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.

  • Additional Informaiton

    • When you calculate the p-value and draw the picture, the p-value is the area in the left tail, the right tail, or split evenly between the two tails.
    • Similarly, for a large p-value like 0.4, as opposed to a p-value of 0.056 (alpha=0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in failing to reject the null hypothesis.
    • The picture of the p-value is as follows:
    • The picture of the p-value is as follows:
    • The picture of the p-value is as follows.

  • 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.

  • Hypothesis testing for a proportion

    • The upper tail area, representing the p-value, is 0.1867.
    • Set up hypotheses and verify the conditions using the null value, p 0 , to ensure $\hat{p}$ is nearly normal under H 0 .
    • If the conditions hold, construct the standard error, again using p 0 , and show the p-value in a drawing.
    • Lastly, compute the p-value and evaluate the hypotheses.
    • The p-value for the test is shaded.

  • 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.

  • 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.

  • 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).

  • Review

    • P ( x > 10 ) = P ( x ≤ 6 )
    • If P ( G | H ) = P ( G ) , then which of the following is correct?
    • If P ( J ) = 0.3, P ( K ) = 0.6, and J and K are independent events, then explain which are correct and which are incorrect.
    • P ( J ) 6= P ( J | K )

  • Summary of Formulas

    • ( lower value,upper value ) = ( point estimate − error bound,point estimate + error bound )
    • error bound = upper value − point estimate OR error bound = (upper value − lower value)/2
    • Use the Normal Distribution for a single population proportion p' = x/n
    • The confidence interval has the format (p' − EBP, p' + EBP) .

  • Complement of an event

    • Rolling a die produces a value in the set {1, 2, 3, 4, 5, 6}.
    • (a) Compute P(Dc) = P(rolling a 1, 4, 5, or 6). ( b) What is P(D) + P(Dc)?
    • (c) Compute P(A) + P(Ac) and P(B) + P(Bc).
    • P(A) = 1 - P(Ac) (2.25)
    • Therefore, P(A) + P(Ac) = 1 and P(B) + P(Bc) = 1.