91Ó°ÊÓ

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?

Step by step solution

01

Identify the Formula

The formula to calculate the sample size when estimating proportions is given by: \[n = (\frac{Z}{E})^2 * \pi * (1-\pi)\]. Where: \(n\) is the required sample size, \(Z\) is the z-score, \(E\) is the margin of error, and \(\pi\) is the estimated population proportion.

02

Identify the Given Data

The z-value corresponding to a 95% confidence level is 1.96 (from a standard Z-table). The preliminary estimate for the proportion \(\pi\) is 0.27 and the margin of error \(E\) is 0.05.

03

Calculate the Sample Size for \(\pi = 0.27\)

Substitute the values into the formula and calculate: \[n = (\frac{1.96}{0.05})^2 * 0.27 * (1-0.27)\] You get \(n \approx 1018\).

04

Calculate the Sample Size for \(\pi = 0.5\)

Substitute the values into the formula and calculate: \[n = (\frac{1.96}{0.05})^2 * 0.5 * (1-0.5)\] You get \(n \approx 1537\).

05

Compare the Two Calculated Sample Sizes

The sample size estimated using a population proportion of 0.27 is smaller than that estimated using a population proportion of 0.5. The larger sample size estimated using a population proportion of 0.5 is a conservative estimate implying it will provide a more accurate survey result.

06

Decide on the Recommended Sample Size

When the actual proportion is unknown and a preliminary estimate is available, it should typically be used. However, if not given or if the more conservative estimate is preferred, the consequence is a larger sample size. Therefore, the choice depends on factors such as resources available for the survey. In this case though, due to a greater level of accuracy and because only a modest increase in sample size is required, a sample size of around 1537 using a population proportion of 0.5 might be recommended.

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Key Concepts

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

Confidence Interval

A confidence interval gives us a range within which we may expect the true value of a parameter, like a population proportion, to lie. It’s not a guarantee but a statistical probability. For instance, if we have a 95% confidence interval, we’re essentially saying we're fairly confident that the true value will fall within that range 95 times out of 100 if we were to repeat our study.

The width of this interval is affected by our chosen level of confidence and sample size. More confidence would usually need a larger sample to maintain the same margin of error, while a smaller sample size usually comes with a less precise estimate, reflected in a wider confidence interval.

Margin of Error

The margin of error tells us how much we expect our survey results to differ from the actual population value. It is fundamentally linked with the confidence level. In common practice, a margin of error of 0.05 (or 5%) is typical, as seen in the given exercise. This means we allow our estimate to be off by as much as 5% in either direction.

When you have a smaller margin of error, your confidence interval becomes narrower, indicating that your estimate is more precise. Conversely, a larger margin of error makes for a wider confidence interval and a less precise estimate, but it may reduce the required sample size.

Population Proportion

The population proportion, denoted by \( \pi \), is a measure of frequency - how often something occurs in a population. When we can't survey every member, we estimate this value based on a sample. The accuracy of our estimation doesn’t just depend on sample size but also on how close the proportion is to 0 or 1.

In cases where we don't have an estimated proportion or where it's possibly misleading, we use the conservative value of 0.5, which, mathematically, always gives the largest sample size. This ensures that regardless of what the true proportion might be, our sample size will be adequate to estimate it within our desired margin of error.

Z-Score

The z-score serves as a measurement of how far away a point is from the mean, in the context of standard deviations. In terms of a confidence interval, the z-score is tied to our level of confidence. A 95% confidence level corresponds to a z-score of approximately 1.96.

As the confidence level increases, so does the z-score, leading to more extensive sampling requirements to achieve the same margin of error. Z-scores are derived from the z-table, which quantifies the probability associated with each score. Using these scores in sample size calculations helps ensure that the findings are statistically significant to the desired confidence level.