T-Test

T-Test - A statistical test involving means of normal populations with unknown standard deviations. The purpose of this test is to show that differences in means are not a result of chance but rather that they are statistically significant disproving the null hypothesis.

Paired Samples T-Test - Uses data in which each case must have scores on two variables. (ie. Which type of chocolate bar was

preferred by the same group of kids) The paired samples t test procedure evaluates whether the mean of the difference between these two variables is significantly different from zero. It is applicable to two types of data problems, repeated measures and matched-object designs.

- Repeated-Measures - a participant is assessed on two occasions or under two conditions on one measure. In the SSPS data file created to conduct a paired-samples t test, each participant has scores on two variables – the first variable represents the first score on the measure. Same goes with the second.

- Matched-subjects Designs – participants are paired, and each is assessed once on a measure.

Assumptions Underlying the Paired-Samples T TestAssumption 1: The difference variable is normally distributed in the population.Assumption 2: The cases represent a random sample from the population.

Other Assumptions:- The measurements are made on an interval or ratio scale.- Members of sample are randomly selected from the defined population.- The standard deviations of the scores for the two groups should be approximately equal.- The populations from which the samples have been drawn are normally distributed.

Example of Paired-Samples T Test:

Consider the situation of voting turnout in American states in elections since 1980.

% turnout in presidential elections by states

Election N Mean Std. Deviation1980 51 55.7 7.31984 51 54.6 6.51988 51 52.1 6.41992 51 57.6 7.41996 51 48.9 7.42000 51 53.8 6.9

Different stories could be told for each of these elections, but let's concentrate on the turnout data for 1980 and 1984. The 1980 election, between incumbent president Jimmy Carter and challenger Ronald Reagan also had a third candidate, John Anderson--who had been a Republican party leader but who ran as an Independent.

Reagan was elected in 1980 and ran for re-election in 1984 against Walter Mondale, but there was no third party candidate in 1984. The mean voting turnout in 1984--when calculated across all the states--dropped compared with 1980. Some say that turnout dropped because there was no third party candidate. Others say that the observed differene between means of only 1.1% points (55.7-54.6) could have been attributable to chance.

We are a writing class, so we didn’t show the computations anymore just so confusion would be avoided.

Treating the 51 states as "Independent" in computing the T-Test produces a test statistic (t) less than one. A test statistic this small falls far short of significance at the customary .05 level, so it suggests that the observed difference in states' voting turnout between 1980 and 1984 is unlikely to have occurred by chance.

But suppose that some systematic process was going on. Suppose that most states tended to demonstrate a slightly lower voting turnout rate between 1980 and 1984--perhaps dropping by about one point. Perhaps this systematic change is lost by only calculating the means for each year.

In fact, because the states are matched on repeated measures, we must use the Paired Samples T-Test in SPSS, which produces this very different result:

Paired Samples StatisticsMean N Std. Deviation Std. Error Mean

Pair 1 % turnout in 1980 election 55.739 51 7.295 1.022

% turnout in 1984 election 54.612 51 6.541 0.916Paired Samples Test: % turnout in 1980 election - % turnout in 1984 election

Paired Differences t df Sig. (2-tailed)

Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference

Lower Upper1.127 2.522 0.353 0.418 1.837 3.193 50 0.002

Independent Samples T-Test (or two-sample) t-test is used to compare the means of two independent samples. (ie. The average height of football and basketball players)

- Each case must have a score on two variables, the grouping variable and the test variable. The grouping variable divides cases into two mutually exclusive groups or categories such as gender, while the test variable defines each case on some quantitative dimension such as verbal comprehension.

Assumptions Underlying the Independent-Samples T TestAssumption 1: The test variable is normally distributed in each of the two populations (as defined by the grouping variable)Assumption 2: The variances of the normally distributed test variable for the test are equal.

Other Assumptions:- The scores in two groups are randomly sampled from their respective populations and are independent of

one another.- The scores in the respective populations are normally distributed.- The variances of scores in the two populations are equal. (Homogeneity of Variance)

Example of Independent-Samples T Test:

A study of the effect of caffeine on muscle metabolism used eighteen male volunteers who each underwent arm exercise tests. Nine of the men were randomly selected to take a capsule containing pure caffeine one hour before the test. The other men received a placebo capsule. During each exercise the subject's respiratory exchange ratio (RER) was measured. (RER is the ratio of CO2 produced to O2 consumed and is an indicator of whether energy is being obtained from carbohydrates or fats).

The question of interest to the experimenter was whether, on average, caffeine changes RER.

The two populations being compared are “men who have not taken caffeine” and “men who have taken caffeine”. If caffeine has no effect on RER the two sets of data can be regarded as having come from the same population.

The results were as follows:

RER(%)

Placebo Caffeine

105 96

119 99

100 94

97 89

96 96

101 93

94 88

95 105

98 88

Mean 100.56 94.22

SD 7.70 5.61

The means show that, on average, caffeine appears to have altered RER from about 100.6 to 94.2, a change of 6.4. However, there is a great deal of variation between the data values in both samples and considerable overlap between them. So is the difference between the two means simply due sampling variation, or does the data provide evidence that caffeine does, on average, reduce RER? The p-value obtained from an independent samples t-test answers this question.

The t-test tests the null hypothesis that the mean of the caffeine treatment equals the mean of the placebo versus the alternative hypothesis that the mean of caffeine treatment is not equal to the mean of the placebo treatment.

Computer output obtained for the RER data gives the sample means and the 95% confidence interval for the difference between the means.

Two Sample T-Test and Confidence Interval

Two sample T for Caffeine vs. Placebo

N Mean StDev SE Mean

Caffeine 9 94.22 5.61 1.9

Placebo 9 100.56 7.70 2.6

95% CI for mu Caffeine - mu Placebo: (-13.1, 0.4)T-Test mu Caffeine = mu Placebo (not =): T = -1.99 P = 0.032 DF = 16Both use Pooled StDev = 6.74

N.B. mu = m = mean