Question

If random samples of the given sizes are drawn from populations with the given proportions: (a) Find the standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) Determine whether the sample sizes are large enough for the Central Limit Theorem to apply. Samples of size 40 from population \(A\) with proportion 0.30 and samples of size 30 from population \(B\) with proportion 0.24

Short answer

The standard error of the difference in sample proportions is to be calculated using the provided formula. The Central Limit theorem applies to this problem because both sample sizes are greater than 30.

Step-by-step solution

Calculate the Standard Error

The Standard Error (SE) of the difference in proportions can be calculated using the following formula: \nSE = \sqrt{\frac{{p_A*(1-p_A)}}{{n_A}} + \frac{{p_B*(1-p_B)}}{{n_B}}} \nwhere \(p_A\) and \(p_B\) are proportions of populations A and B respectively and \(n_A\) and \(n_B\) are sizes of samples drawn respectively. Using the given values, \nSE = \sqrt{\frac{{0.30*(1-0.30)}}{{40}} + \frac{{0.24*(1-0.24)}}{{30}}}

Check if the Central Limit Theorem applies

The Central Limit Theorem applies if the sample sizes n_A and n_B are both greater than 30. In this instance n_A = 40 and n_B=30.

Key concepts

Standard Error

In statistics, when dealing with the precision of sample estimates, we frequently refer to the standard error (SE). The standard error is a measure of the variability or dispersion of a sample statistic, like the mean or proportion, from the actual population parameter. To help you understand this, imagine you are aiming at a target. Each shot that's fired represents a sample estimate, and each shot likely lands in different spots around the center. The standard error tells us how close these shots (estimates) tend to group around the true value (center of the target).

The smaller the standard error, the closer our sample statistic is expected to be to the population parameter; conversely, a larger standard error indicates more variability and less precision. When comparing two sample proportions, such as \( \hat{p}_A - \hat{p}_B \), we calculate the standard error of the difference to understand the variability in that difference. The formula provided in the exercise, \( SE = \sqrt{\frac{{p_A*(1-p_A)}}{{n_A}} + \frac{{p_B*(1-p_B)}}{{n_B}}} \), encapsulates how both sample sizes and population proportions affect the standard error of the difference.

For the given exercise, using the formula, we compute SE to assess the variability between two independent sample proportions from different populations. Such calculations are crucial in inferential statistics as they underlie the confidence we can have in our conclusions about population differences based on sample data.

Sample Proportions

Sample proportions reflect the fraction of subjects in a sample that exhibit a particular trait or outcome. For example, if you're examining voter preferences in a political survey, the proportion of people favoring a certain candidate within the sample is a sample proportion. Think of it like slicing a pie; the sample proportion represents the size of one slice that matches your criteria, in relation to the entire pie (the sample).

In the context of the exercise, two different sample proportions are being compared: \( \hat{p}_A \) for the first sample from population A, and \( \hat{p}_B \) for the second sample from population B. The importance of sample proportions lies in their use as estimates for the true population proportions. When you're working with finite samples, you don't have full visibility of the entire population, so these sample proportions act as glimpses into the broader picture.

It's essential to use the proper formulae to calculate sample proportions and interpret them correctly. Small differences in proportions may reflect substantial shifts in opinions or behaviors when projected onto large populations, which is why statistical techniques, such as the Central Limit Theorem, are employed to analyze the larger implication of the findings.

Distribution of Differences

The concept of the distribution of differences arises when we compare two sample statistics, for instance, sample means or proportions. It refers to the expected pattern of variation we would observe in the differences between these statistics if we repeated our sampling process numerous times. This is like performing multiple experiments and tracking how much the outcomes vary from one another. It's not the differences in the individual samples themselves that we are interested in, but rather the differences in their estimates of the central tendency (like means or proportions).

In our case, we calculate the distribution of differences for sample proportions \( \hat{p}_A - \hat{p}_B \) to understand how much the proportion from Population A might differ from that of Population B. This is particularly informative in hypothesis testing or when making inferences about whether two populations differ significantly in certain traits or behaviors. The standard error calculated in the exercise is critical since it dictates how stretched or squeezed the distribution of differences will be.

Additionally, for the distribution of differences to be reliable and for us to apply the Central Limit Theorem confidently, our samples must meet certain size criteria, typically n > 30. As noted in the exercise, one of our samples is exactly at this threshold, meaning we should be cautious interpreting the results, as the CLT might be marginally satisfied, affecting our analysis's robustness.