Question

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?

Short answer

Estimated sample size based on the preliminary proportion estimate \(p=0.27\) is 724, while estimated sample size using a conservative proportion estimate \(p=0.5\) is 961. Since the latter estimate guarantees we will be within a margin of error for a wider range of proportion estimates, we recommend a sample size of 961.

Step-by-step solution

Preparation

Understand that the sample size estimation formula \( n = \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \) will be used in this problem. For a 95% confidence level, the Z score is approximately 1.96. The desired margin of error is 0.05. We will need to perform this calculation twice, first using a preliminary proportion estimate of 0.27, then using a conservative estimate of 0.5.

Calculate sample size using preliminary proportion estimate

Substitute the values into the sample size calculation formula. Using Z =1.96 (corresponding to 95% confidence level), p = 0.27, and E = 0.05, we find: \( n = \frac{(1.96)^2 * 0.27 * (1 - 0.27)}{(0.05)^2} \) = 723. Whether we round this up or down depends on whether or not we want a margin of error strictly less than 0.05 or not. Normally, the sample size is rounded up to ensure that the margin error will not be exceeded. So, the estimated sample size is 724.

Calculate sample size using conservative proportion estimate

Repeat the calculation but now, use a conservative estimate of 0.5 for the proportion, p. We get: \( n = \frac{(1.96)^2 * 0.5 * (1 - 0.5)}{(0.05)^2} \) = 961. Again, rounding up to ensure the margin error is not exceeded, the sample size is 961.

Compare sample sizes and make a recommendation

It's visible that estimation with conservative proportion \(p=0.5\) yields a larger sample size. And since this size will ensure a margin of error not exceeding 0.05 for a broader range of proportion values, it's advisable to use the conservative estimate.

Key concepts

Statistics and Data Analysis

Engaging in statistics and data analysis means diving into the world of collecting, analyzing, interpreting, presenting, and organizing data. It's a key component of research across various fields, from marketing to medicine. One aspect of statistical analysis is estimating population parameters, such as a proportion or average, based on a sample. A common task is determining sample size, which is the number of observations or replicates to include in a statistical sample.

When dealing with proportions, like in the survey on consumers preferring diesel cars, analysts aim for a sample sufficiently large to provide conclusive results. A sample that's too small may fail to represent the broader population accurately, leading to unreliable conclusions. However, too large a sample may be impractical or expensive. Thus, finding the right balance is essential, and that's where the formula for sample size determination becomes crucial. It takes into account the estimated proportion, the desired confidence level, and the margin of error, all of which weigh heavily on the reliability and validity of the study's findings.

Confidence Level

The confidence level is a fundamental concept in statistics, expressing the degree to which we trust that the true parameter (such as a proportion) lies within a specified range. It’s often confused with probability, but it's not exactly the same. When saying we have a 95% confidence level, like in our diesel car preference example, we mean if we were to take many random samples and create confidence intervals from these samples, we would expect about 95% of those intervals to contain the true population parameter.

A higher confidence level indicates greater certainty but requires a larger sample size, leading to more resources utilized in data collection. For the context of our problem, a 95% confidence level has been chosen, which corresponds to a Z-score of 1.96. This choice strikes a balance between a certain level of assurance in our results and practical sample size considerations.

Margin of Error

The margin of error plays a critical role in determining sample size, representing the maximum expected difference between the true population parameter and a sample estimate. Think of it as a buffer zone around your sample estimate; within this zone, you're acknowledging that the true value may lie. It is influenced by both the sample size and the variability within the data. The margin of error is inversely related to the square root of the sample size, meaning as sample size increases, the margin of error decreases.

The smaller the margin of error you can tolerate, the larger your sample size needs to be. In the exercise solution, a margin of error of .05 is set, reflecting a tolerable variance of 5 percentage points from the sampled proportion to the true population proportion. This choice is a balance between accuracy and the feasibility of surveying a large number of individuals. As sample size calculation demonstrates, using a smaller margin of error would necessitate a larger sample, so it must be carefully selected to meet the research's requirements without undue burden.