Question

Americans are becoming more conscious of the importance of good nutrition, and some researchers believe that we may be altering our diets to include less red meat and more fruits and vegetables. To test this theory, a researcher decides to select hospital nutritional records for subjects surveyed 10 years ago and to compare the average amount of beef consumed per year to the amounts consumed by an equal number of subjects she will interview this year. She knows that the amount of beef consumed annually by Americans ranges from 0 to approximately 104 pounds. How many subjects should the researcher select for each group if she wishes to estimate the difference in the average annual per- capita beef consumption correct to within 5 pounds with \(99 \%\) confidence?

Short answer

Based on the given information and calculations, the researcher should select a sample size of 110 subjects for each group to estimate the difference in the average annual per-capita beef consumption correct to within 5 pounds with 99% confidence.

Step-by-step solution

Approximate the standard deviation

We will approximate the standard deviation by dividing the range of beef consumed annually (104 pounds) by 4. \(\sigma \approx \frac{104}{4} = 26\)

Find the critical z-score

The researcher wants to estimate the difference with \(99\%\) confidence. So, the critical z-score \(Z_{\alpha/2}\) can be found using the standard normal table or a calculator. For \(99\%\) confidence, \(\alpha = 1 - 0.99 = 0.01\). Thus, we need to find \(Z_{0.005}\) (since we are looking for a two-tailed score). \(Z_{\alpha/2} = Z_{0.005} = 2.576\)

Apply the formula for calculating sample size

Now we can apply the formula for estimating the sample size using the calculated values: \(n = \frac{(Z_{\alpha/2} * \sigma)^2}{E^2}\) \(n = \frac{(2.576 * 26)^2}{5^2}\) \(n \approx 109.75\) Since we cannot have a fraction of a subject, we round up to the nearest whole number: \(n = 110\)

Conclusion

The researcher should select 110 subjects for each group in order to estimate the difference in the average annual per-capita beef consumption correct to within 5 pounds with \(99 \%\) confidence.

Key concepts

Confidence Intervals

When researchers want to estimate a population parameter, such as an average or difference in averages, they often use a confidence interval. A confidence interval is a range of values, derived from sample data, that likely contains the true value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval.

For instance, if a researcher wishes to estimate the average beef consumption with a 99% confidence interval, this means that, theoretically, if the study were to be repeated multiple times, 99% of the intervals calculated from those studies would contain the true average beef consumption. The 99% confidence interval is wider than, say, a 95% confidence interval, reflecting the higher certainty required and consequently a larger sample size to reduce the margin of error.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of sample size determination, standard deviation plays a pivotal role because it influences the width of the confidence interval. The larger the standard deviation, the more scattered the data is, and the greater the sample size needed to estimate the population parameter with a specified level of confidence and precision. The researcher in our exercise has approximated the standard deviation using the range of beef consumption, which is a rough estimate considering the actual data variability could be different. A more accurate standard deviation, if known, could potentially lead to a different sample size requirement.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the bell-shaped curve of a normal distribution, the total area under the curve is equal to 1, and the mean, median, and mode of the distribution are all equal.

This distribution is particularly important in statistics because of the central limit theorem, which states that, under many conditions, independent random variables summed together will converge to a normal distribution regardless of the original distribution of the variables. This makes the normal distribution a fundamental concept in calculating confidence intervals and sample sizes, as we usually assume that the sample means will be normally distributed around the population mean. In our exercise, the researcher uses the properties of the normal distribution to determine the critical z-score, which is necessary to compute the confidence interval and sample size.