Hypothesis Tests for Means
A free Statistics and Data Analysis lesson from the “Inference and Conclusions” unit, with a worked example and practice problems including step-by-step solutions.
A test for a mean compares a sample average to a claimed population average. The p-value helps decide whether the sample mean is unusually far from the claim. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Test a claim about a population mean
- Use sample mean and standard error
- Write a conclusion about a mean in context
Worked example
Problem. A package claims the average fill is 12 ounces. A sample gives p-value 0.04 for the claim that the true mean is less than 12. At alpha = 0.05, what decision fits?
- Worked Example: First identify exactly what the question is asking: A package claims the average fill is 12 ounces. A sample gives p-value 0.04 for the claim that the true mean is less than 12. At alpha = 0.05, what decision fits?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Compare p-value to alpha.
- 0.04 is below 0.05, so reject the null.
Answer: reject the null hypothesis
Practice problems
1. Practice case A: A hypothesis test for average score is about:
Choices: population mean · population proportion · sample size only · placebo effect
Show solution
- Warm-up: First identify exactly what the question is asking: A hypothesis test for average score is about:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A mean is an average.
- Mean tests are about population means.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: population mean
2. Practice case B: A claim says the mean score is 75 points. Which null fits?
Choices: p = 75 · mu > 75 · x = 75 · mu = 75
Show solution
- Warm-up: First identify exactly what the question is asking: A claim says the mean score is 75 points. Which null fits?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- The null usually includes equality.
- Use mu for a population mean.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: mu = 75
3. Practice case C: A researcher suspects the true mean is greater than 50. Which alternative fits?
Choices: mu = 50 · mu < 50 · mu > 50 · p > 50
Show solution
- Warm-up: First identify exactly what the question is asking: A researcher suspects the true mean is greater than 50. Which alternative fits?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- The alternative states the suspected direction.
- Use mu for means.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: mu > 50
4. Practice case D: For alpha = 0.10 and p-value = 0.07, what is the decision?
Choices: increase p-value · reject the null hypothesis · fail to reject the null hypothesis · prove the null
Show solution
- Warm-up: First identify exactly what the question is asking: For alpha = 0.10 and p-value = 0.07, what is the decision?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Reject when p-value is below alpha.
- The listed p-values are below their alpha levels.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: reject the null hypothesis
5. Practice case E: For alpha = 0.01 and p-value = 0.03, what is the decision?
Choices: fail to reject the null hypothesis · reject the null hypothesis · prove the alternative · drop the context
Show solution
- Warm-up: First identify exactly what the question is asking: For alpha = 0.01 and p-value = 0.03, what is the decision?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Fail to reject when p-value is above alpha.
- The evidence is not strong enough.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: fail to reject the null hypothesis
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