F-Test

Examples of F-Test in the following topics:

• Variance Estimates

  • The $F$-test can be used to test the hypothesis that the variances of two populations are equal.
  • This $F$-test is known to be extremely sensitive to non-normality.
  • $F$-tests are used for other statistical tests of hypotheses, such as testing for differences in means in three or more groups, or in factorial layouts.
  • However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the $F$-test is relatively robust.
  • Discuss the $F$-test for equality of variances, its method, and its properties.

• The F-Test

  • An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.
  • An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.
  • Exact F-tests mainly arise when the models have been fitted to the data using least squares.
  • The F-test is sensitive to non-normality.
  • This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).

• The One-Way F-Test

  • The $F$-test as a one-way analysis of variance assesses whether the expected values of a quantitative variable within groups differ from each other.
  • The $F$ test as a one-way analysis of variance is used to assess whether the expected values of a quantitative variable within several pre-defined groups differ from each other.
  • If the $F$-test is performed at level $\alpha$ we cannot state that the treatment pair with the greatest mean difference is significantly different at level $\alpha$.
  • Note that when there are only two groups for the one-way ANOVA $F$-test, $F=t^2$ where $t$ is the Student's $t$-statistic.
  • Explain the purpose of the one-way ANOVA $F$-test and perform the necessary calculations.

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • The test states: if the function $f$ is twice differentiable at a critical point $x$ (i.e.
  • If $f''(x) < 0$ then f(x) has a local maximum at $x$.
  • Now, by the first derivative test, $f(x)$ has a local minimum at $x$.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Summary

  • A One-Way ANOVA hypothesis test determines if several population means are equal.
  • The distribution for the test is the F distribution with 2 different degrees of freedom.
  • A Test of Two Variances hypothesis test determines if two variances are the same.
  • The distribution for the hypothesis test is the F distribution with 2 different degrees of freedom.

• The Integral Test and Estimates of Sums

  • The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
  • The integral test for convergence is a method used to test infinite series of non-negative terms for convergence.
  • The infinite series $\sum_{n=N}^\infty f(n)$ converges to a real number if and only if the improper integral $\int_N^\infty f(x)\,dx$ is finite.
  • for every $\varepsilon > 0$, and whether the corresponding series of the $f(n)$ still diverges.
  • The integral test applied to the harmonic series.

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • For a function of two variables, suppose that $M(x,y)= f_{xx}(x,y)f_{yy}(x,y) - \left( f_{xy}(x,y) \right)^2$.
  • If $M(a,b)>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a local minimum of $f$.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• Randomization Tests: Two or More Conditions

  • Compute a randomization test for differences among more than two conditions.
  • The method of randomization for testing differences among more than two means is essentially very similar to the method when there are exactly two means.
  • The first step in a randomization test is to decide on a test statistic.
  • The F ratio is computed not to test for significance directly, but as a measure of how different the groups are.
  • Therefore, the proportion of arrangements with an F as large or larger than the F of 2.06 obtained with the data is

• 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.

• Restricting Domains to Find Inverses

  • $f^{-1}(x)$ is defined as the inverse function of $f(x)$ if it consistently reverses the $f(x)$ process.
  • More concisely and formally, $f^{-1}x$ is the inverse function of $f(x)$ if $f({f}^{-1}(x))=x$.
  • Without any domain restriction, $f(x)=x^2$ does not have an inverse function as it fails the horizontal line test.
  • But if we restrict the domain to be $x > 0$ then we find that it passes the horizontal line test and therefore has an inverse function.  
  • This function fails the horizontal line test and therefore does not have an inverse.