Student's t-test

(noun)

Any statistical hypothesis test in which the test statistic follows a Student's $t$ distribution if the null hypothesis is supported.

Related Terms

Examples of Student's t-test in the following topics:

  • Comparing Two Sample Averages

    • Student's t-test is used in order to compare two independent sample means.
    • A t-test is any statistical hypothesis test in which the test statistic follows Student's t distribution, as shown in , if the null hypothesis is supported.
    • If using Student's original definition of the t-test, the two populations being compared should have the same variance.
    • If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances.
    • This is a plot of the Student t Distribution for various degrees of freedom.

  • The t-Test

    • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
    • A t-test is any statistical hypothesis test in which the test statistic follows a Student's t-distribution if the null hypothesis is supported.
    • When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student's t-distribution.
    • All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal.
    • Writing under the pseudonym "Student", Gosset published his work on the t-test in 1908.

  • Mann-Whitney U-Test

    • It has greater efficiency than the $t$-test on non-normal distributions, such as a mixture of normal distributions, and it is nearly as efficient as the $t$-test on normal distributions.
    • The $U$-test is more widely applicable than independent samples Student's $t$-test, and the question arises of which should be preferred.
    • As it compares the sums of ranks, the Mann–Whitney test is less likely than the $t$-test to spuriously indicate significance because of the presence of outliers (i.e., Mann–Whitney is more robust).
    • For distributions sufficiently far from normal and for sufficiently large sample sizes, the Mann-Whitney Test is considerably more efficient than the $t$.
    • Overall, the robustness makes Mann-Whitney more widely applicable than the $t$-test.

  • The t-Distribution

    • Student's $t$-distribution arises in estimation problems where the goal is to estimate an unknown parameter when the data are observed with additive errors.
    • 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.
    • 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 $T$:
    • This distribution is important in studies of the power of Student's $t$-test.

  • Wilcoxon t-Test

    • The Wilcoxon $t$-test assesses whether population mean ranks differ for two related samples, matched samples, or repeated measurements on a single sample.
    • It can be used as an alternative to the paired Student's $t$-test, $t$-test for matched pairs, or the $t$-test for dependent samples when the population cannot be assumed to be normally distributed.
    • The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples.
    • In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$.
    • Other names may include the "$t$-test for matched pairs" or the "$t$-test for dependent samples."

  • Assumptions

    • Assumptions of a $t$-test depend on the population being studied and on how the data are sampled.
    • Most $t$-test statistics have the form $t=\frac{Z}{s}$, where $Z$ and $s$ are functions of the data.
    • If using Student's original definition of the $t$-test, the two populations being compared should have the same variance (testable using the $F$-test or assessable graphically using a Q-Q plot).
    • If the sample sizes in the two groups being compared are equal, Student's original $t$-test is highly robust to the presence of unequal variances.
    • Welch's $t$-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.

  • t-Test for Two Samples: Paired

    • Paired-samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice.
    • $t$-tests are carried out as paired difference tests for normally distributed differences where the population standard deviation of the differences is not known.
    • Paired samples $t$-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" $t$-test).
    • Because half of the sample now depends on the other half, the paired version of Student's $t$-test has only $\frac{n}{2-1}$ degrees of freedom (with $n$ being the total number of observations.
    • Paired-samples $t$-tests are often referred to as "dependent samples $t$-tests" (as are $t$-tests on overlapping samples).

  • Multivariate Testing

    • Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
    • A generalization of Student's $t$-statistic, called Hotelling's $T$-square statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.
    • Hotelling's $T^2$ statistic follows a $T^2$ distribution.
    • Hotelling's $T$-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's $t$-distribution.
    • In particular, the distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a $t$-test.

  • Distribution Needed for Hypothesis Testing

    • 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:
    • If you are testing a single population proportion, the distribution for the test is for proportions or percentages:

  • t-Test for One Sample

    • The $t$-test is the most powerful parametric test for calculating the significance of a small sample mean.
    • The formula for the $t$-statistic $T$ for a one-sample test is as follows:
    • Under $H_0$ the statistic $T$ will follow a Student's distribution with $19$ degrees of freedom: $T\sim \tau \cdot (20-1)$.
    • Determine the so-called $p$-value of the value $t$ of the test statistic $T$.
    • The Student's distribution gives $T\left( 19 \right) =1.729$ at probabilities $0.95$ and degrees of freedom $19$.