IntroStatsCalc

One-Proportion Z-Test

Hypothesis test for a population proportion (1-PropZTest).

Inputs

Alternative hypothesis Ha
Significance level α

Results update as you type.

Results

Reject H₀: z = 2, p-value = 0.0455

The p-value (0.0455) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the population proportion is different from 0.5.

Standard normal curve. The shaded p-value region is both tails beyond z = 2, with area 0.0455. Dashed lines mark the critical values −1.96 and 1.96.−4−2024z = 2critical −1.96critical 1.96

Results

p̂ = x/n0.6
Test statistic z2
p-value0.0455
Standard error0.05
Critical value(s)−1.96, 1.96
Decision at α = 0.05Reject H₀

On the TI-84

STAT → TESTS → 5:1-PropZTest. Enter p0, x, n, pick the alternative, Calculate.

1-PropZTest 0.5, 60, 100, ≠

Show steps
  1. Hypotheses. H₀: p = 0.5   Ha: p ≠ 0.5
  2. Sample proportion. p̂ = 60 ÷ 100 = 0.6
  3. Test statistic. z = (p̂ − p0) ÷ √(p0(1 − p0) ÷ n) = (0.6 − 0.5) ÷ √(0.5 · 0.5 ÷ 100) = 0.1 ÷ 0.05 = 2
  4. p-value. Two-tailed test, so double the tail area beyond the test statistic: p-value = 0.0455. Since 0.0455 ≤ 0.05, reject H₀.
  5. Critical-value method. At α = 0.05, the rejection region is z ≤ −1.96 or z ≥ 1.96. The test statistic z = 2 is in the rejection region, so we reject H₀. (Same decision as the p-value method.)
  6. Conclusion. The p-value (0.0455) is less than or equal to α = 0.05, so the data give enough evidence to conclude that the population proportion is different from 0.5.

About this calculator

When to use this

Use a one-proportion z-test to check a claim about the percent of a population, such as "more than half of students prefer online classes." x is the number of successes in a random sample of size n, and p0 is the claimed proportion.

Notice the standard error uses p0 (the value assumed true under H0), not p̂. That's different from the confidence interval, which uses p̂.

More inference calculators: CI for a Mean · CI for a Proportion · Sample Size · Z-Test (mean) · T-Test (mean) · 2-Sample T-Test · Paired T-Test · 2-Prop Z-Test