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 100 from population \(A\) with proportion 0.20 and samples of size 50 from population \(B\) with proportion 0.30

Short answer

The standard error of the distribution of differences in sample proportions is approximately 0.07. Yes, the sample sizes are large enough for the Central Limit Theorem to apply.

Step-by-step solution

Finding the Standard Error

To start, the formula for the standard error of two proportions is formulated as follows: \(\sigma_{\hat{p}_{A}-\hat{p}_{B}}=\sqrt{\frac{{p_{A}(1-p_{A})}}{{n_{A}}}} + \frac{{p_{B}(1-p_{B})}}{{n_{B}}}}\). Here, \(p_{A}\) and \(p_{B}\) are the proportions in the two populations and \(n_{A}\) and \(n_{B}\) are the sizes of the samples drawn from the two populations. Substituting the given values into this formula will yield the standard error.

Calculation

Substituting given values, we get \(\sigma_{\hat{p}_{A}-\hat{p}_{B}}=\sqrt{\frac{{0.20(1-0.20)}}{{100}} + \frac{{0.30(1-0.30)}}{{50}}}\). This simplifies to \(\sigma_{\hat{p}_{A}-\hat{p}_{B}}=\sqrt{\frac{{0.16}}{{100}} + \frac{{0.21}}{{50}}}\), and further simplification gives the answer for the standard error.

Apply the Central Limit Theorem

The Central Limit Theorem can be applied if both \(n_{A}p_{A}\) and \(n_{B}p_{B}\) are greater than or equal to 5. In this case, \(n_{A}p_{A}=100*0.20=20\) and \(n_{B}p_{B}=50*0.30=15\). Both results are greater than 5, hence, the conditions for the Central Limit Theorem are satisfied, and it can be applied here.

Key concepts

Standard Error

When dealing with statistics, one crucial concept to grasp is standard error. It measures the amount of variability or dispersion in a sampling distribution. Specifically, in the context of differences between sample proportions, the standard error quantifies the expected distribution of these differences if you were to repeat the sampling process many times. Think of it as a gauge of how much the statistic you calculate from a sample, like a sample mean or proportion, would differ from the true population value.

To calculate the standard error for the difference between two sample proportions, denoted as \(\sigma_{\hat{p}_{A}-\hat{p}_{B}}\), you virtually blend the formulas for the standard error of both samples. The formula is \(\sigma_{\hat{p}_{A}-\hat{p}_{B}}=\sqrt{\frac{{p_{A}(1-p_{A})}}{{n_{A}}} + \frac{{p_{B}(1-p_{B})}}{{n_{B}}}}\), where \(p_{A}\) and \(p_{B}\) represent the known proportions in the populations, and \(n_{A}\) and \(n_{B}\) are the sample sizes. The square root encapsulates both variability and sample size, reflecting how they influence the precision of the sample proportions.

Distribution of Sample Proportions

The distribution of sample proportions describes how the proportion values are spread out for all possible random samples of a certain size from a population. It's like creating a map of all the outcomes you might expect if you took many, many samples. This distribution tends to form a shape that is approximately normal (bell-shaped) when the sample size is large enough, which is particularly handy as it allows statisticians to use the normal distribution's properties to make inferences about the population.

In real-world scenarios, you rarely have the luxury of polling an entire population. Therefore, when you select a sample and calculate its proportion, this acts as an estimate for the population proportion. The central limit theorem reassures us that with a big enough sample size, these estimates become more reliable. The distribution's standard deviation, or its standard error, helps determine how much the sample proportion will likely differ from the actual population proportion.

Sampling Distributions

The foundational bedrock for many statistical inference techniques is the concept of sampling distributions. This concept imagines what would happen if you took every possible sample of a particular size from the population and then mapped out the frequency of the sample statistics. The sampling distribution represents the probability distribution of a given statistic based on a random sample and it is crucial in the field of inferential statistics.

The beauty of the sampling distribution lies in its predictable nature, as dictated by the central limit theorem. This theorem states that no matter what the shape of the original population distribution is, the sampling distribution of the sample mean (or proportion) will approximate a normal distribution if the sample size is large enough. This property holds true as long as the samples are independent and identically distributed (i.i.d.). A practical use of the sampling distribution is it allows the calculation of confidence intervals, which are ranges within which the true population parameter is likely to lie. With larger samples, the wider range of possible sample means (or proportions) narrows down, making your estimate more precise.