Statistics for Behavioral Science

Chapter 5, 6, 7, 8

Alpha

Level of Significance

What is alpha if it isn’t specified?

a = 0.05

When to use z-test

When you know the population standard deviation (sigma) and the sample size is large (n > 40)

When to use the t-test

When you don’t know the population standard deviation (sigma) and have a small sample size (typically n < 40). )

When to use a two-sample z-test

It is used when you have a large sample size and know the population standard deviation (n > 100)

When to use a two-sample t-test

It is used when you have a large sample size and know the population standard deviation (n < 100)

Null hypothesis

Population mean is the SAME as that of the sample. There are no mean differences between groups

(H0: mu = x)

Alternative hypothesis

The population mean is different from the sample. There are mean differences between groups

(H1: mu ≠ x)

Type 1 Error

False positive result, rejecting a true null hypothesis. (A fire alarm going off with no fire) p = alpha

Type 2 Error

False negative; failing to reject a null hypothesis (A fire alarm not going off during a fire) p = beta

Effect Size (d)

difference between our means

Region of Rejection

The area in a statistical distribution where, if the null hypothesis falls inside the range of your t-test results, we fail to reject the null hypothesis

When do you use the two-tailed test

When the problem does not have a direction (lower/higher)

When do you use the one-tailed test

When you have a direction present in the problem

P-value

The probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis (H0) is true

What do you use when know the standard deviation and the sample size is anything

Z-test

What do you do when there is a Large Sample (n ≥ 40) and the standard deviation is unknown

Use t-test (Z-test approximation is possible but not preferred)

What do you do when there is a small Sample (n < 30) and standard deviation is unknown

Use t-test (t-distribution is required).

Two sample t-test types

Pooled variance t-test & Separate variance t-test

Steps to decide to use pooled-variance t test or separate variance t test

Three steps in order: sample size large

Steps to decide whether to use a t-test or a z-test

1. Do you have sigma? Yes, do z. No, ask the second question.

How big is n? If n is 40 or more, do z. If less than 40, t.

Null hypothesis distribution

A map of what results are not likely by chance

Critical z-score

Z scores that serve as the boundary of the area which is determined by the alpha

P-level

If the p-level is bigger than alpha then we fail to reject the null hypothesis; if the p-level is smaller then as reject the null hypothesis

Reducing Type 1 Error

lower your alpha level (from .05 —> to .01)

Reducing Type 2 Error

use a one-tailed rather than a two-tailed test

Determining Homogeneity of Variance (HOV)

If you

Levene’s test

If the p-value si greater than 0.05, then you can assume HOV; if you have less than 0.05 then you do not have HOV

Estimated effect size (g)

How to find the difference between means

The standard error of the difference between means

If you know the population standard deviations

The standard error of the difference between the means would look like this:

Comparing 2 sample means with a z test

(unusual)

Pooled variance t-test (Sp²)

Two sample variances can be pooled together to form a single estimate of the population variance. We also assume that these two populations have same variance (HOV)

Pooled variances t-test formula (for unequal sample sizes)

Denominator of pooled variances and separate variances

It is the estimated standard error of the difference between means

Pooled t-test for equal sample sizes

Confidence intervals for the difference between two population means

One tailed test

Two tailed test

Tcalc and Tcv approach

That's when we find significance by comparing tcalc with the tcv.