t-distribution

(noun)

a family of continuous probability disrtibutions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard devition is unknown

Related Terms

Examples of t-distribution in the following topics:

  • t Distribution Table

    • T distribution table for 31-100, 150, 200, 300, 400, 500, and infinity.

  • The t-Distribution

    • The $t$-distribution with $n − 1$ degrees of freedom is the sampling distribution of the $t$-value when the samples consist of independent identically distributed observations from a normally distributed population.
    • The $t$-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
    • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
    • Student's $t$-distribution with $\nu$ degrees of freedom can be defined as the distribution of the random variable $T$:
    • Note that the $t$-distribution becomes closer to the normal distribution as $\nu$ increases.

  • Introducing the t distribution

    • Several t distributions are shown in Figure 5.11.
    • Each row in the t table represents a t distribution with different degrees of freedom.
    • Just like the normal distribution, all t distributions are symmetric.
    • Each row represents a different t distribution.
    • The t distribution with 18 degrees of freedom.

  • The t distribution as a solution to the standard error problem

    • When estimating the mean and standard error from a small sample, the t distribution is a more accurate tool than the normal model.
    • Use the t distribution for inference of the sample mean when observations are independent and nearly normal.
    • To proceed with the t distribution for inference about a single mean, we must check two conditions.
    • When examining a sample mean and estimated standard error from a sample of n independent and nearly normal observations, we use a t distribution with n − 1 degrees of freedom (df).
    • For example, if the sample size was 19, then we would use the t distribution with df = 19 − 1 = 18 degrees of freedom and proceed exactly as we did in Chapter 4, except that now we use the t table.

  • Introduction to one-sample means with the t distribution

    • First, a large sample ensures that the sampling distribution of $\bar{x}$ is nearly normal.
    • The standard error estimate will not generally be accurate for smaller sample sizes, and this motivates the introduction of the t distribution, which we introduce in Section 5.3.2.
    • We will see that the t distribution is a helpful substitute for the normal distribution when we model a sample mean $\bar{x}$ that comes from a small sample.
    • While we emphasize the use of the t distribution for small samples, this distribution may also be used for means from large samples.

  • t Distribution

    • State the difference between the shape of the t distribution and the normal distribution
    • This distribution is called the Student's t distribution or sometimes just the t distribution.
    • The t distribution is therefore leptokurtic.
    • The t distribution approaches the normal distribution as the degrees of freedom increase.
    • A comparison of t distributions with 2, 4, and 10 df and the standard normal distribution.

  • Student Learning Outcomes

  • Estimating the Target Parameter: Interval Estimation

    • The distribution of the resulting statistic, $t$, is not Normal and fits the $t$-distribution.
    • There is a different $t$-distribution for each sample size $n$.
    • In order to specify a specific $t$-distribution, we use its degrees of freedom, which is denoted by $df$, and $df= n-1$.
    • A confidence interval for is calculated by: $\bar{x}\pm t^{*}\frac{s}{\sqrt{n}}$, where $t^*$ is the critical value for the $t(n-1)$ distribution.
    • A plot of the $t$-distribution for several different degrees of freedom.

  • Confidence Interval, Single Population Mean, Standard Deviation Unknown, Student's-t

    • For each sample size n, there is a different Student's-t distribution.
    • The mean for the Student's-t distribution is 0 and the distribution is symmetric about 0.
    • The Student's-t distribution has more probability in its tails than the Standard Normal distribution because the spread of the t distribution is greater than the spread of the Standard Normal.
    • For the TI-84+ you can use the invT command on the DISTRibution menu.
    • The notation for the Student's-t distribution is (using T as the random variable) is

  • Distribution Needed for Hypothesis Testing

    • Earlier in the course, we discussed sampling distributions.
    • Particular distributions are associated with hypothesis testing.
    • Perform tests of a population mean using a normal distribution or a student's-t distribution.
    • (Remember, use a student's-t distribution when the population standard deviation is unknown and the distribution of the sample mean is approximately normal. ) In this chapter we perform tests of a population proportion using a normal distribution (usually n is large or the sample size is large).
    • If you are testing a single population mean, the distribution for the test is for means: