t-test

Examples of t-test in the following topics:

• 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.
  • For example, in the $t$-test comparing the means of two independent samples, the following assumptions should be met:
  • 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).
  • Welch's $t$-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.

• Comparing Two Sample Averages

  • Student's t-test is used in order to compare two independent sample means.
  • The result is a t-score test statistic.
  • 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.
  • In this case, we have two independent samples and would use the unpaired form of the t-test.
  • Paired sample 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).

• 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.
  • Gosset devised the t-test as a cheap way to monitor the quality of stout.
  • Gosset's work on the t-test was published in Biometrika in 1908.
  • The form of the test used when this assumption is dropped is sometimes called Welch's t-test.

• 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).
  • A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication .
  • Paired-samples $t$-tests are often referred to as "dependent samples $t$-tests" (as are $t$-tests on overlapping samples).

• t-Test for Two Samples: Independent and Overlapping

  • Two-sample t-tests for a difference in mean involve independent samples, paired samples, and overlapping samples.
  • The two sample t-test is used to compare the means of two independent samples.
  • In the latter case the estimated t-statistic must either be tested with modified degrees of freedom, or it can be tested against different critical values.
  • The two-sample t-test is probably the most widely used (and misused) statistical test.
  • In this case, we have two independent samples and would use the unpaired form of the 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."

• Alternatives to the t-Test

  • When the normality assumption does not hold, a nonparametric alternative to the $t$-test can often have better statistical power.
  • The $t$-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances.
  • The Welch's $t$-test is a nearly exact test for the case where the data are normal but the variances may differ.
  • For exactness, the $t$-test and $Z$-test require normality of the sample means, and the $t$-test additionally requires that the sample variance follows a scaled $\chi^2$ distribution, and that the sample mean and sample variance be statistically independent.
  • The nonparametric counterpart to the paired samples $t$-test is the Wilcoxon signed-rank test for paired 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.
  • Because measures of this type are usually highly correlated, it is not advisable to conduct separate univariate $t$-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (type I error).
  • Hotelling's $T^2$ statistic follows a $T^2$ 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.

• Quantitative or Qualitative Data?

  • Paired and unpaired t-tests and z-tests are just some of the statistical tests that can be used to test quantitative data.
  • A t-test is any statistical hypothesis test in which the test statistic follows a 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 t distribution .
  • This fact makes it more convenient than the t-test, which has separate critical values for each sample size.
  • Plots of the t distribution for several different degrees of freedom.

• Tips for Testing Series

  • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
  • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
  • When testing the convergence of a series, you should remember that there is no single convergence test which works for all series.
  • Integral test: For a positive, monotone decreasing function $f(x)$ such that $f(n)=a_n$, if $\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty$ then the series converges.
  • The integral test applied to the harmonic series.