What is a t-test?A t-test (also known as Student's t-test) is a tool for evaluating the means of one or two populations using hypothesis testing. A t-test may be used to evaluate whether a single group differs from a known value (a one-sample t-test), whether two groups differ from each other (an independent two-sample t-test), or whether there is a significant difference in paired measurements (a paired, or dependent samples t-test). Show How are t-tests used?First, you define the hypothesis you are going to test and specify an acceptable risk of drawing a faulty conclusion. For example, when comparing two populations, you might hypothesize that their means are the same, and you decide on an acceptable probability of concluding that a difference exists when that is not true. Next, you calculate a test statistic from your data and compare it to a theoretical value from a t-distribution. Depending on the outcome, you either reject or fail to reject your null hypothesis. What if I have more than two groups?You cannot use a t-test. Use a multiple comparison method. Examples are analysis of variance (ANOVA), Tukey-Kramer pairwise comparison, Dunnett's comparison to a control, and analysis of means (ANOM). t-Test assumptionsWhile t-tests are relatively robust to deviations from assumptions, t-tests do assume that:
For two-sample t-tests, we must have independent samples. If the samples are not independent, then a paired t-test may be appropriate. Types of t-testsThere are three t-tests to compare means: a one-sample t-test, a two-sample t-test and a paired t-test. The table below summarizes the characteristics of each and provides guidance on how to choose the correct test. Visit the individual pages for each type of t-test for examples along with details on assumptions and calculations.
The table above shows only the t-tests for population means. Another common t-test is for correlation coefficients. You use this t-test to decide if the correlation coefficient is significantly different from zero. One-tailed vs. two-tailed testsWhen you define the hypothesis, you also define whether you have a one-tailed or a two-tailed test. You should make this decision before collecting your data or doing any calculations. You make this decision for all three of the t-tests for means. To explain, let’s use the one-sample t-test. Suppose we have a random sample of protein bars, and the label for the bars advertises 20 grams of protein per bar. The null hypothesis is that the unknown population mean is 20. Suppose we simply want to know if the data shows we have a different population mean. In this situation, our hypotheses are: $ \mathrm H_o: \mu = 20 $ $ \mathrm H_a: \mu \neq 20 $ Here, we have a two-tailed test. We will use the data to see if the sample average differs sufficiently from 20 – either higher or lower – to conclude that the unknown population mean is different from 20. Suppose instead that we want to know whether the advertising on the label is correct. Does the data support the idea that the unknown population mean is at least 20? Or not? In this situation, our hypotheses are: $ \mathrm H_o: \mu >= 20 $ $ \mathrm H_a: \mu < 20 $ Here, we have a one-tailed test. We will use the data to see if the sample average is sufficiently less than 20 to reject the hypothesis that the unknown population mean is 20 or higher. See the "tails for hypotheses tests" section on the t-distribution page for images that illustrate the concepts for one-tailed and two-tailed tests. How to perform a t-testFor all of the t-tests involving means, you perform the same steps in analysis:
What is the appropriate test statistic to use to determine the correlation coefficient?The formula for the test statistic is t=r√n−2√1−r2. The value of the test statistic, t, is shown in the computer or calculator output along with the p-value. The test statistic t has the same sign as the correlation coefficient r. The p-value is the combined area in both tails.
Is tCorrelation is a statistic that describes the association between two variables. The correlation statistic can be used for continuous variables or binary variables or a combination of continuous and binary variables. In contrast, t-tests examine whether there are significant differences between two group means.
Is tTo determine if a correlation coefficient is statistically significant you can perform a t-test, which involves calculating a t-score and a corresponding p-value.
What statistic is used for correlation?Correlation is measured by a statistic called the correlation coefficient, which represents the strength of the putative linear association between the variables in question. It is a dimensionless quantity that takes a value in the range −1 to +13.
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