1. Toward statistical inference
Sampling variability
Toward statistical inference

2. Parameter vs. statistic
Number that describes a population
Ex: mean, variance
Learn about it by taking sample - statistical inference
Number that describes a sample
Value is known after examining the sample
Value changes from sample to sample

3. Sampling variability
If you take a simple random sample from a population, every sample will likely be different.
Thus, the value of a statistic varies with repeated random sampling.
If we take many such samples, we notice an emerging pattern - the sampling distribution

4. Sampling distribution
Portrays how the statistic changes from sample to sample
We can get an idea of it by
Taking many random samples of the same size, n, from the same population
Calculating the sample proportion for each sample
Examining a histogram of all the sample proportions obtained.

5. Estimating population proportion
What proportion of the population meets a certain criterion?
View this as people/objects marked with “1” (if meet criterion) and “0” (if don’t meet criterion)
Take sample, add up the “1”s, divide by n - yields sample proportion

6. Population proportion (real life examples)
What percentage of California voters voted for Arnold?
What percentage of items coming off the assembly line have a certain defect?
What percentage of seeds (in an agricultural field trial) sprouted?

7. Simulation
To understand shape of sampling distribution, we can use simulation
Use computer to take random draws from population, graph results
For population of “0”s and “1”s, we can use the computer to generate from a binomial distribution, where n is sample size and p is population percentage meeting our criterion

8. Mean of a statistic
The mean is the center of the sampling distribution for the statistic.
It is the average value of the statistic over many samples.
If the mean of the sampling distribution is equal to the parameter of interest, the statistic is unbiased.

9. Variability of a statistic
What is the spread of the sampling distribution?
Spread is smaller for larger sample sizes - more data helps you understand the parameter better
Variability of a statistic is unaffected by the size of the population as long as the population is much bigger than the sample size, n.

10. Goldenrod and the gall fly

14. Cautions
The sampling distribution only shows how the statistic varies due to chance.
Sampling distribution doesn’t show possible bias due to undercoverage, nonresponse, lack of realism, etc.