Examples of functional test in the following topics:
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- Recognize whether a function has an inverse by using the horizontal line test
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- The vertical line test is used to determine whether a curve on an $xy$-plane is a function
- Apply the vertical line test to determine which graphs represent functions.
- The vertical line test demonstrates that a circle is not a function.
- Thus, it fails the vertical line test and does not represent a function.
- Explain why the vertical line test shows, graphically, whether or not a curve is a function
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- An easy way to check if a function is a one-to-one is by graphing it and then performing the horizontal line test.
- One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.
- Notice it fails the horizontal line test.
- The graph of the function,$f(x)=\left | x-2 \right |$, fails the horizontal line test and is therefore not a one-to-one function.
- The graph of the function $f(x)=x^2$ fails the horizontal line test and is therefore NOT a one-to-one function.
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- The purpose of the scrotum is to provide the testes with a chamber of appropriate temperature for optimal sperm production.
- The left testis is usually lower than the right, which may
function to avoid compression in the event of impact.
- This asymmetry may also allow
more effective cooling of the testes.
- The function of the scrotum appears to be to keep the temperature of the testes slightly lower than that of the rest of the body.
- However, temperature regulation may not be the only function of the scrotum.
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- The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
- In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
- The test states: if the function $f$ is twice differentiable at a critical point $x$ (i.e.
- 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
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- non-parametric statistics (in the sense of a statistic over data, which is defined to be a function on a sample that has no dependency on a parameter), whose interpretation does not depend on the population fitting any parameterized distributions.
- Anderson–Darling test: tests whether a sample is drawn from a given distribution.
- Median test: tests whether two samples are drawn from distributions with equal medians.
- Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function.
- Squared ranks test: tests equality of variances in two or more samples.
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- 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.
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- A $z$-test is a test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution.
- A $Z$-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution.
- We then calculate the standard score $Z = \frac{(T-\theta)}{s}$, from which one-tailed and two-tailed $p$-values can be calculated as $\varphi(-Z)$ (for upper-tailed tests), $\varphi(Z)$ (for lower-tailed tests) and $2\varphi(\left|-Z\right|)$ (for two-tailed tests) where $\varphi$ is the standard normal cumulative distribution function.
- For larger sample sizes, the $t$-test procedure gives almost identical $p$-values as the $Z$-test procedure.
- If the variation of the test statistic is strongly non-normal, a $Z$-test should not be used.
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- The result is a t-score test statistic.
- Although the t-test will be explained in great detail later in this textbook, it is important for the reader to have a basic understanding of its function in regard to comparing two 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.
- 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).
- 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.
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- 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.
- This can be tested using a normality test, or it can be assessed graphically using a normal quantile plot.
- 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).
- The data used to carry out the test should be sampled independently from the two populations being compared.