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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 9596 confidence? Compute the required sample size first using .27 as a preliminary estimate of \(p\) and then using the conservative value of .5. How do the two sample sizes compare? What sample size would you recommend for this study?

Step by step solution

01

Compute Sample Size Using p=0.27

Using \(Z = 1.96\) for 95% confidence level, \(p = 0.27\), and \(E = 0.05\), the sample size \(n = (1.96^2 * 0.27* (1-0.27)) / 0.05^2\). Compute this expression to determine the sample size.

02

Compute Sample Size Using p=0.5

Do the same computation as in Step 1, now with \(p = 0.5\). Compute the expression \(n = (1.96^2 * 0.5 * (1-0.5)) / 0.05^2\) to find the sample size.

03

Comparing Sample Sizes

Compare the results of the two computations. It is usual that the sample size computed with \(p = 0.5\) is larger because it is the most conservative estimate as it maximizes the term \(p*(1-p)\) in the formula.

04

Make Final Recommendation

Given the comparison in Step 3, it can be decided which sample size to recommend. Usually, the larger the sample size, the more accurately it estimates the true population proportion; however, practical constraints such as cost and time may impact this decision.

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Key Concepts

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

Sample Size Calculation

Calculating the sample size is crucial in planning a survey because it determines how accurately the survey results will reflect the population. In general, a larger sample size provides a more accurate estimate of the population parameter.
To calculate the sample size for estimating a proportion, a specific formula is utilized:

• When the preliminary estimate of the proportion, \( p \), is known, use the formula: \[ n = \frac{Z^2 \times p \times (1-p)}{E^2} \] Here, \( n \) is the sample size, \( Z \) is the z-score from the standard normal distribution corresponding to the desired confidence level, \( E \) is the margin of error, and \( p \) is the estimated proportion.
• If no approximate value of \( p \) is available, use a conservative estimate of \( p = 0.5 \), which maximizes \( p(1-p) \) and thus the required sample size.

The ability to estimate the correct sample size helps in managing resources efficiently and ensuring the reliability of the survey.

Proportion Estimation

Proportion estimation is about determining what fraction of your population fits a certain characteristic. For instance, in the given problem, you want to estimate how many people might prefer diesel cars.
To estimate a population proportion, you typically use survey data to calculate a sample proportion (\( \hat{p} \)). You then use this sample proportion to make inferences about the population proportion.

• A critical step in the process is to ensure your sample is representative of the population; otherwise, your estimate may be biased.
• The formula employed for proportion estimation involves statistical techniques that assume normally distributed outcomes as per the Central Limit Theorem, making it fundamental in survey design.

Accurate proportion estimation allows for effective prediction and decision-making based on survey results.

Confidence Level

The confidence level is an important concept in statistics, often expressed as a percentage, which tells us how sure we can be about a parameter estimated from a sample. In most social science research, common confidence levels are 90%, 95%, and 99%.
For a 95% confidence interval, one would be 95% certain that the true proportion falls within the range of the interval. This means that if we took 100 different samples and calculated an interval for each sample, we expect that about 95 of these intervals would contain the population proportion.

• Tighter confidence levels (e.g., 99%) require a larger sample size, while looser levels (e.g., 90%) need a smaller sample.
• The z-score used in sample size calculations is determined by the confidence level, with 1.96 being the z-score for a 95% confidence interval.

This coherence in measurements boosts credibility and trust in the research outcomes.

Statistical Survey Design

Designing a statistical survey involves several decisions that will influence its accuracy and reliability. One key aspect is determining the sample size, as with the given exercise.
A well-designed survey also includes:

• Drafting clear and unbiased questions to ensure honest responses.
• Choosing a method for data collection, such as online surveys, interviews, or paper questionnaires.
• Deciding on a sampling method, such as random sampling, which helps ensure the generalization of results to the larger population.

Effective survey design mitigates errors and biases, providing results that are more likely to represent the entire population accurately.
By focusing on these aspects, surveys can be powerful tools for gathering valuable insights.