H2 Maths 6.4 - Sample Mean Distribution and CLT

Step 1 - Population versus sample

We start with a population - every possible value of a quantity - with true mean mu and variance sigma squared.

Step 1 - Population versus sample

Because surveying the whole population is impractical, we draw a simple random sample of size n.

Step 1 - Population versus sample

The sample gives us two key statistics: the sample mean x-bar and the unbiased sample variance s squared.

Step 2 - Sample mean as a random variable

Here is the key idea: treat the sample mean capital X-bar as a random variable, not just a single computed number.

Step 2 - Sample mean as a random variable

If the individual observations have mean mu and variance sigma squared, then the expected value of X-bar equals mu, and the variance of X-bar equals sigma squared over n.

Step 2 - Sample mean as a random variable

Larger samples reduce the spread of X-bar - the standard error shrinks as root n grows.

Step 3 - The Central Limit Theorem

If the parent population is already normal, X-bar is exactly normal for any sample size.

Step 3 - The Central Limit Theorem

But the Central Limit Theorem says: even when the parent population is not normal, X-bar is approximately normal once n is large enough - typically n greater than or equal to 30.

Step 3 - The Central Limit Theorem

This is what makes the normal model so useful in hypothesis testing.

Step 4 - Standardise and find a probability

JC students have revision hours with mean 6.4 and standard deviation 1.8.

Step 4 - Standardise and find a probability

For a random sample of 36 students, find the probability that the sample mean exceeds 7 hours.

Step 4 - Standardise and find a probability

First write the model line, then standardise to get a Z-score, then read the probability from the normal table.

Step 5 - Unbiased estimates from summarised data

Exams often give summarised data - the sum of shifted values - rather than the raw observations.

Step 5 - Unbiased estimates from summarised data

In a sample of 40 students, the sums of x minus 8 and the sums of x minus 8 squared are given as 16 and 92.

Step 5 - Unbiased estimates from summarised data

Undo the shift to recover x-bar, then use the standard s-squared formula with divisor n minus 1.