Source: Quiz 7 Solutions, Principles of Statistics I, Texas A&M University
Tags: hypothesis testing, z-test for proportions, population proportion, sample proportion, p-hat, significance level, null hypothesis, one-tailed test, CDC, STAT 211
When the parameter of interest is a population proportion (not a mean), the z-test formula changes: the denominator uses p₀ from the null hypothesis instead of a sample standard deviation. The logic of setting up hypotheses, computing a test statistic, finding the p-value, and making a decision stays identical. Proportion problems typically involve percentages, rates, or "what fraction of the population" questions.
Population proportion (p)
The true fraction of the population that has a given characteristic. This is the parameter you are testing.
Sample proportion (p̂)
The fraction observed in the sample. Calculated as p̂ = X / n, where X is the number of "successes" in the sample and n is the sample size.
Null hypothesis value (p₀)
The specific proportion stated in H₀. This value is used in the denominator of the test statistic, not p̂.
Standard error of p̂ (under H₀)
√(p₀(1 − p₀) / n). This measures the expected variability of the sample proportion assuming the null is true.
Use this test when:
The variable is categorical (yes/no, meets requirement/doesn't, etc.)
You are testing a claim about the population proportion
The sample is large enough that np₀ ≥ 5 and n(1 − p₀) ≥ 5 (normal approximation conditions)
z* = (p̂ − p₀) / √(p₀(1 − p₀) / n)
Key difference from the mean test: the denominator is built entirely from p₀ and n. You do not use the sample proportion in the denominator.
The process is the same as for means:
State H₀ and H₁ (with p₀ as the value and the direction based on the research question)
Calculate the sample proportion p̂ = X / n
Compute the test statistic z*
Find the p-value using the appropriate tail direction
Compare p-value to α and state your conclusion in context
Proportion problems often use phrasing like:
"Is the proportion less than..." → left-tailed, H₁: p < p₀
"Is the proportion greater than..." → right-tailed, H₁: p > p₀
"Is the proportion different from..." → two-tailed, H₁: p ≠ p₀
Setup: According to the CDC, 49.4% of American adults 18 and over meet aerobic physical activity guidelines (p₀ = 0.494). A gym wants to know if the proportion of its members meeting the guidelines is less than the national average. They sample n = 40 members and find X = 28 meet the guidelines. Test at α = 0.01.
Step 1: Hypotheses
H₀: p = 0.494
H₁: p < 0.494
(Left-tailed because the gym suspects their proportion is lower.)
Step 2: Sample proportion
p̂ = 28 / 40 = 0.7
Step 3: Test statistic
z* = (0.7 − 0.494) / √(0.494 × 0.506 / 40)
z* = 0.206 / √(0.249964 / 40)
z* = 0.206 / √0.006249
z* = 0.206 / 0.0791 ≈ 2.606
Step 4: p-value
Left-tailed test, so p-value = P(Z < 2.606).
Since 2.606 is far to the right, P(Z < 2.606) = 0.995. The p-value is approximately 0.995.
Step 5: Decision
0.995 > 0.01 → fail to reject H₀.
Conclusion: We do not have sufficient evidence to conclude that the proportion of gym members who meet the physical activity requirement is less than the national average.
The sample proportion (0.7 = 70%) is actually higher than the national rate (49.4%). We were testing whether the gym's rate was lower. The data goes in the opposite direction of the alternative hypothesis, which is why the p-value is so large. The test statistic is positive even though we had a left-tailed test, placing it in the wrong tail entirely.
Z-test statistic for a population proportion:
z* = (p̂ − p₀) / √(p₀(1 − p₀) / n)
Sample proportion:
p̂ = X / n
Standard error under H₀:
SE = √(p₀(1 − p₀) / n)
What stays the same:
The hypothesis testing framework (H₀, H₁, test stat, p-value, decision, conclusion)
The decision rule (compare p-value to α)
The conclusion phrasing
What changes:
The parameter: μ vs. p
The test statistic denominator: σ/√n for means vs. √(p₀(1−p₀)/n) for proportions
The numerator: (x̄ − μ₀) for means vs. (p̂ − p₀) for proportions
The standard error for proportions uses the null value p₀, not the sample value p̂
⚠️ In the proportion z-test, always use p₀ (the null hypothesis value) in the denominator, not p̂. This is probably the most common computational error on proportion problems.
⚠️ Make sure you convert percentages to decimals before computing. If the problem says "49.4%," use 0.494 in your formula.
⚠️ Check the direction of your test against the data. If p̂ is larger than p₀ but you have a left-tailed test, your z* will be positive and your p-value will be very large, meaning you fail to reject. This is correct, not an error.
⚠️ The normal approximation conditions (np₀ ≥ 5 and n(1−p₀) ≥ 5) should be checked before running the test, though Quiz 7 does not explicitly test this.
⚠️ The significance level in the proportion problem was α = 0.01, not 0.05. Always read the problem carefully for the given α.
Q: Write the z-test statistic formula for a population proportion.
A: z* = (p̂ − p₀) / √(p₀(1 − p₀) / n)
Q: In a sample of 50 people, 30 say yes. What is p̂?
A: p̂ = 30 / 50 = 0.60.
Q: Why do we use p₀ rather than p̂ in the denominator of the proportion z-test?
A: Because the test statistic measures how far the sample result is from the null, assuming H₀ is true. Under H₀, the standard error is based on p₀, the hypothesised value.
Q: You test H₀: p = 0.5 vs. H₁: p < 0.5, and your sample proportion is 0.62. What will happen to your p-value?
A: The z* will be positive (sample proportion is above the null value), so P(Z < z*) will be close to 1. The p-value will be very large, and you will fail to reject H₀. The data goes against the direction of the alternative.
Q: What is the key difference between the denominators of the mean z-test and the proportion z-test?
A: The mean z-test uses σ/√n (or s/√n), where σ or s comes from the data. The proportion z-test uses √(p₀(1−p₀)/n), built entirely from the null hypothesis value and the sample size.
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