Question

Catching a Cold Do well-rounded people get fewer colds? A study in the Chronicle of Higher Education found that people who have only a few social outlets get more colds than those who are involved in a variety of social activities. \({ }^{21}\) Suppose that of the 276 healthy men and women tested, \(n_{1}=96\) had only a few social outlets and \(n_{2}=105\) were busy with six or more activities. When these people were exposed to a cold virus, the following results were observed: \begin{tabular}{lcc} \hline & Few Social Outlets & Many Social Outlets \\ \hline Sample Size & 96 & 105 \\ Percent with Colds & \(62 \%\) & \(35 \%\) \\ \hline \end{tabular} a. Construct a \(99 \%\) confidence interval for the difference in the two population proportions. b. Does there appear to be a difference in the population proportions for the two groups? c. You might think that coming into contact with more people would lead to more colds, but the data show the opposite effect. How can you explain this unexpected finding?

Short answer

Explain. Yes, there is a significant difference between the proportions of people getting colds in both groups. The 99% confidence interval for the difference in population proportions is (22.48, 23.05), which does not include 0. This suggests that there is a significant difference in the population proportions for the two groups, with the group with many social outlets having a lower proportion of people getting colds compared to the group with few social outlets. A possible explanation could be that increased exposure to various germs and bacteria through social interaction helps build immunity to common colds, though further research would be needed to confirm this hypothesis.

Step-by-step solution

Calculate Sample Proportions and Standard Deviation

For this, we'll use the number of people in each group, and the given percentages. Sample proportion for the first group with few social outlets: $$p_1 = \frac{62}{100} \times 96 = 59.52$$ Sample proportion for the second group with many social outlets: $$p_2 = \frac{35}{100} \times 105 = 36.75$$ Now, let's calculate the standard deviation of the difference between these sample proportions: $$s_p = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$$ Plugging in the numbers, we get $$s_p = \sqrt{\frac{59.52(1 - 59.52)}{96} + \frac{36.75(1 - 36.75)}{105}}$$ $$s_p \approx 0.110$$

Construct the 99% Confidence Interval

We are going to use the normal distribution to build the confidence interval. Following the formula for the difference between the two population proportions, we have: $$CI = (p_1 - p_2) \pm z \times s_p$$ We are looking for a 99% confidence interval, which means our z-value from a table or calculator is 2.576. Using these numbers, we get: $$CI = (59.52 - 36.75) \pm 2.576 \times 0.110$$ $$CI = 22.77 \pm 0.283$$ So, our 99% confidence interval is approximately \((22.48, 23.05)\).

Analyze the results

Now we have the confidence interval, let's analyze the results: a. Our confidence interval for the difference in population proportions is \((22.48, 23.05)\). b. Since the confidence interval does not include 0, there appears to be a significant difference in the population proportions for the two groups. c. A possible explanation for the unexpected finding could be that people with more social interactions have a better immune system due to the exposure to various germs and bacteria in small doses, helping them build immunity to common colds. However, this is just a hypothesis and would need further research to support it.

Key concepts

Population Proportion

When we discuss 'population proportion,' we refer to the percentage of individuals in a population who exhibit a particular trait or characteristic. In the context of the given exercise, two population proportions were examined: the proportion of healthy individuals engaging in few social outlets who contracted colds, and the proportion of those involved in many social activities who caught colds. This allowed researchers to compare the likelihood of getting a cold between these two distinct groups, giving insights into how social habits can impact health.

To be specific, population proportion is represented by the 'p' in our statistical formulas, and in our exercise, we had two of these proportions: for the fewer social outlets (\( p_1 \) and for the many social outlets (\( p_2 \) groups. Accurately estimating these proportions is critical since they form the basis for constructing the confidence interval and testing statistical significance.

Sample Size

The 'sample size' is the number of individuals included in a sample and is denoted by 'n.' It is a crucial factor in determining the precision of population proportion estimates. With a larger sample size, we can be more confident that our sample proportion (\( \bar{x} \) is closer to the true population proportion (\( p \). In our study, sample sizes of 96 and 105 were considered sufficient to detect a difference in the proportion of those getting colds based on the number of social activities individuals are involved in.

A larger sample size can also decrease the margin of error in the confidence interval, making it narrower and thus providing a more accurate estimate of the population parameter. However, increasing sample size comes with cost and time implications, which researchers must balance against the benefits of increased precision.

Normal Distribution

The assumption of a 'normal distribution' underlies many statistical methods, including the construction of confidence intervals for population proportions. A normal distribution is a bell-shaped curve that is symmetric about the mean, and most values cluster around the central peak, tapering off symmetrically towards the extremes.

In the exercise, the normal distribution is used to approximate the distribution of the sample proportions when constructing the confidence interval because it allows us to use the standard normal 'z' values to multiply by our standard deviation. This approach is appropriate when sample sizes are large enough, employing the Central Limit Theorem, which states that the sampling distribution becomes approximately normal irrespective of the original distribution as the sample size increases.

Standard Deviation

In statistics, 'standard deviation' measures the amount of variation or dispersion from the average. When dealing with proportions, it tells us how much we can expect the sample proportion to differ from the true population proportion purely due to random sampling variability.

In our cold study, the standard deviation helps us quantify the uncertainty associated with the difference between the two sample proportions (\( p_1 - p_2 \). In turn, this is a key component in calculating the confidence interval. The smaller the standard deviation, the tighter our confidence interval will be. This illustrates that our estimates are more precise, whereas a larger standard deviation implies a greater degree of uncertainty about our estimate.

Statistical Significance

The concept of 'statistical significance' pertains to the likelihood that the observed differences or relationships in the data are not due to random chance. Testing for statistical significance involves determining if an observed effect is likely to be present in the population from which the sample was drawn.

In the context of the given problem, the difference between the two sample proportions was evaluated for significance using a confidence interval approach. Because the 99% confidence interval for the difference in population proportions did not include zero, the result was considered statistically significant, suggesting that the difference observed in the samples likely reflects a real difference in the population. This indicates that the number of social activities may have a genuine effect on the likelihood of catching a cold.