Sample means, standard errors and the central limit theorem.
Example 1
Independent observations have mean 20 and standard deviation 32. A random sample has size 64. Find the standard deviation of the sample mean.
Show worked answer
Answer: 4
- Standard error =√nσ.
- SE=√6432=4.
Example 2
A normal population has mean 40 and standard deviation 56. For independent samples of size 64, find P(sample mean >56.8). Give your answer to 4 decimal places.
Show worked answer
Answer: 0.0082
- The sample mean is normal with standard error √6456=7.
- Standardise the sample mean: z=2.4.
- Find 1−Φ(2.4).
- Rounded answer: 0.0082.
Example 3
A population is not normal but has finite variance. For sufficiently large independent random samples of size 120, which quantities have an approximately normal distribution under the central limit theorem?
Show worked answer
Answer: Sample means from repeated sampling
- The theorem concerns averages over repeated samples of the stated size.
- It does not turn the original observations into a normal population, and the sample size is fixed.