Hypothesis Testing

Hypothesis testing is a method for using sample data to judge a claim about a population. You state a null hypothesis H₀ (no effect, no change) and an alternative hypothesis Hₐ (what you suspect). Then you ask how surprising the data would be if H₀ were true. Very surprising data count as evidence against H₀.

Before this lesson: Confidence Intervals, Sampling Distribution of a Sample Proportion and Standard Normal Distribution

The logic of a significance test

A hypothesis test works like a trial. The null hypothesis is “innocent until proven guilty.” You assume it is true, then look at the evidence. If the evidence would be very unlikely under H₀, you reject H₀. If not, you fail to reject it.

The null hypothesis H₀ says nothing special is going on: a parameter equals a claimed value. The alternative hypothesis Hₐ is the claim you want evidence for. Both are always about a population parameter, such as p or μ, never about a sample statistic like p̂ or x̄.

A test statistic measures how far the sample result is from the H₀ value, in standard errors. The p-value is the probability of a result at least that extreme if H₀ is true. A small p-value means the data are hard to explain by chance alone.

Hypothesis testing formula and steps

\text{test statistic} = \dfrac{\text{statistic} - \text{null value}}{\text{standard error}}

• For a proportion: z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}}, using the null value p₀ in the standard error.
• For a mean: t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}, with df = n − 1 (see t-distribution).

Three forms of the alternative

Four steps

1. State: write H₀ and Hₐ, define the parameter, and choose the significance level α (often 0.05).
2. Plan: name the test and check its conditions.
3. Do: compute the test statistic and the p-value.
4. Conclude: if p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀. Then answer the question in context.

The significance level α is the cutoff you set before looking at the data. A p-value at or below α is called statistically significant.

Hypothesis testing examples

Example 1: Write hypotheses for a mean

A phone maker claims its battery lasts 10 hours on average. A consumer group suspects the true mean is less. Write the hypotheses.

H_0: \mu = 10

H_a: \mu \lt 10

Answer: H₀: μ = 10 and Hₐ: μ < 10, where μ is the true mean battery life in hours for all phones of this model. This is a one-sided test.

Example 2: Write hypotheses for a proportion

A coin is flipped 200 times to see whether it is fair. Write the hypotheses.

H_0: p = 0.5

H_a: p \ne 0.5

Answer: H₀: p = 0.5 and Hₐ: p ≠ 0.5, where p is the true proportion of heads for this coin. This is a two-sided test.

Example 3: Compute a z test statistic

A city says 40% of its high school students walk to school. A random sample of 150 students finds 72 who walk. Find the test statistic for H₀: p = 0.40.

\hat{p} = \dfrac{72}{150} = 0.48

0.40 \cdot 0.60 = 0.24

\dfrac{0.24}{150} = 0.0016

\text{SE} = \sqrt{0.0016} = 0.04

0.48 - 0.40 = 0.08

z = \dfrac{0.08}{0.04} = 2

Answer: z = 2. The sample proportion is 2 standard errors above the claimed 40%.

Example 4: Make a decision

Use Example 3 with Hₐ: p > 0.40 and α = 0.05. Find the p-value and state a conclusion.

P(Z \ge 2) = \text{normalcdf}(2, 10^{99})

\approx 0.0228

0.0228 \lt 0.05

Answer: the p-value is about 0.0228, which is less than 0.05, so we reject H₀. There is convincing evidence that more than 40% of the city’s high school students walk to school.

Example 5: Checking a cereal label (word problem)

A cereal box says it holds 500 g. An inspector weighs a random sample of 16 boxes and finds x̄ = 496.5 g and s = 6 g. The weights show no strong skew or outliers. Test H₀: μ = 500 against Hₐ: μ < 500 at α = 0.05.

\sqrt{16} = 4

\text{SE} = \dfrac{6}{4} = 1.5

496.5 - 500 = -3.5

t = \dfrac{-3.5}{1.5} \approx -2.333

df = 16 - 1 = 15

P(t \le -2.333) = \text{tcdf}(-10^{99}, -2.333, 15)

\approx 0.017

0.017 \lt 0.05

Answer: t ≈ −2.333 and the p-value is about 0.017. Since 0.017 < 0.05, reject H₀. There is convincing evidence that the true mean weight of these boxes is less than 500 g.

Common mistakes

Related methods

Tests and confidence intervals

A two-sided test at α = 0.05 matches a 95% confidence interval. If the H₀ value is outside the interval, the test rejects H₀ (see the confidence interval for a mean). An interval also tells you how big the effect might be, which a test alone does not.

What comes next

The next lesson, P-Values, explains how to read and interpret a p-value in detail. Later lessons give the full test for each situation: one proportion, two proportions, one mean, paired data and two means.

Practice problems

1. A school claims 80% of its graduates go to college. A reporter thinks the true rate is lower. Write the hypotheses.
2. A restaurant says its mean wait time is 12 minutes. A manager wants to know if it is different. Write the hypotheses.
3. Is “H₀: x̄ = 50” a correct null hypothesis? Explain.
4. For H₀: p = 0.5 with n = 100 and p̂ = 0.56, find the z test statistic.
5. For H₀: μ = 20 with n = 25, x̄ = 21.5 and s = 5, find the t test statistic.
6. A test gives a p-value of 0.03, and α = 0.05. What is the decision?
7. A test gives a p-value of 0.12, and α = 0.05. What is the decision?
8. Why do statisticians say “fail to reject H₀” instead of “accept H₀”?
9. A drug company asks whether its new drug shortens the mean recovery time from 7 days. Write Hₐ and say whether the test is one-sided or two-sided.
10. A test of Hₐ: p < p₀ gives z = −1.8. Find the p-value and the decision at α = 0.05.

H_0: p = 0.80

H_a: p \lt 0.80

H_0: \mu = 12

H_a: \mu \ne 12

0.5 \cdot 0.5 = 0.25

\dfrac{0.25}{100} = 0.0025

\text{SE} = \sqrt{0.0025} = 0.05

0.56 - 0.5 = 0.06

z = \dfrac{0.06}{0.05} = 1.2

\text{SE} = \dfrac{5}{\sqrt{25}} = \dfrac{5}{5}

= 1

21.5 - 20 = 1.5

t = \dfrac{1.5}{1} = 1.5

0.03 \lt 0.05

0.12 \gt 0.05

P(Z \le -1.8) = \text{normalcdf}(-10^{99}, -1.8)

\approx 0.0359

0.0359 \lt 0.05

Hypothesis testing FAQ

What is the difference between the null and alternative hypothesis?

The null hypothesis H₀ says there is no effect or no change, and it always contains an equal sign. The alternative Hₐ is what you are looking for evidence of, using <, > or ≠.

What does it mean to reject the null hypothesis?

It means the data would be very unlikely if H₀ were true, since the p-value is at or below α. So you have convincing evidence for Hₐ.

What is a significance level?

The significance level α is the cutoff for “unlikely” that you choose before collecting data. The most common choice is α = 0.05, and 0.01 and 0.10 are also used.

How do you know if a test is one-sided or two-sided?

Look at the question. Words like “more than,” “less than” or “increase” give a one-sided test. Words like “different” or “changed” give a two-sided test.

Summary

• H₀ states that a parameter equals a claimed value; Hₐ states what you suspect (<, > or ≠).
• The test statistic is (statistic − null value) ÷ standard error.
• The p-value is the probability of a result at least this extreme if H₀ is true.
• If p-value ≤ α, reject H₀; otherwise, fail to reject H₀.
• Always state the conclusion in context, about the population parameter.

Previous lesson: Confidence Interval for a Difference in Means

Next lesson: P-Values

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