Inference when σ is unknown

Question 1

How to estimate σ from sample

Answer 1

Has to an unbiased estimate of σ

• Take the deviations from M rather than μ

Question 2

Degrees of freedom

Answer 2

Number of deviation scores contributing to SS minus the number of restrictions (parameters e.g. 2 tests = 2 restrictions)
Number of deviation scores which are free to vary independently (only when known)
n - 1

Unbiased estimate of σ^2 can be obtained by dividing the sum of (X−M)^2 by the degrees of freedom (df) associated with the deviations

Question 3

Properties of t distribution

Answer 3

Family of curves, dependent of df
For any df, t distribution has a mean of 0 (symmetrical and unimodal)
As sample size increases, the t-distribution approaches the normal distribution
For df < ∞, t distribution has narrower peak and fatter tails than normal distribution, and hence a larger standard deviation

Question 4

Areas under normal and t distributions

Answer 4

Area under normal curve between Z = 1.0 is 68%

For df = 5, area under t distribution between t = +/-1.0 is 63.6%

For df = 20, area under t distribution between t = +/-1.0 is 67%

Question 5

Size of effect

Answer 5

A statistical measure that indicates the magnitude or strength of a relationship between variables in a study

• Confidence intervals convey info on the size of effect