T Distribution and T-Test (PDF)

T distribution and T-test Introduction In health sciences, researchers often need to compare groups or assess the significance of observed differences. This is where statistical tools like the t-test come into play. The t-test is a fundamental tool for hypothesis testing when the sample...

T distribution and T-test Introduction In health sciences, researchers often need to compare groups or assess the significance of observed differences. This is where statistical tools like the t-test come into play. The t-test is a fundamental tool for hypothesis testing when the sample size is small or when the population standard deviation is unknown. To understand the t-test fully, it's crucial to grasp the concept of the tdistribution. Definition of T Distribution The t-distribution, also known as Student's t-distribution, is a probability distribution that is symmetric and bell-shaped, much like the normal distribution. However, unlike the normal distribution, the t-distribution has heavier tails, making it suitable for smaller sample sizes. t-distribution approaches vs. standard normal distribution There are different types of t-distributions, each characterised by its degrees of freedom (df). The degrees of freedom in a t-distribution depend on the sample size and play a crucial role in determining the shape of the distribution. As the sample size increases, the t-distribution approaches the shape of the standard normal distribution. Assumptions of T-Test 1. Normality: The data within each group should be approximately normally distributed. 2. Independence: Observations within each group must be independent of each other. 3. Homogeneity of variance: The variance within each group should be approximately equal. Types of t-test 1.one-Sample T-Test 2.independent-Samples T-Test 3.Paired-Samples T-Test (Dependent-Samples T-Test) One-sample T-Test Definition: The one-sample t-test compares the mean of a single sample to a known or hypothesised population mean to determine if there is a statistically significant difference between them. Use: It is used when researchers have a single sample and want to determine if the sample mean is significantly different from a known or hypothesised population mean. Example: Investigating whether the mean body mass index (BMI) of a sample of individuals differs significantly from the population mean BMI. Independent-Samples T-Test Definition: The independent-samples t-test compares the means of two independent groups to determine if there is a statistically significant difference between them. Example: Comparing the mean blood pressure levels between patients receiving a new drug and those receiving a placebo. Paired-Samples T-Test (Dependent-Samples T-Test) Definition: The paired-samples t-test compares the means of two related groups to determine if there is a statistically significant difference between them. Each observation in one group is paired or matched with a corresponding observation in the other group. Use: It is used when comparing the means of the same group at two different time points or under two different conditions, such as before and after an intervention or treatment. Example: Assessing the difference in cholesterol levels before and after a dietary intervention within the same group of participants. What if I have more than two groups or more than two time points?