Independent Samples

Statistical Null Hypothesis

H₀: μ_x - μ_y = 0

Statistical Alternative Hypothesis

H_A: μ_x - μ_y ≠ 0

Random Sampling Distribution of (Χ - Υ )

A theoretical relative frequency distribution of all the values of (X - Y) that would be obtained by chance from an infinite number of samples of a particular size drawn from two populations, μ_x and μ_y

μ_x-y

Symbol for the mean of the

sampling distribution of (X - Y)

μ_x-y = 0 when μ_x - μ_y = 0

Estimated Standard error of the

difference between two means

An estimate of the standard deviation of the sampling distribution of (X - Y)

S_x-y

Estimated Standard error of the

difference between two means

Assumption of homogeneity of variance

The assumption that σ²/_x = σ²/_y 

that the variances in the two populations

from which the samples are drawn are the same.

Pooled estimate of the population variance (s²_p)

A single estimate of the population variance based on combining the data from both samples.

(n_x - 1) + (n_y - 1)

degrees of freedom for the pooled estimate

of the population variance obtained

from two independent samples.

S_x-y

symbol for the estimated standard error of (X - Y)

directional (one-tailed) test

the alternative hypothesis: H_A states that

μ_x  differs from μ_y in one particular direction

(and the critical region is located

in only one tail of the sampling distribution)

Effect size

an estimate of the degree to which the treatment effect

is present in the population, expressed as a

"number...free of the original measurement unit"

(Cohen's) d

a measure of effect size; expresses the difference between

(μ_x - μ_y) _true and (μ_x - μ_y)_hyp

relative to the population standard deviation

(Hedges's) g

a measure of effect size; expresses the difference between X and Y relative to S_p,

the pooled estimate of the population standard deviation

r

point-biserial between groups r;

a correlation measure of effect size =

We can determine the amount of power for a particular sample size or the sample size required to achieve a desired level of power if we can:

1. The risk (α level) we are willing to take of rejecting a true null hypothesis

2. The magnitude of the smallest discrepancy that, if it exists, we wish to be reasonably sure of discovering.

Power of a test

the probability of rejecting a false null hypothesis

Central limit theorem

regardless of the shape of the population(s) of scores from which random samples are drawn, the sampling distribution of X (or X - Y) more and more resembles a normal distribrution as sample size increases.

When we test the hypothesis about the difference between two means, the statistical conditions

must be fundamentally the same:

1. Each sample is drawn at random from it's respective population.

2. The two random samples must be independently selected.

3. Samples are drawn with replacement.

4. The sampling distribution of (X - Y) follows the normal curve.

Random sampling model

a procedure that ensures that a sample is obtained in a way that all samples of the same size have an equal chance of being selected from the population.

Random Assignment

any procedure that allows chance

to divide an available group of subjects or observations into two or more sub-groups.