Objective Basic Principles Examples
The Central Limit Theorem module targets the following cognitive tasks:
| Task Skills | Concepts | |
|---|---|---|
| CLT-1: | Understand the sampling distribution of sample means | |
| CLT-2: | Compute the mean of the sampling distribution of sample means | Understand the mean of the sampling distribution of sample means |
| CLT-3: | Compute the standard deviation of the sampling distribution of sample means | Understand the standard deviation of the sampling distribution of sample means |
| CLT-4: | Describe the shape of the sampling distribution of sample means | Understand the shape of the sampling distribution of sample means |
| CLT-5: | Use the CLT to answer probability questions involving the sample mean | Understand how the CLT can be used to answer probability questions involving the sample mean |
| CLT-6: | Understand how the sample size affects the standard error | |
| CLT-7: | Understand why large sample sizes are desirable |
If the parent distribution is normal, the sampling distribution of the mean is normal with the same mean and a standard deviation that is reduced by a factor of the square root of n (the sample size) relative to the parent distribution. Remarkably, this conclusion holds approximately even if the parent distribution is not normal. This latter result is called the Central Limit Theorem.
Example #1 illustrates the central limit theorem when the underlying distribution is normal or chi-square.
Example #2 (Geometric Parent Distribution) illustrates the central limit theorem when the underlying distribution is uniform, bowtie, right wedge, left wedge, and triangular.
Self-test