Statistics and sampling (MR, unit 9)

Sampling distribution of the mean
Calculate mean of each sample.
Count number of times each value occurs and plot results as a distribution.

Properties:

• Results are very close to the normal distribution, a statistical rule called the central limit theorem. Larger samples are obviously even closer.
• The mean of the sampling distribution (u) = the population mean
• The sampling distribution has a standard deviation called the standard error (the dispersion of values around the mean).

Standard error of the mean:

s is the sample standard deviation (the sample based estimate of the standard deviation of the population, as this is usually not known)

n is the size (number of items) of the sample.

Confidence levels, limits & intervals
68% of population lies with sample mean +/- 1 SE
95% of population lies with sample mean +/- 1.96 SE
99% of population lies with sample mean +/- 2.58 SE

68%, 95%, 99% = confidence levels (degrees of certainty)
Edge of ranges = confidence limits
The ranges themselves = confidence intervals

Sampling distribution of a proportion
Many surveys deal not with a population means but with proportions (attitudes, or the percentage of time an event occurs).
Arrange as you would for sampling distribution of a population. Then use following calculation

Standard error of a proportion =
p = the proportion in the sample (taken as an estimate of the population)
q = 1-p
n = size of sample

Example: 285 people out of sample of 400 regularly travel by bus
se = square route (pq/n)
se = square route (285/400 * 1-0.7125 / 400)
se = 0.0226
99% confidence interval = 2.58*se
0.7125 +/-(2.58*0.0226)
With 99% confidence we can say that between 65-77% of people regularly travel by bus