163 Distribution of Sample Means (3 of 4)
Learning Objectives
- Describe the sampling distribution of sample means.
Shape of the Sampling Distribution of Means
Now we investigate the shape of the sampling distribution of sample means. When we discussed the sampling distribution of sample proportions, we learned that this distribution is approximately normal if np ≥ 10 and n(1 – p) ≥ 10. In other words, we had a guideline based on sample size for determining the conditions under which we could use a normal curve to do probability calculations for sample proportions.
Now we investigate these questions:
- When will the distribution of sample means be approximately normal?
- Does it depend on the size of the sample?
- What happens if the distribution of the variable in the population is heavily skewed?
The following simulation video helps us investigate these questions.
WalkThrough Simulation
Learn By Doing
Comment
Are you surprised that a variable with a skewed distribution in the population can have a sampling distribution that is approximately normal? This discovery is probably the single most important result presented in introductory statistics courses. It is called the central limit theorem, which says that for large samples, the sampling distribution of sample means is approximately normal. This theorum is important! Inference procedures, such as hypothesis tests and confidence intervals, are based on a normal model for the sampling distribution. The central limit theorem assures us that we can use a normal probability model for sample means without knowing anything about the shape of the distribution of the variable in the population. All we have to do is collect large samples.
How large a sample size do we need to assume that sample means will be normally distributed? It really depends on the population distribution, as we saw in the simulation. The more skewed the distribution in the population, the larger the samples we need in order to use a normal model for the sampling distribution.
The general guideline is that samples of size greater than 30 will have a fairly normal distribution regardless of the shape of the distribution of the variable in the population. But if a population is strongly skewed, it is safer to use larger samples.
Learn By Doing
The distribution of incomes is strongly skewed to the right for individuals in the U.S. The following histograms represent mean income from 200 samples randomly selected from the U.S. population. One histogram is based on samples of size of n = 4, one on samples of size of n = 40, and one on samples of size of n = 100.
https://assessments.lumenlearning.com/assessments/3680
Summary
- Let’s say we have a quantitative data set from a population with mean μ and standard deviation σ.The model for the theoretical sampling distribution of means of all random samples of size n has the following properties:
- The mean of the sampling distribution of means is μ.
- The standard deviation of the sampling