## t-test (Sample, Pooled or Unpaired and Paired)

The t-test is a statistical test used to determine if there is a significant difference between the means of two groups. There are three types of t-tests: the independent samples t-test, the paired samples t-test, and the pooled t-test.

**Independent samples t-test (unpaired t-test):**This test is used when the two groups being compared are independent, meaning that the individuals in one group are not related to the individuals in the other group. The independent samples t-test assumes that the two groups have equal variances.

Here are the steps to perform an independent samples t-test:Pharmaceutical example of t-test

a. State the null hypothesis and the alternative hypothesis.

b. Calculate the mean and standard deviation of each group.

c. Calculate the t-test statistic using the formula: t = (mean1 – mean2) / (sqrt((s1^2/n1) + (s2^2/n2))), where mean1 and mean2 are the means of the two groups, s1 and s2 are the standard deviations of the two groups, and n1 and n2 are the sample sizes of the two groups.

d. Calculate the degrees of freedom (df) using the formula: df = n1 + n2 – 2.

e. Determine the p-value from a t-distribution table or statistical software.

f. Compare the p-value to the chosen significance level (usually 0.05) to determine if the null hypothesis can be rejected.

**Paired samples t-test:**This test is used when the two groups being compared are related, meaning that the individuals in one group are related to the individuals in the other group. The paired samples t-test assumes that the differences between the pairs are normally distributed.

Here are the steps to perform a paired samples t-test:

a. State the null hypothesis and the alternative hypothesis.

b. Calculate the difference between each pair of observations.

c. Calculate the mean and standard deviation of the differences.

d. Calculate the t-test statistic using the formula: t = (mean of differences) / (s / sqrt(n)), where s is the standard deviation of the differences and n is the number of pairs.

e. Calculate the degrees of freedom (df) using the formula: df = n – 1.

f. Determine the p-value from a t-distribution table or statistical software.

g. Compare the p-value to the chosen significance level (usually 0.05) to determine if the null hypothesis can be rejected.

**Pooled t-test:**This test is used when the two groups being compared have equal variances, but the sample sizes are not equal.

Here are the steps to perform a pooled t-test:

a. State the null hypothesis and the alternative hypothesis.

b. Calculate the mean and standard deviation of each group.

c. Calculate the pooled standard deviation using the formula: s = sqrt(((n1-1)s1^2 + (n2-1)s2^2) / (n1 + n2 – 2)).

d. Calculate the t-test statistic using the formula: t = (mean1 – mean2) / (s * sqrt((1/n1) + (1/n2))), where mean1 and mean2 are the means of the two groups, s is the pooled standard deviation, and n1 and n2 are the sample sizes of the two groups.

e. Calculate the degrees of freedom (df) using the formula: df = n1 + n2 – 2.

f. Determine the p-value from a t-distribution table or statistical software.

g. Compare the p-value to the chosen significance level (usually 0.05) to determine if the null hypothesis can be rejected.

#### Pharmaceutical example of t-test

Here are some examples of t-tests in the pharmaceutical field:

- A researcher wants to compare the efficacy of two different pain medications in reducing pain in patients with osteoarthritis. The researcher randomly assigns patients to either Group A (medication A) or Group B (medication B). After a period of time, the researcher measures the pain scores of both groups using a pain scale. The researcher can use an independent samples t-test to determine if there is a significant difference in the mean pain scores between the two groups.
- A pharmaceutical company wants to compare the bioavailability of two different formulations of a drug. The company randomly assigns subjects to receive either Formulation A or Formulation B, and then measures the concentration of the drug in the subjects’ bloodstreams at various time points. The company can use a paired samples t-test to determine if there is a significant difference in the mean concentration of the drug between the two formulations at each time point.
- A researcher wants to compare the mean weight of tablets produced by two different manufacturing processes. The researcher takes a sample of tablets from each process and weighs them. The researcher can use a pooled t-test to determine if there is a significant difference in the mean weight of tablets produced by the two processes.

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