One-Sample Tests for Proportions
In the previous lesson, we used a one-sample Z test to compare a sample mean with a population mean.
In many practical situations, however, we are interested in proportions or percentages, not averages.
This lesson explains how to perform a one-sample hypothesis test for proportions.
What Is a One-Sample Proportion Test?
A one-sample proportion test is used to determine whether the proportion of a population is significantly different from a claimed or expected value.
This test is commonly used in:
- Opinion polls
- Quality control
- Marketing studies
- A/B testing (basic form)
When Can We Use This Test?
A one-sample proportion Z test is appropriate when:
- The data represents successes and failures
- The sample is random
- The sample size is large enough
A common rule of thumb:
np ≥ 10 and n(1 − p) ≥ 10
Setting Up the Hypotheses
Let p represent the true population proportion.
| Hypothesis | Meaning |
|---|---|
| H₀ | p equals the claimed proportion |
| H₁ | p differs from the claimed proportion |
As before, the alternative hypothesis can be:
- Two-tailed (≠)
- Left-tailed (<)
- Right-tailed (>)
The Test Statistic for Proportions
The Z test statistic for proportions is:
Z = ( p̂ − p₀ ) ÷ √[ p₀(1 − p₀) / n ]
- p̂ = sample proportion
- p₀ = claimed population proportion
- n = sample size
Deep Numerical Example (Step-by-Step)
A company claims that 60% of customers prefer its product.
A survey of 200 customers finds that 102 prefer the product.
- Sample size (n) = 200
- Sample successes = 102
- Claimed proportion (p₀) = 0.60
- Significance level (α) = 0.05
Step 1: Calculate the Sample Proportion
p̂ = 102 ÷ 200 = 0.51
Step 2: State the Hypotheses
H₀: p = 0.60
H₁: p ≠ 0.60
Step 3: Calculate the Standard Error
√[ p₀(1 − p₀) / n ] = √[ 0.60 × 0.40 / 200 ]
= √(0.24 / 200) = √0.0012 ≈ 0.0346
Step 4: Calculate the Z Value
Z = (0.51 − 0.60) ÷ 0.0346
Z ≈ −2.60
Step 5: Make the Decision
For α = 0.05 (two-tailed):
- Critical Z values = ±1.96
Since −2.60 < −1.96, it falls in the rejection region.
Decision: Reject the null hypothesis
Interpretation in Plain Language
There is sufficient statistical evidence to conclude that the true customer preference proportion is different from 60%.
The company’s claim is not supported by the data.
P-Value Interpretation
Using the p-value approach:
- If p-value ≤ α → Reject H₀
- If p-value > α → Fail to reject H₀
For Z = −2.60, the p-value is less than 0.01, which is smaller than 0.05.
Common Mistakes to Avoid
- Using sample proportion in the denominator
- Ignoring sample size conditions
- Confusing percentage with proportion
- Misinterpreting “fail to reject” as proof
Quick Check
Why do we use p₀ in the standard error formula?
Practice Quiz
Question 1:
What does p̂ represent?
Question 2:
When do we reject the null hypothesis?
Question 3:
Can this test be used for very small samples?
Mini Practice
A survey claims that 40% of users prefer dark mode.
- n = 150
- Number of users preferring dark mode = 54
- α = 0.05
Test whether the claim is valid.
What’s Next
In the next lesson, we will move to Two-Sample Independent t Tests, which compare means from two different groups.