5.7 Sampling Distributions for Sample Means

Formulas

For these problems you are given a sample mean. The average of sample means is equal to the given mean. 

Source: AP Statistics Formula Sheet

For the standard deviation, when you deal with sample statistics, you must use standard error (SE). This version of standard deviation, which is used for population proportions in this case, is another comparison that must be memorized just like the chart in section 5.1.

Source: The AP Statistics CED

Normal Condition: Central Limit Theorem

When you are working with numerical data and you want to estimate the mean of a population, you can use the sampling distribution of the sample mean (x̄) to make inferences about the population mean. However, before you can solve the problem, you must first assure the sampling distribution is normally distributed using the Central Limit Theorem. 😄

One important property of the sampling distribution of the sample mean is that it is approximately normal, provided the sample size is large enough. This means that even if the population distribution is not normal, the sampling distribution of the sample mean can be modeled using a normal distribution if the sample size is large enough.

The "large enough" sample size is often taken to be 30 or greater. This is known as the Central Limit Theorem, which states that the sampling distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the shape of the population distribution. ⬭

Practice Problem

Suppose that you are conducting a study to estimate the average income of small business owners in your state. You decide to use a simple random sample of 100 small business owners, and you collect data on their annual incomes. After analyzing the data, you find that the sample mean income is $50,000 per year with a standard deviation of$10,000. 💰

a) Explain what the sampling distribution for the sample mean represents and why it is useful in this situation.

b) Suppose that the true population mean income for small business owners in your state is actually $45,000 per year. Describe the shape, center, and spread of the sampling distribution for the sample mean in this case.

c) Explain why the Central Limit Theorem applies to the sampling distribution for the sample mean in this situation.

d) Discuss one potential source of bias that could affect the results of this study, and explain how it could influence the estimate of the population mean income.

Answer

a) The sampling distribution for the sample mean represents the distribution of possible values for the sample mean if the study were repeated many times. It is useful in this situation because it allows us to make inferences about the population mean based on the sample data.

b) If the true population mean income for small business owners in your state is $45,000 per year, the sampling distribution for the sample mean would be approximately normal with a center at$45,000 and a spread that depends on the sample size and the variability of the population.

c) The Central Limit Theorem applies to the sampling distribution for the sample mean in this situation because the sample size (n = 100) is large enough for the distribution to be approximately normal, even if the population is not normally distributed.

d) One potential source of bias in this study could be selection bias, which occurs when certain groups of individuals are more or less likely to be included in the sample. For example, if the sample is drawn from a list of small business owners who have registered with a particular organization, it could be biased toward those who are more financially successful or who are more likely to be active members of the organization. This could lead to an overestimate of the population mean income for small business owners in the state. 

On the other hand, if the sample is drawn from a list of small business owners who have applied for a particular loan program, it could be biased toward those who are less financially successful or who are more likely to be in need of financial assistance. This could lead to an underestimate of the population mean income.