Question

Use StatKey or other technology to generate a bootstrap distribution of sample proportions and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample proportion as an estimate of the population proportion \(p\). Proportion of home team wins in soccer, with \(n=120\) and \(\hat{p}=0.583\)

Short answer

The short answer will depend on the actual data obtained in Step 1. Therefore, it can't be provided without the bootstrap distribution. However, the process will provide an understanding of whether the theoretical assumptions of standard error made by the Central Limit Theorem hold true for the given data.

Step-by-step solution

Calculate Bootstrap Standard Error

First, generate a bootstrap distribution of sample proportions. This process will be done by resampling the given data numerous times, and each time calculating the sample proportion. Collect all these new proportions in order to form the bootstrap distribution. Once done, calculate the standard deviation of this bootstrap distribution, which serves as the estimated standard error.

Calculate Theoretical Standard Error

Use the Central Limit Theorem (CLT) to calculate the theoretical standard error. According to the CLT, the standard error is calculated as \(SE = \sqrt{\dfrac{\hat{p}(1 - \hat{p})}{n}}\), where \(n\) is the sample size and \(\hat{p}\) is the sample proportion. Substituting \(n = 120\) and \(\hat{p} = 0.583\) into the formula, solve for \(SE\).

Compare Standard Errors

Compare the bootstrap standard error from Step 1 with the theoretical standard error from Step 2. The comparison will give an insight into how well the results from the actual data align with the theoretical model.

Key concepts

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that describes how the distribution of sample means becomes increasingly normal as the sample size grows, regardless of the population's distribution. When students deal with real-world data, the CLT allows them to make inferences about population parameters using sample statistics.

Consider an example where a student is interested in the proportion of home team wins in soccer. If they take multiple samples, the means of those samples tend to form a normal distribution according to the CLT. This is especially helpful because normal distributions have well-understood properties, allowing us to make predictions and construct confidence intervals. For a large enough sample size, the sample mean will approximate a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. The formula given by the CLT to calculate standard error of the sample proportion is an essential tool for students to understand the variability of their estimates.

Standard Error

The standard error (SE) of a statistic is a measure of the variability or spread of that statistic if we were to take many samples from a population. It is crucial for interpreting the accuracy of an estimate from a single sample. In the context of our soccer wins example, the standard error gives an indication of how much the sample proportion of home team wins (hat{p}) might vary from sample to sample.

To calculate the standard error according to the Central Limit Theorem, one can use the formula SE = sqrt{frac{hat{p}(1 - hat{p})}{n}}, where n is the sample size. This formula takes into account the observed sample proportion and the size of the sample to provide an estimate of the standard error. Understanding this concept is essential since the SE is often used in constructing the confidence intervals and it gives a sense of how precise our estimate of the population parameter is.

Sample Proportion

Sample proportion (hat{p}) is a statistic that estimates the true proportion (p) of a certain characteristic within a population based just on a sample. In research studies or exercises like the example given, using the sample proportion is practical and common because it's not always feasible to gather data from an entire population. For instance, a sample of 120 soccer games was used to estimate the proportion (hat{p}) of home team wins.

The sample proportion is calculated by dividing the number of times the characteristic of interest is observed by the total number of observations in the sample. When calculating the bootstrap distribution, this sample proportion is used repeatedly as we simulate resampling from the population to create numerous new samples. This demonstrates the variability of the sample proportion and how it can serve as an estimate for the population proportion. Accurate use and interpretation of the sample proportion are essentials skills for students, as it can affect conclusions about the population based on their collected data.