          What is it: The t-Test is used for comparing two samples and checking if they came from the same population. No time sequence is maintained, each of the two samples are collapsed into their descriptive statistics.

Why use it: A t-Test is any statistical hypothesis test in which the test statistic has a Student's t-distribution if the null hypothesis is true. It is applied when sample sizes are small enough that using an assumption of normality and the associated z-test leads to incorrect inference. Among the most frequently used t-Tests are:

• A test of the null hypothesis that the means of two normally distributed populations are equal.
• A test of whether the mean of a normally distributed population has a value specified in a null hypothesis.
• A test of whether the slope of a regression line differs significantly from 0.

Once a t value is determined, a p-value can be found using a table of values from Student's t-distribution.

Where to use it: The t-Test can give a statistical basis for whether a sample is from a population or whether multiple samples indicate their populations are equal.

When to use it: There are two main t-Tests:

• Paired t-Test: used when each data point in one group corresponds to a matching data point in the other group.
• Unpaired t-Test: used whether or not the groups contain matching datapoints:
• Two-sample assuming equal variances
• Two-sample assuming unequal variances

Paired t-Test: The paired t-Test is used to investigate the relationship between two groups where there is a meaningful one-to-one correspondence between the data points in one group and those in the other, e.g. a variable measured at the same time points under experimental and control conditions. It is NOT sufficient that the two groups simply have the same number of datapoints!

Unpaired t-Test: The unpaired t-test does not require that the two groups be paired in any way, or even of equal sizes. A typical example might be comparing a variable in two experimental groups of patients, one treated with drug A and one treated with drug B. Such situations are common in medicine where an accepted treatment already exists and it would not be ethical to withhold this from a control group. Here, we wish to know if the differences between the groups are "real" (statistically significant) or could have arisen by chance. The calculations involved in an unpaired t-Test are slightly more complicated than for the paired test. Note that the unpaired t-Test is equivalent to one-way ANOVA used to test for a difference in means between two groups. To perform an unpaired t-test:

How to use it: Calculating a single sample t-Test consists of the following steps:

1. Check the data to confirm the data forms a normal distribution (approximately) using a histogram.
2. Determine the significance level (alpha) that you want and the comparison mean value that you want to test against.
3. Calculate the mean and standard deviation of your data.
4. Calculate the Test Statistic using the following method:
Test Statistic = (Data mean – Comparison mean)/((standard deviation)/Square root (number of samples))
5. For testing the Null Hypothesis: Data Mean = Comparison Mean, Calculate the lower critical value by LCV = - Table E value for alpha, degrees of freedom (N-1), and Upper Critical Value by UCV = Table E value for alpha, degrees of freedom (N-1).
6. Evaluate Null Hypothesis by comparing the Test Statistic to the LCV and UCV. If the Test Statistic is < LCV or > UCV, reject the Null Hypothesis in favor of the Alternate Hypothesis (Data Mean is NOT = Comparison Mean). Otherwise, fail to reject the Null Hypothesis, there is insufficient evidence to show that the means are different.
7. For testing the Null Hypothesis: Data Mean = Comparison Mean, Calculate the critical value = - Table D value for (alpha x 2), degrees of freedom (N-1).
8. Evaluate Null Hypothesis by comparing the Test Statistic to the Critical Value. If the Test Statistic is < Critical Value, reject the Null Hypothesis in favor of the Alternate Hypothesis (Data Mean is < Comparison Mean). Otherwise, fail to reject the Null Hypothesis, there is insufficient evidence to show that the data mean is < the comparison mean.
9. For testing the Null Hypothesis: Data Mean = Comparison Mean, Calculate the critical value = Table D value for (alpha x 2), degrees of freedom (N-1).
10. Evaluate Null Hypothesis by comparing the Test Statistic to the Critical Value. If the Test Statistic is > Critical Value, reject the Null Hypothesis in favor of the Alternate Hypothesis (Data Mean is > Comparison Mean). Otherwise, fail to reject the Null Hypothesis, there is insufficient evidence to show that the data mean is > the comparison mean.

Important Notes: Assumptions for t-test:

• Normal distribution of data, tested by using a normality test, such as Shapiro-Wilk and Kolmogorov-Smirnov test.
• Equality of variances, tested by using either the F test, the more robust Levene's test, Bartlett's test, or the Brown-Forsythe test.
• Sample size should not differ hugely between the groups.
• Samples may be independent or dependent, depending on the hypothesis and the type of samples:
• Independent samples are usually two randomly selected groups
• Dependent samples are either two groups matched on some variable (for example, age) or are the same people being tested twice (called repeated measures)

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