Question

A small amount of the trace element selenium, \(50-200\) micrograms ( \(\mu \mathrm{g}\) ) per day, is considered essential to good health. Suppose that a random sample of 30 adults was selected from each of two regions of the United States and that each person's daily selenium intake was recorded. The means and standard deviations of the selenium daily intakes for the two groups are shown in the table. Find a \(95 \%\) confidence interval for the difference in the mean selenium intakes for the two regions. Interpret this interval.

Short answer

Answer: The confidence interval represents the range in which we can reasonably expect the true difference of the mean selenium intakes between the two regions to be, with a 95% confidence level. It can help us determine whether there is a significant difference in mean selenium intakes between the two regions.

Step-by-step solution

Gather the necessary data

In order to calculate the 95% confidence interval for the difference in mean selenium intakes for the two regions, we need the following data: - Sample means for both regions (\(\bar{x_1}\) and \(\bar{x_2}\)) - Sample standard deviations for both regions (\(s_1\) and \(s_2\)) - Sample size (30 adults in each region, so \(n_1 = n_2 = 30\)) - Z-score for 95% confidence level (\(z = 1.96\))

Plug the values into the formula

Now we need to plug the gathered data into the confidence interval formula: $$ CI = (\bar{x_1} - \bar{x_2}) \pm 1.96 * \sqrt{\frac{s_1^2}{30} + \frac{s_2^2}{30}} $$

Solve for the confidence interval

After plugging in the values, solve for the confidence interval. This will give us the lower and upper bounds of the 95% confidence interval for the difference in mean selenium intakes for the two regions.

Interpret the confidence interval

The confidence interval shows the range in which we can reasonably expect the true difference of the mean selenium intakes between the two regions to be, with a 95% confidence level. If the interval contains zero, it might indicate that there is no significant difference between the mean selenium intakes of the two regions. If the interval doesn't contain zero, it means that there is a significant difference between the two population means. This can provide insights into whether the selenium intake levels in these two regions are significantly different or not.

Key concepts

Statistics Education

Understanding the basic principles of statistics is essential for interpreting data and making informed decisions. One fundamental concept in statistics is the confidence interval, which quantifies the uncertainty associated with an estimate. To grasp this idea, one should start by familiarizing themselves with terms such as sample mean, which represents the average of a set of observations, and standard deviation, which measures the amount of variation or dispersion from the average.

When teaching statistical methods, it's vital to ensure that students not only learn how to calculate these measures but also understand their meanings. For example, in exercises involving health-related data, like daily selenium intake, context is crucial. Educators can highlight the real-world significance of such analyses, emphasizing how they can impact public health decisions. This approach can help students see the relevance of statistics in their daily lives and foster a deeper engagement with the material.

Confidence Interval Interpretation

Interpreting a confidence interval (CI) involves understanding what it can tell us about a population based on sample data. A CI for the difference between two means, as seen in the selenium intake problem, gives us a range where the true difference likely falls. However, it's not just about the numbers. The confidence level, in this case, 95%, indicates that if we were to take many samples and build intervals from them, 95% of these intervals would contain the true difference.

It's important to explain to students that a CI includes a margin of error, represented by the term \(1.96 \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\) in the selenium example. This margin helps account for the natural variability in the data. If a CI contains zero, as mentioned in the problem's solution, the difference might not be statistically significant – meaning that any observed difference could simply be due to sampling variation rather than a true difference in the populations.

Sample Mean and Standard Deviation

In research and statistics, the terms sample mean and standard deviation are closely related and key to understanding data sets. The sample mean \(\bar{x}\) is calculated by adding all the numbers in a data set and dividing by the number of observations. It is a central measure that represents the 'average' value.

The standard deviation \(s\), on the other hand, is a measure of variability. It indicates how much the values in the sample differ from the mean. In the case of the selenium intake exercise, we have two sample means and standard deviations to compare – one for each region. Understanding these concepts is crucial for analyzing whether the daily selenium intake differs significantly between regions, or if it can be attributed to natural fluctuations within each sample. Educators should stress that higher standard deviations indicate more spread in the data, which can impact the width of the confidence interval and the conclusions drawn from it.