Question

Who Smokes More: Male Students or Female Students? Data 1.1 on page 4 includes lots of information on a sample of 362 college students. The complete dataset is available at StudentSurvey. We see that 27 of the 193 males in the sample smoke while 16 of the 169 females in the sample smoke. (a) What is the best point estimate for the difference in the proportion of smokers, using male proportion minus female proportion? Which gender smokes more in the sample? (b) Find and interpret a \(99 \%\) confidence interval for the difference in proportions.

Short answer

The best estimate for the difference in the proportion of smokers, using male proportion minus female proportion, is 0.045, suggesting more male students smoke. The 99% confidence interval for the difference in proportions is (0.001, 0.089)

Step-by-step solution

Calculate Proportions of Smokers in Each Group

First, the proportions of smokers within each group need to be calculated. This is done by dividing the number of smokers in each group by the total number of individuals in that group. For males, the proportion is calculated as \(27 ÷ 193 = 0.140\). For females, the proportion is \(16 ÷ 169 = 0.095\)

Calculate the Difference in Proportions

The difference in proportions (male proportion minus female proportion) is then calculated as \(0.140 - 0.095 = 0.045\)

Identify Which Gender Smokes More

By comparing the calculated proportions, it can be seen that the proportion of male students who smoke is greater than the proportion of female students who smoke in this sample.

Compute 99% Confidence Interval for the Difference in Proportions

The confidence interval can be calculated using the formula for the difference between two independent proportions. Here, z=2.58 for a 99% confidence level. Using the standard error for difference \(\sqrt{( p1*(1 - p1) / n1 ) + ( p2*(1 - p2) / n2 )}\), substituting appropriate values yields 0.023. The confidence interval is then found to be \[0.045 \pm 2.58 * 0.023\], which results in (0.001, 0.089).

Interpret the Confidence Interval

The 99% confidence interval suggests that, if many samples were taken and the 99% confidence interval computed for each sample, then 99% of those intervals would contain the true difference in smoking proportions between male and female students. In this case, we are 99% confident that the true difference in smoking proportions between male and female students in the population is between 0.001 and 0.089.

Key concepts

Point Estimate

When we talk about a point estimate in statistics, we refer to a single value that serves as the best guess or most plausible value of an unknown population parameter. It acts as a snapshot, providing a quick look at what the data seems to indicate about the population.

In the context of the exercise provided, the point estimate is concerned with the difference in smoking rates between two subsets of a population – male and female college students. It's determined by simply calculating the proportion of smokers in each group and taking the difference. The calculated point estimate tells us which group has a higher rate of smoking according to our sample. In this case, with males smoking at a rate of 0.140 and females at 0.095, males appear to smoke more with a point estimate difference of 0.045.

This type of analysis is crucial because it provides an immediate understanding of which direction the difference lies and how large that difference is in the sample. However, because it is based on a sample, it might not precisely represent the entire population. That's why it's important to also calculate a confidence interval, which we'll discuss next.

Confidence Interval

The term 'confidence interval' is a range of values, derived from the data, within which we can say with a certain level of confidence that the true population parameter lies. It is a measure of the reliability or precision of our point estimate – the wider the interval, the less precise our estimate, and vice versa.

In the exercise's step-by-step solution, you have already seen how to calculate a 99% confidence interval for the difference in proportions. This high level of confidence (99%) is chosen to reflect greater certainty. With the calculated interval ranging from 0.001 to 0.089, we're saying there's a 99% likelihood that the true difference in smoking rates between males and females in the general population falls somewhere within these two bounds.

It's crucial for students to understand that a confidence interval provides context for the point estimate. It does not simply tell us the range where the true difference is; it also indicates how confident we can be in our estimate. The interpretation given in the solution elaborates on this: If many samples were taken and the process repeated, the true difference would lie within these bounds 99% of the time. This concept is fundamental for drawing conclusions from sample data, as any single study could be an outlier due to chance.

Proportion Calculation

Proportion calculation is at the heart of many statistical analyses. It is the process of determining what fraction of a group has a particular characteristic. To calculate a proportion, you simply take the number of individuals with the characteristic and divide it by the total number in the group.

In our example, we calculate the proportion of smokers among males by dividing the number of male smokers (27) by the total number of males (193), yielding 0.140. We do the same for females to arrive at 0.095. By comparing these proportions, we gain insight into the behavior of the two groups within the sample.

Understanding how to calculate and interpret proportions is essential, as these values form the basis for further statistical methods, like hypothesis testing and the creation of confidence intervals. For example, if students need to analyze whether a new teaching method is more effective, they would first calculate the proportion of students who improved their grades under each method before making comparisons and conclusions. Therefore, proficiency in proportion calculation is a fundamental skill in the toolkit of anyone working with statistics.