Student's t-distribution

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

• Student Learning Outcomes

  • By the end of this chapter, the student should be able to:

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

  • This problem led him to "discover" what is called the Student's-t distribution.
  • 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.
  • A probability table for the Student's-t distribution can also be used.
  • The notation for the Student's-t distribution is (using T as the random variable) is

• 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.
  • Student's $t$-distribution (or simply the $t$-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
  • 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.

• 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:

• Summary of Formulas

  • Use the Normal Distribution for Means (Section 7.2) $EBM = z_{\frac{\alpha }{2}} \cdot (\frac{\sigma }{\sqrt{n}})$
  • Use the Student's-t Distribution with degrees of freedom df = n − 1.
  • $EBM = t_{\frac{\alpha }{2}} \cdot \frac{s}{\sqrt{n}}$
  • Use the Normal Distribution for a single population proportion p' = x/n

• 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.

• 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.

• Multivariate Testing

  • 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.
  • It is proportional to the $F$-distribution.

• 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.
  • Because of a contractual agreement with the brewery, he published the article under the pseudonym "Student. " That is why the t test is called the "Student's t test. "
  • The t distribution is therefore leptokurtic.
  • The t distribution approaches the normal distribution as the degrees of freedom increase.

• Assumptions

  • Typically, $Z$ is designed to be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas $s$ is a scaling parameter that allows the distribution of $t$ to be determined.
  • $s^2$ follows a $\chi^2$ distribution with $p$ degrees of freedom under the null hypothesis, where $p$ is a positive constant.
  • Each of the two populations being compared should follow a normal distribution.
  • 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.