KEVIN BERNHARDTTROY BUCKNERBRIAN GALVIN

Testing Differences Between Means: The Basics

Tests to use for comparing means

When comparing…

Note: ANOVA/Multiple Regression can be used with only 2 means – personal preference.

Number of means

Type of test

2 means t-test

3 or more means ANOVA/Multiple Regression

H0 and Ha

There is a better way of comparing means other than saying, “They’re 1.5 SD apart. That’s quite a difference.”

H0: Null hypothesis. What we are planning to reject.

Ha: Alternative/Research hypothesis. Defendable statement based off data presented.

Mutually exclusive – both cannot be true. Reject one, regard other as tenable.

Comparing means....

Alternative use of z-score equation: z = (x-x_bar)/σ Goal: Compare between 2 means…..

“What’s the likelihood that you would obtain a sample average of X if the population average is x_bar?”

NOTE: “Population” does not always mean statistical population.

Variable Previous use New use

x individual observation Sample mean

x_bar sample mean Population mean (...of sample means)

σ population SD (or sample)

Standard error of means

Standard Error

Standard error of mean can be estimated with the following equation: Sample SD (σ preferred) Sample size

Courtesy of www.discover6sigma.org

 

Standard Error = manipulated observations

Interpreting Standard Error of the Mean

In terms of σ, we can accept H0. Attribute 0.5 SD to sampling error.

Except....

Interpreting Standard Error of the Mean (cont.)

Standard deviation is of individual observations.Comparison is between means.

Therefore SE is used. Standard deviation > Standard Error Affects distribution, not means.

Net Effect: Sample mean is further from population mean (2 SE), therefore we cannot accept H0 immediately.

Setting the alpha level....

p: Probability of mistakenly rejecting H0. With p, you are saying that you are willing to make

this mistake 5% of the time (p = 0.05)

Calculating the probability of obtaining a given sample mean: NORM.DIST(value, mean, standard deviation,

cumulative) TRUE: total area to the left of “value” (aka the sample

mean) FALSE: probability that “value will occur

Creating the graph...

Using the t-Test vs. z-Test

Defining the decision ruleNull Hypothesis vs. Alternative HypothesisBoth cannot be trueAlpha – error rate you have adoptedNormally 5%Critical value is the criterion associated with

the error rate

Finding Critical Value for a z-Test

NORM.INV(area we’re interested in under the curve that represents the distibution, mean of the distribution, standard error of the mean)

Finding Critical Value for a t-Test

Used when you don’t know the population standard deviation.

T.INV (probability you’re interested in, degrees of freedom)

Comparing Critical Values

t-Test has slightly less statistical power than the z-Test, because critical value is farther from the mean due to thicker tails.     

Statistical Power

When the mean is below (or above) the critical value, then the null hypothesis is false and the alternative hypothesis is therefore true.  

Statistical power depends on the position of the alternative hypothesis curve.  

Beta

Beta = 1 – power.

If we would accept a true hypothesis 60% of the time (power), then beta is 1 - .60 = 40%.

t-Test vs z-Test?

Use t-Test when the sample size is under 30, and z-Test when it is over 30+