The t-distribution is used in the estimation of the mean of a normally distributed population when the sample size is small. It provides a better estimate of the standard error of the mean than the normal distribution when the sample size is small. The t-distribution is used to calculate a t-statistic, which is compared to a critical value from the t-distribution to test hypotheses about the population mean.
The class of π‘ distributions (or π distributions) are all bell shaped functions with mean zero, but higher variance than the standard normal. A π‘ distribution is characterized by a whole number index called the number of β degrees of freedomβ or βdfβ, which. For instance $t_{55}$ means π‘ distribution with 55 degrees of freedom.
There are several calculations related to the t-distribution:
Probability density function (PDF): The PDF of the t-distribution is given by the formula:
where x is the value of the random variable, df is the degrees of freedom, and Gamma() is the gamma function.
Cumulative distribution function (CDF): The CDF of the t-distribution cannot be expressed in a simple closed-form formula, but it can be calculated using software or tables. The CDF gives the probability that a random variable from the t-distribution is less than or equal to a given value.
Critical values: The t-distribution is used to calculate critical values for hypothesis tests and confidence intervals. The critical values are obtained from t-tables or calculated using software based on the specified degrees of freedom and significance level.
Confidence intervals: The t-distribution is used to construct confidence intervals for the population mean when the sample size is small. The formula for a 95% confidence interval is:
where xΜ is the sample mean, s is the sample standard deviation, n is the sample size, and tΞ±/2 is the critical value from the t-distribution with (n-1) degrees of freedom and significance level Ξ±/2.
Hypothesis testing: The t-distribution is used to test hypotheses about the population mean when the population standard deviation is unknown and the sample size is small. The t-statistic is calculated as:
where $xΜ$ is the sample mean, ΞΌ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. The t-statistic follows a t-distribution with (n-1) degrees of freedom. The null hypothesis is rejected if the calculated t-statistic is greater than the critical value from the t-distribution with (n-1) degrees of freedom and significance level Ξ±.
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Sulav Jung Hamal - 2024/01/23