AP Statistics Significance Tests for a Proportion Lesson (Free PDF)

AP Statistics math lesson: Significance Tests for a Proportion – Maths PDF

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This test proportion lesson for AP Statistics covers significance tests for a proportion (Common Core AP STAT 6.4, AP STAT 6.5, AP STAT 6.6, AP STAT 6.7, HSS-IC.A.2, HSS-IC.B.5), with a free PDF to print.

A significance test for a proportion compares H0: p = p0 with an alternative using z = (p-hat – p0)/sqrt(p0(1-p0)/n). The p-value is the chance, if H0 is true, of a result this extreme. If it is below alpha, reject H0. A Type I error rejects a true H0; a Type II error fails to reject a false H0.

AP Statistics · Common Core State Standards: AP STAT 6.4, AP STAT 6.5, AP STAT 6.6, AP STAT 6.7, HSS-IC.A.2, HSS-IC.B.5

Key facts in this test proportion lesson

Key facts

H0 states no effect (p = p0); Ha is p > p0, p < p0, or p not equal to p0.
z = (p-hat – p0) / sqrt(p0(1 – p0)/n); check random, 10%, and large counts with p0.
p-value: probability of a result at least as extreme as observed, assuming H0 is true.
p-value < alpha: reject H0 (convincing evidence); otherwise fail to reject; never 'accept H0'.
Type I error: reject true H0 (probability alpha). Type II: fail to reject false H0; power = 1 – P(Type II).

What is a significance test?

A significance test uses sample data to decide between two competing claims about a population parameter. The null hypothesis \(H_0\) is the claim of no effect or no change (such as \(p=0.45\)). The alternative hypothesis \(H_a\) is what we are trying to find evidence for (\(p > 0.45\), \(p < 0.45\), or \(p\neq 0.45\)).

The four-step process for a test.

Example: a basketball player usually makes 45% of her three-point shots. After training, she makes 58 of 100. Is there evidence that her true percentage has increased? \(H_0: p=0.45\) and \(H_a: p > 0.45\), where \(p\) is the true proportion of three-pointers she makes after training.

How do you compute the test statistic and p-value?

For a one-sample \(z\) test for a proportion:

\[z=\frac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\]

Conditions: random sample, 10% condition, and large counts using \(p_0\): \(np_0\) and \(n(1-p_0)\) both at least 10. Here \(np_0=45\) and \(n(1-p_0)=55\).

\(\hat{p}=\frac{58}{100}=0.58\), \(SE=\sqrt{\frac{0.45(0.55)}{100}}=0.0497\), \(z=\frac{0.58-0.45}{0.0497}=2.61\).

Right-tailed p-value for z = 2.61: 0.0045.

The p-value is the probability, assuming \(H_0\) is true, of getting a sample result at least as extreme as the one observed. Here \(P(Z>2.61)=0.0045\). For a two-sided test, double the tail area.

A two-sided p-value adds both tails.

How do you reach a conclusion?

Compare the p-value with the significance level \(\alpha\) (often 0.05). If \(p\text{-value}<\alpha\), reject \(H_0\): there is convincing evidence for \(H_a\). Otherwise fail to reject \(H_0\): there is not convincing evidence. Here \(0.0045>0.05\), so we fail to reject \(H_0\). The data do not give convincing evidence that her true three-point percentage is above 0.45. We never say that we “accept” \(H_0\).

What are Type I and Type II errors?

Type I and Type II errors.

A Type I error rejects a true \(H_0\); its probability is \(\alpha\). A Type II error fails to reject a false \(H_0\). The power of a test is the probability of rejecting a false \(H_0\); it increases with a larger sample size, a larger \(\alpha\), or a parameter farther from \(p_0\).

Test proportion lesson: questions and answers

What does a p-value of 0.03 mean?

If the null hypothesis were true, there would be only a 3% chance of getting a sample result as extreme as, or more extreme than, the one we observed. It is not the probability that H0 is true.

Why do we say ‘fail to reject’ instead of ‘accept’ H0?

A test can only show that the data are inconsistent with H0. A large p-value means the sample is plausible under H0, but other values of p might also be plausible.

How do I choose between one-sided and two-sided alternatives?

Use the direction in the question before collecting data. If the claim is ‘greater than’ or ‘less than,’ use a one-sided test; if it asks whether the parameter is ‘different’ from a value, use a two-sided test.

What is the relationship between alpha and a Type I error?

The significance level alpha is the probability of making a Type I error when H0 is true. Choosing alpha = 0.05 means a 5% chance of rejecting a true null.

How can I increase the power of a test?

Use a larger sample size, a larger significance level, or study a situation where the true parameter is farther from the null value. Lower variability also raises power.

How to use this test proportion lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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