.
To manage this margin, we aim for precision without overstretching resources. It helps in deciding how many people need to be surveyed to get accurate results.
The smaller the margin of error, the more precise your results, but it often requires a larger sample size. The balance is in achieving precision without unnecessary effort.
Confidence Level
The confidence level is an expression of how sure we are of our estimate's precision. In the problem, a 95% confidence level signifies a very high degree of certainty.
  • A 95% confidence level means that if we were to take 100 different samples and compute their intervals, about 95 of them would contain the true population proportion.
  • This helps in making reliable decisions and is a commonly used standard in social sciences.
Higher confidence levels imply greater certainty but also require larger sample sizes, which might not always be practical.
The exercise emphasizes the importance of balancing desired certainty with logistical feasibility. Confidence levels are not about guaranteeing correctness; they provide a probabilistic framework in which we make decisions.
Proportion Estimation
Proportion estimation refers to determining what fraction of the population falls into a particular category. It's vital for predicting outcomes.
  • In our case, we estimate the proportion of voters who support Chavez. Since no prior data is available, we assume a 50% proportion, which represents maximum variability.
  • This assumption makes the calculation conservative, ensuring a buffer for estimation errors.
By estimating proportions correctly, we can plan strategies more accurately and make informed decisions.
Assuming 50% when no prior information exists is a standard approach, allowing estimation under uncertain conditions.
Critical Value
Critical value, often denoted by "z," is a factor derived from the confidence level. It's essential for measuring how many standard deviations a point is above or below the mean in a standard normal distribution.
  • For a 95% confidence level, a standard z-value is approximately 1.96.
  • It helps determine the spread in the data and hence is crucial in calculating the margin of error.
When calculating sample size, the critical value determines how "wide" our confidence interval needs to be.
Using a higher critical value means wider confidence intervals, providing a more conservative estimate. The critical value ensures our estimates are set with an adequately chosen level of confidence and precision.

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Most popular questions from this chapter

Critical value What critical value t* from Table B would you use for a 99% confidence interval for the population mean based on an SRS of size 58? If possible, use technology to find a more accurate value of t*. What advantage does the more accurate df provide?

93% confidence Find z* for a 93% confidence interval using Table A or your calculator. Show your method.

Plagiarizing An online poll posed the following question: It is now possible for school students to log on to Internet sites and download homework. Everything from book reports to doctoral dissertations can be downloaded free or for a fee. Do you believe that giving a student who is caught plagiarizing an F for their assignment is the right punishment? Of the 20,125 people who responded, 14,793 clicked 鈥淵es.鈥� That鈥檚 73.5% of the sample. Based on this sample, a 95% confidence interval for the percent of the population who would say "Yes" is \(73.5 \% \pm\) 0.61%. Which of the three inference conditions is violated? Why is this confidence interval worthless?

Sisters and brothers (3.1, 3.2) How strongly do physical characteristics of sisters and brothers correlate? Here are data on the heights (in inches) of 11 adult pairs: \(^{8}\) $$\begin{array}{llllllllll}\text{Brother:} \quad{71} & {68} & {66} & {67} & {70} & {71} & {70} & {73} & {72} & {65} & {66} \\\ \text{Sister:}\quad\quad{69} & {64} & {65} & {63} & {65} & {62} & {65} & {64} & {66} & {59} & {62} \\ \hline\end{array}$$ (a) Construct a scatterplot using brother鈥檚 height as the explanatory variable. Describe what you see. (b) Use your calculator to compute the least-squares regression line for predicting sister鈥檚 height from brother鈥檚 height. Interpret the slope in context. (c) Damien is 70 inches tall. Predict the height of his sister Tonya. (d) Do you expect your prediction in (c) to be very accurate? Give appropriate evidence to support your answer.

Critical value What critical value t* from Table B would you use for a 90% confidence interval for the population mean based on an SRS of size 77? If possible, use technology to find a more accurate value of t*. What advantage does the more accurate df provide?
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