Introduction to Statistics 2 - Standard Errors and Confidence Intervals

Textual data

Useful for bookkeeping, but may not be for analysis

Normal Distribution

A function that represents the distribution of variables as a symmetrical bell-shaped graph.

X

Distribution of observations

x

Single observation

x with line over it

Single sample mean

Standardised normal distribution

A purely theoretical probability distribution that reflects a specific normal curve for the standardised value, Z.
Measured in units of the standard deviation
Mean = 0
SD = 1

Transforming to the Standardised Normal Z Distribution

Z = (X-μ)/σ
μ = mean
Subtract the mean then divide by the SD

Point estimate

Single value that serves as an estimate of a population parameter

Confidence Interval

The range of values within which a population parameter is estimated to lie
Larger samples give smaller confidence intervals
Large percentage of confidence would produce a large interval
(see slides for equation)
Using the t-distribution
95% CI = mean +/- (t value for that confidence level) x SE
Only accounts for random sampling variation

Standard Error of the Mean

Variation between sample means
Measures the imprecision of sample mean as an estimate of population mean
S = Standard deviation
n = sample size

T distribution

Closely related to a standard normal - but with heavier tails
Shape depends on the degrees of freedom (df)
With very large degrees of freedom, can assume the normal distribution
Bottom half is the standard error of the mean

T value

The value of a t distribution with n-1 degrees of freedom that marks the 2.5% tail area

Generalisation

The extent to which findings and conclusions from a particular experiment can be broadly applied to the population. This is possible if the sample of people is representative of the population.

Inference

Generalising from random sample to population (from which random sample is drawn)
- includes hypothesis testing and estimating confidence intervals
Standard error of difference in means

Confidence interval for difference in means

Measure of spread

SD, IQR

Measure of precision of estimate of population parameters using data from a random sample

SE, 95% CI

Standard error of difference in means