Testing differences between means_The Basics

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K E V I N B E R N H A R D T T R O Y B U C K N E R B R I A N G A L V I N 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)/s ? 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) s population SD (or sample) Standard error of means Standard Error ? Standard error of mean can be estimated with the following equation: ? Sample SD (s preferred) ? Sample size Courtesy of www.discover6sigma.org ? Standard Error = manipulated observations Interpreting Standard Error of the Mean ? In terms of s, 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 rule ? Null Hypothesis vs. Alternative Hypothesis ? Both cannot be true ? Alpha – error rate you have adopted ? Normally 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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