testing differences between means_the basics
Post on 20-Jun-2015
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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+
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