P-Value Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
A hypothesis test produces z=2.3z=2.3 for a two-tailed test. Calculate the p-value and interpret it at both α=0.05\alpha=0.05 and α=0.01\alpha=0.01.

Solution

  1. 1
    Two-tailed p-value: p=2×P(Z>2.3)=2×(1−0.9893)=2×0.0107=0.0214p = 2 \times P(Z > 2.3) = 2 \times (1 - 0.9893) = 2 \times 0.0107 = 0.0214
  2. 2
    At α=0.05\alpha=0.05: p=0.0214<0.05p=0.0214 < 0.05 → Reject H0H_0 (result is statistically significant)
  3. 3
    At α=0.01\alpha=0.01: p=0.0214>0.01p=0.0214 > 0.01 → Fail to reject H0H_0 (result is not significant at 1% level)
  4. 4
    Interpretation: there's a 2.14% probability of getting a test statistic at least as extreme as 2.3 if H0H_0 is true

Answer

p=0.0214p=0.0214. Significant at α=0.05\alpha=0.05 but not at α=0.01\alpha=0.01.
The p-value is the probability of obtaining evidence as extreme or more extreme than observed, assuming H₀ is true. Small p-values (< α) indicate data is unlikely under H₀. The same p-value can lead to different conclusions depending on the chosen significance level.

About P-Value

The probability of observing a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis H0H_0 is true.

Learn more about P-Value →

More P-Value Examples

Example 2 hard

Correct the following misconceptions about p-values: (a) 'p=0.03 means there's a 3% chance H₀ is tru

Example 3 easy

A one-tailed test has [formula]. Find the p-value and determine if we reject [formula] at [formula].

Example 4 hard

Study A: [formula], [formula], effect size [formula] (tiny). Study B: [formula], [formula], effect s

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