Question

We saw in Exercise 6.221 on page 466 that drinking tea appears to offer a strong boost to the immune system. In a study extending the results of the study described in that exercise, \(^{58}\) blood samples were taken on five participants before and after one week of drinking about five cups of tea a day (the participants did not drink tea before the study started). The before and after blood samples were exposed to e. coli bacteria, and production of interferon gamma, a molecule that fights bacteria, viruses, and tumors, was measured. Mean production went from \(155 \mathrm{pg} / \mathrm{mL}\) before tea drinking to \(448 \mathrm{pg} / \mathrm{mL}\) after tea drinking. The mean difference for the five subjects is \(293 \mathrm{pg} / \mathrm{mL}\) with a standard deviation in the differences of \(242 .\) The paper implies that the use of the t-distribution is appropriate. (a) Why is it appropriate to use paired data in this analysis? (b) Find and interpret a \(90 \%\) confidence interval for the mean increase in production of interferon gamma after drinking tea for one week.

Short answer

The paired data is appropriate in this case since we are observing the same subjects before and after the experiment and thus the 'before' and 'after' data are dependent on each other. The \(90\%\) confidence interval for the mean increase in interferon gamma after drinking tea for one week is \(64 \mathrm{pg} /\mathrm{mL}\) to \(522 \mathrm{pg} /\mathrm{mL}\). This means we are \(90\%\) confident that the true mean increase in interferon gamma production due to tea drinking for a week in the entire population of similar individuals would be in this interval.

Step-by-step solution

Understanding why paired data is appropriate

In this experiment, the same subjects are being tested before and after drinking tea. The subjects are the same, so their immune responses may be affected by many factors that are specific to them, like their age, health condition, lifestyle etc. Thus, each 'before' data has its unique corresponding 'after' data, and the two pieces of data are dependent. Hence, using paired data for analysis is appropriate for this experiment.

Calculating the margins of error

The margin of error for a given confidence level can be calculated using the formula: \[E = t * \frac{s}{\sqrt{n}}\] where: -s is the standard deviation of the differences, which is \(242\) -n is the number of subjects, which is \(5\) -t is the t-score, which corresponds to the desired confidence level in the t-distribution table. For a \(90\%\) confidence level and \(5-1 = 4\) degrees of freedom, the t-score is approximatively \(2.132\). Thus, the margin of error E is approximately \(2.132 * \frac{242}{\sqrt{5}} = 229.0\)

Calculating the confidence interval

Using the calculated Margin of Error (E), the confidence interval is: \[\text{CI} = \text{mean difference} \pm E\] The mean difference is given as \(293 \mathrm{pg} /\mathrm{mL}\), and E is approximated as \(229\). Thus the \(90\%\) confidence interval is: \[\text{CI} = 293 \pm 229\] So, the \(90\%\) confidence interval is: \[\text{CI} = [64, 522]\]

Interpreting the confidence interval

The \(90\%\) confidence interval for the mean increase in production of interferon gamma after one week of drinking tea is from \(64 \mathrm{pg} /\mathrm{mL}\) to \(522 \mathrm{pg} /\mathrm{mL}\). This means that we are \(90\%\) sure that the true mean difference in interferon gamma production (after - before) for the entire population of individuals similar to those in the study is somewhere between \(64 \mathrm{pg} /\mathrm{mL}\) and \(522 \mathrm{pg} /\mathrm{mL}\).

Key concepts

Confidence Interval

When researchers conduct experiments and collect data, they often need to make inferences about a population based on a sample. A confidence interval gives a range of values that likely contain the unknown parameter of the population, like a mean or a difference in means. It's constructed from the sample data and provides an estimate of the parameter that includes a level of certainty—or confidence—often expressed as a percentage, such as 90%, 95%, or 99%.

In our tea-drinking study, we analyzed the impact of tea on immune response by measuring interferon gamma production before and after tea consumption in a small group. We used the interval to estimate the mean increase of interferon gamma across the population from which our sample of tea drinkers was drawn. It's important to remember that the confidence interval is not a definitive range but rather indicates the reliability of our estimate. A 90% confidence interval suggests there is a 90% chance that the interval calculated from our sample data contains the true mean difference for the entire population.

This is vital information for biomedical research where identifying substances that significantly impact health measurements like interferon gamma production can lead to important health recommendations or further studies.

t-Distribution

The t-distribution, also known as Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped, like the standard normal distribution. However, it is more spread out, having thicker tails, which account for the additional variability that's present when sample sizes are small.

As the sample size increases, the t-distribution approaches the normal distribution. This property makes it extremely useful for analyzing situations where the sample size is small and the population standard deviation is unknown, as is the case in our study of tea drinkers. Since we only have 5 participants, and we do not know the population standard deviation of interferon gamma production changes, we rely on the t-distribution to estimate the true mean change.

The degrees of freedom (df), which in this case are the number of subjects minus one (df = n - 1), determine the exact shape of the t-distribution. With more degrees of freedom, the t-distribution looks more like the standard normal distribution. This factor is also used when determining the t-score to calculate our confidence interval.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation means that the values are spread out over a wider range.

In the context of paired data analysis, as with our tea study, standard deviation plays a crucial role. It measures how much the differences within each pair (before and after measurements) deviate from the mean difference. This gives us insight into the consistency of the effect of tea on interferon gamma production among the participants.

In our solution, we calculated the standard deviation of the differences, not just the standard deviation of before or after measurements, because we are interested in the variability of changes stimulated by tea drinking, which is reflective of the paired nature of our data.