1. SAMPLING DISTRIBUTIONS. SAMPLING VARIABILITY
COMMON NEED: estimate population parameters: mean or proportion or spread
EXAMPLE. Producers of clothing need to know average dimensions of human body and their variability. Food industry needs to know average consumption of different foods, etc.
Process: Collect sample (info) compute relevant statistic make inference about the parameter. For example, use to approximate μ.
Question. If we collect 2 different samples, are we likely to get the same sample means?
Sample 1: sample mean:
likely
different
Sample 2: sample mean:
Statistics from different samples have different values, i.e. they vary from sample to sample!

2. SAMPLING DISTRIBUTIONS, CONTD.
Values of a statistic computed from different samples will be different.
E.g. values of computed using different samples will be different.
Question: If we repeat sampling many times, what distribution of these sample means would we get?
That distribution is called the SAMPLING DISTRIBUTION OF
E.g. Want to estimate probability of getting H on a coin. Toss the coin 20 times, get . Then, toss it again 20 times, get another value of , etc.
Question: If we repeat sampling many times, what distribution of these sample proportions would we get?
That distribution is called the SAMPLING DISTRIBUTION OF .

3. CHARACTERISTICS OF A SAMPLING DISTRIBUTION
Sampling distribution has its own features (parameters) like center and spread. We need to know them for inference.
Unbiasedness.
Def. A statistic that is used to estimate a parameter is said to be unbiased for that parameter if the sampling distribution of this statistic centers at the true parameter value.
i.e. The statistic, on average does not over or under-estimate the parameter.
Sampling distribution of a biased statistic
Sampling distribution of an unbiased statistic
True parameter value

4. SPREAD OF AN UNBIASED STATISTIC
Which of two unbiased statistics is a better estimator of the unknown parameter? The one with larger of smaller spread?
The one with smaller spread!
Unbiased statistic with larger spread
Unbiased statistic with smaller spread
True parameter value

5. “GOOD” STATISTICS A statistic good for estimation is one that
does not systematically over- or under– estimates the parameter, and
is “always” close to the parameter it estimates.
DESIARABLE PROPERTIES OF STATISTICS:
UNBIASED FOR THE PARAMETER IT ESTIMATES;
SMALL VARIANCE (error).
EXAMPLES: The best statistic for estimation of population mean is sample mean
The best statistic for estimation of the population proportion is the sample proportion