Problem StatementIn our sample dataset, students reported their typical time to run a mile, and whether or not they were an athlete. Suppose we want to know if the average time to run a mile is different for athletes versus non-athletes. This involves testing whether the sample means for mile time among athletes and non-athletes in your sample are statistically different (and by extension, inferring whether the means for mile times in the population are significantly different between these two groups). You can use an Independent Samples t Test to compare the mean mile time for athletes and non-athletes. Show The hypotheses for this example can be expressed as: H0: µnon-athlete − µathlete = 0 ("the difference of the means is equal to zero") where µathlete and µnon-athlete are the population means for athletes and non-athletes, respectively. In the sample data, we will use two variables: Athlete and MileMinDur. The variable Athlete has values of either “0” (non-athlete) or "1" (athlete). It will function as the independent variable in this T test. The variable MileMinDur is a numeric duration variable (h:mm:ss), and it will function as the dependent variable. In SPSS, the first few rows of data look like this: Before the TestBefore running the Independent Samples t Test, it is a good idea to look at descriptive statistics and graphs to get an idea of what to expect. Running Compare Means (Analyze > Compare Means > Means) to get descriptive statistics by group tells us that the standard deviation in mile time for non-athletes is about 2 minutes; for athletes, it is about 49 seconds. This corresponds to a variance of 14803 seconds for non-athletes, and a variance of 2447 seconds for athletes1. Running the Explore procedure (Analyze > Descriptives > Explore) to obtain a comparative boxplot yields the following graph: If the variances were indeed equal, we would expect the total length of the boxplots to be about the same for both groups. However, from this boxplot, it is clear that the spread of observations for non-athletes is much greater than the spread of observations for athletes. Already, we can estimate that the variances for these two groups are quite different. It should not come as a surprise if we run the Independent Samples t Test and see that Levene's Test is significant. Additionally, we should also decide on a significance level (typically denoted using the Greek letter alpha, α) before we perform our hypothesis tests. The significance level is the threshold we use to decide whether a test result is significant. For this example, let's use α = 0.05. 1When computing the variance of a duration variable (formatted as hh:mm:ss or mm:ss or mm:ss.s), SPSS converts the standard deviation value to seconds before squaring. Running the TestTo run the Independent Samples t Test:
Syntax
OutputTablesTwo sections (boxes) appear in the output: Group Statistics and Independent Samples Test. The first section, Group Statistics, provides basic information about the group comparisons, including the sample size (n), mean, standard deviation, and standard error for mile times by group. In this example, there are 166 athletes and 226 non-athletes. The mean mile time for athletes is 6 minutes 51 seconds, and the mean mile time for non-athletes is 9 minutes 6 seconds. The second section, Independent Samples Test, displays the results most relevant to the Independent Samples t Test. There are two parts that provide different pieces of information: (A) Levene’s Test for Equality of Variances and (B) t-test for Equality of Means. A Levene's Test for Equality of of Variances: This section has the test results for Levene's Test. From left to right:
The p-value of Levene's test is printed as ".000" (but should be read as p < 0.001 -- i.e., p very small), so we we reject the null of Levene's test and conclude that the variance in mile time of athletes is significantly different than that of non-athletes. This tells us that we should look at the "Equal variances not assumed" row for the t test (and corresponding confidence interval) results. (If this test result had not been significant -- that is, if we had observed p > α -- then we would have used the "Equal variances assumed" output.) B t-test for Equality of Means provides the results for the actual Independent Samples t Test. From left to right:
Note that the mean difference is calculated by subtracting the mean of the second group from the mean of the first group. In this example, the mean mile time for athletes was subtracted from the mean mile time for non-athletes (9:06 minus 6:51 = 02:14). The sign of the mean difference corresponds to the sign of the t value. The positive t value in this example indicates that the mean mile time for the first group, non-athletes, is significantly greater than the mean for the second group, athletes. The associated p value is printed as ".000"; double-clicking on the p-value will reveal the un-rounded number. SPSS rounds p-values to three decimal places, so any p-value too small to round up to .001 will print as .000. (In this particular example, the p-values are on the order of 10-40.) C Confidence Interval of the Difference: This part of the t-test output complements the significance test results. Typically, if the CI for the mean difference contains 0 within the interval -- i.e., if the lower boundary of the CI is a negative number and the upper boundary of the CI is a positive number -- the results are not significant at the chosen significance level. In this example, the 95% CI is [01:57, 02:32], which does not contain zero; this agrees with the small p-value of the significance test. Decision and ConclusionsSince p < .001 is less than our chosen significance level α = 0.05, we can reject the null hypothesis, and conclude that the that the mean mile time for athletes and non-athletes is significantly different. Based on the results, we can state the following:
What is the dependent variable in tThe independent variable is whatever makes the two scores different (for example, time or temperature of the testing space). The dependent variable is their score on the measure. To perform a dependent means t-test you will need to have the two scores in two separate columns that are both numeric in type.
Does tIn other words, a t test is used when we wish to compare two means (the scores must be measured on an interval or ratio measurement scale). We would use a t test if we wished to compare the reading achievement of boys and girls. With a t test, we have one independent variable and one dependent variable.
What is an example of a dependent tFor example, you could use a dependent t-test to understand whether there was a difference in smokers' daily cigarette consumption before and after a 6 week hypnotherapy programme (i.e., your dependent variable would be "daily cigarette consumption", and your two related groups would be the cigarette consumption values ...
What are the independent and dependent variables in paired tThe dependent variable is quantitative in nature. The independent variable is qualitative in nature, that is, the levels represent different categories. The independent variable has two and only two levels.
What is the dependent variable in a oneFor a one-sample t-test, there will only be one variable's data to be entered into SPSS Statistics: the dependent variable, dep_score, which is the depression score.
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