p-value

(noun)

The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

Related Terms

Examples of p-value in the following topics:

  • Determining the Solution

    • Then pA and pB are the desired population proportions.
    • P' A − P' B = 0.1 − 0.06 = 0.04.
    • Half the p-value is below -0.04 and half is above 0.04.
    • Compare α and the p-value: α = 0.01 and the p-value = 0.1404. α < p-value.
    • The p-value is p = 0.1404 and the test statistic is 1.47.

  • Generating the exact null distribution and p-value

    • If the hypothesis test is one-sided, then the p-value is represented by a single tail area.
    • If the test is two-sided, compute the single tail area and double it to get the p-value, just as we have done in the past.
    • Compute the exact p-value to check the consultant's claim that her clients' complication rate is below 10%.
    • We can compute the p-value by adding up the cases where there are 3 or fewer complications:
    • This exact p-value is very close to the p-value based on the simulations (0.1222), and we come to the same conclusion.

  • Comparing Two Independent Population Proportions

    • Half the $p$-value is below $-0.04$ and half is above 0.04.
    • Compare $\alpha$ and the $p$-value: $\alpha = 0.01$ and the $p\text{-value}=0.1404$.
    • $\alpha = p\text{-value}$.
    • Make a decision: Since $\alpha = p\text{-value}$, do not reject $H_0$.
    • This image shows the graph of the $p$-values in our example.

  • The Henderson-Hasselbalch Equation

    • The Henderson–Hasselbalch equation connects the measurable value of the pH of a solution with the theoretical value pKa.
    • $-p{ K }_{ a }=-pH+log(\frac { [A^{ - }] }{ [HA] } )$
    • $pH=p{ K }_{ a }+log(\frac { { [A }^{ - }] }{ [HA] } )$
    • ${ 10 }^{ pH-p{ K }_{ a } }=\frac { [base] }{ [acid] }$
    • $pH=p{ K }_{ a }+log(\frac { { [NH_3}] }{ [NH_4^+] } )$

  • Expected Values of Discrete Random Variables

    • The probability distribution of a discrete random variable $X$ lists the values and their probabilities, such that $x_i$ has a probability of $p_i$.
    • The probabilities $p_i$ must satisfy two requirements:
    • The sum of the probabilities is 1: $p_1+p_2+\dots + p_i = 1$.
    • Suppose random variable $X$ can take value $x_1$ with probability $p_1$, value $x_2$ with probability $p_2$, and so on, up to value $x_i$ with probability $p_i$.
    • If all outcomes $x_i$ are equally likely (that is, $p_1 = p_2 = \dots = p_i$), then the weighted average turns into the simple average.

  • Microbial Growth at Low or High pH

    • Pure water has a pH very close to 7 at 25°C.
    • Solutions with a pH less than 7 are said to be acidic, and solutions with a pH greater than 7 are said to be basic or alkaline .
    • The pH scale is traceable to a set of standard solutions whose pH is established by international agreement.
    • Neutrophiles are organisms that thrive in neutral (pH 7) environments; extromophiles are organisms that thrive in extreme pH environments.
    • A pH scale with annotated examples of chemicals at each integer pH value

  • Summary of Functions

    • X takes on the values x = 0,1, 2, 3, ...
    • X may take on the values x= 0, 1, ..., up to the size of the group of interest.
    • (The minimum value for X may be larger than 0 in some instances. )
    • X takes on the values x = 0, 1, 2, 3, ...
    • This formula is valid when n is "large" and p "small" (a general rule is that n should be greater than or equal to 20 and p should be less than or equal to 0.05).

  • Transformation of Radiative Transfer

    • First let's write momenta in the primed frame in terms of its values in the unprimed frame, we have
    • $\displaystyle p'_t = \gamma \left ( p_t - \beta p_x \right ) \\ p'_x = \gamma \left ( p_x - \beta p_t \right ) \\ p'_y = p_y \\ p'_z = p_z .$
    • $\displaystyle \gamma \left ( 1 - \beta \frac{p_x}{p_t} \right ) = \frac{\gamma \left ( p_t - \beta p_x \right )}{p_t} = \frac{p_t'}{p_t}.$
    • $\displaystyle d^3 {\bf p}' = \frac{p_t'}{p_t} d^3 {\bf p} ~\textrm{so}~ \frac{ d^3 {\bf p}}{p_t} ~\textrm{is invariant. }$
    • Let's substitute the values of $x'_A$ and $t'_A$ in terms of $x_A$ and $t_A$ to yield

  • Difference Quotients

    • Its "input value" is its argument, usually a point ("P") expressible on a graph.
    • Forward difference: $\Delta F(P) = F(P + \Delta P) - F(P)$;
    • Central difference: $\delta F(P) = F(P + P) - F(P - P)$
    • Backward difference: $\nabla F(P) = F(P) - F(P - \Delta P)$.
    • $\frac{\Delta F(P)}{\Delta P} = F(P + \Delta P) - \frac{F(P)}{\Delta P} = [\nabla F(P + \Delta P)]\Delta P$

  • pOH and Other p Scales

    • However, because these values are often very small for weak acids and weak bases, the p-scale is used to simplify these numbers and make them more convenient to work with.
    • Quite often we will see the notation pKa or pKb, which refers to the negative logarithms of Ka or Kb, respectively.
    • Relation between p[OH] and p[H] (brighter red is more acidic, which is the lower numbers for the pH scale and higher numbers for the pOH scale; brighter blue is more basic, which is the higher numbers for the pH scale and lower numbers for the pOH scale).
    • This lesson introduces the pH scale and discusses the relationship between pH, [H+], [OH-] and pOH.
    • Convert between pH and pOH scales to solve acid-base equilibrium problems.