# The T-Distribution and T-Test

**Topics:**Statistics, Normal distribution, Statistical hypothesis testing

**Pages:**4 (1244 words)

**Published:**April 5, 2011

“In probability and statistics, Student's t-distribution (or simply the t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small” (Narasimhan , 1996). Similar to the normal distribution, the t-distribution is symmetric and bell-shaped, but has heavier tails, meaning that it is more likely to produce values far from its mean. This makes the t-distribution useful for understanding statistical behaviors of random quantities. It plays a role in a number of widely-used statistical analyses, including the Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. Statistical significance is “measuring the likelihood that an event occurs by chance” (Statistical Assessment Service, 2000). In statistics, a result is called statistically significant if it is unlikely to have occurred at random. The amount of evidence required to accept that an event is unlikely to have arisen by chance is known as the significance level or p-value: “the p-value measures consistency by calculating the probability of observing the results from your sample of data or a sample with results more extreme, assuming the null hypothesis is true” (Simon, 2007); or in simpler terms, a p-value is a measure of how much evidence there is against the null hypothesis. The null hypothesis, traditionally represented by the symbol HO, “represents the hypothesis of no change or no effect” (Simon, 2007). If the obtained p-value is small then it can be said either the null hypothesis is false or an unusual event has occurred. A confidence interval “gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given...

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