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You want to know whether the mean petal length of iris flowers differs according to their species. You find two different species of irises growing in a garden and measure 25 petals of each species. You can test the difference between these two groups using a t-test and null and alterative hypotheses.
When to use a t-testA t-test can only be used when comparing the means of two groups (a.k.a. pairwise comparison). If you want to compare more than two groups, or if you want to do multiple pairwise comparisons, use an ANOVA test or a post-hoc test. The t-test is a parametric test of difference, meaning that it makes the same assumptions about your data as other parametric tests. The t-test assumes your data:
If your data do not fit these assumptions, you can try a nonparametric alternative to the t-test, such as the Wilcoxon Signed-Rank test for data with unequal variances. What type of t-test should I use?When choosing a t-test, you will need to consider two things: whether the groups being compared come from a single population or two different populations, and whether you want to test the difference in a specific direction. One-sample, two-sample, or paired t-test?
One-tailed or two-tailed t-test?
In your test of whether petal length differs by species:
Performing a t-testThe t-test estimates the true difference between two group means using the ratio of the difference in group means over the pooled standard error of both groups. You can calculate it manually using a formula, or use statistical analysis software. T-test formulaThe formula for the two-sample t-test (a.k.a. the Student’s t-test) is shown below.
 In this formula, t is the t-value, x1 and x2 are the means of the two groups being compared, s2 is the pooled standard error of the two groups, and n1 and n2 are the number of observations in each of the groups. A larger t-value shows that the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups. You can compare your calculated t-value against the values in a critical value chart to determine whether your t-value is greater than what would be expected by chance. If so, you can reject the null hypothesis and conclude that the two groups are in fact different. T-test function in statistical softwareMost statistical software (R, SPSS, etc.) includes a t-test function. This built-in function will take your raw data and calculate the t-value. It will then compare it to the critical value, and calculate a p-value. This way you can quickly see whether your groups are statistically different. In your comparison of flower petal lengths, you decide to perform your t-test using R. The code looks like this: t.test(Petal.Length ~ Species, data = flower.data) Download the data set to practice by yourself. Sample data set Interpreting test resultsIf you perform the t-test for your flower hypothesis in R, you will receive the following output: The output provides:
From the output table, we can see that the difference in means for our sample data is −4.084 (1.456 − 5.540), and the confidence interval shows that the true difference in means is between −3.836 and −4.331. So, 95% of the time, the true difference in means will be different from 0. Our p-value of 2.2e–16 is much smaller than 0.05, so we can reject the null hypothesis of no difference and say with a high degree of confidence that the true difference in means is not equal to zero. Presenting the results of a t-testWhen reporting your t-test results, the most important values to include are the t-value, the p-value, and the degrees of freedom for the test. These will communicate to your audience whether the difference between the two groups is statistically significant (a.k.a. that it is unlikely to have happened by chance). You can also include the summary statistics for the groups being compared, namely the mean and standard deviation. In R, the code for calculating the mean and the standard deviation from the data looks like this:
flower.data %>% group_by(Species) %>% summarize(mean_length = mean(Petal.Length), sd_length = sd(Petal.Length)) |
What experiment could be designed to test whether the two populations are one species or two
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