1. Section 8.2: The Sampling Distribution of a Sample Mean

2. General Properties of the Sampling Distribution of Mean
Let xbar denote the mean of the observations in a random sample of size n from a population having mean μ and standard deviation σ. Then the following rules hold:

3. Rule 3: When the population distribution is normal, the sampling distribution of mean is also normal for any sample size n.
Rule 4: (Central Limit Theorem): When n is sufficiently large, the sampling distribution of mean is well approximated by a normal curve, even when the population distribution is not itself normal. (Can be safely applied if n exceeds 30)

5. Example
Male bluethroats have a complex song, which is thought to be used to attract female birds. Let x denote the duration of a randomly selected song (in seconds) for a male bluethroat. Suppose that the mean value of song duration is μ = 13.8 sec and that the standard deviation of a song duration is σ = 11.8 sec.

6. The authors of the paper noted that the song length distribution is not normal. The sampling distribution of mean based on a random sample of n = 25 song durations then also has mean value of 13.8 sec.
That is the sampling distribution of mean is centered at 13.8.

7. The standard deviation of xbar is

8. Because the population distribution is not normal and because the sample size is not larger than 30, we cannot assume that the sampling distribution of mean is normal in shape.