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.
Specify a population mean under H0 and a directional or two-sided alternative before examining the result. For a one-sample t test, calculate SE=s/√n, then t=(x-bar−mu0)/SE with df=n−1. Check randomization/independence and distribution conditions. Use the appropriate t distribution to obtain a p-value, or compare with the supplied critical values. Reject when p<=alpha; otherwise fail to reject. A p-value measures as-extreme-or-more-extreme results under H0, not the probability H0 is true. For independent groups standardize the difference of means with its two-sample SE and appropriate degrees of freedom; paired samples require a difference analysis.
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. An independent random sample has n=16, mean 54 and sample SD 8. Test H0: mu=50. Calculate t.
- SE=8/√16=2.
- t=(54−50)/2=2.
Answer: 2
Lesson preview: this page shows 5 practice items. The interactive activity contains 8 practice items and 8 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. A claim sets population mean fill at 50 mL. Investigators suspect a change in either direction. Which hypotheses match?
Choices: H0: mu!=50; Ha: mu=50 · H0: mu=50; Ha: mu>50 only · H0: mu=50; Ha: mu!=50 · H0: xbar=50; Ha: xbar!=50
Show solution
- Hypotheses concern the population parameter.
- Either-direction suspicion requires a two-sided alternative.
Answer: H0: mu=50; Ha: mu!=50
2. An independent random sample has n=16, mean 54 and sample SD 8. Test H0: mu=50. Calculate t.
Show solution
- SE=8/√16=2.
- t=(54−50)/2=2.
Answer: 2
3. For the one-sample mean test based on n=16, what df should be used?
Show solution
- The one-sample t test has n−1 degrees of freedom.
Answer: 15
4. For this two-sided test at alpha=.05, the critical values are ±2.131. The test statistic is t=2. Choose the decision.
Choices: Reject H0 · Prove H0 true · Prove every package has a different fill · Fail to reject H0
Show solution
- Compare |2| with 2.131.
- The statistic does not reach the critical boundary.
Answer: Fail to reject H0
5. A two-sided mean test reports p=.009. What does that mean under its model?
Choices: The chance the sample mean is correct is .009 · Exactly .9% of individual observations are unusual · If the null mean were true, a statistic at least as extreme in either direction would occur with probability .009 · The probability the null is true is .009
Show solution
- The probability is conditional on H0 and includes as-extreme-or-more-extreme statistics.
Answer: If the null mean were true, a statistic at least as extreme in either direction would occur with probability .009
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Independent groups: complete a Welch t analysis
Two independent random samples from large approximately normal populations have means 18 and 14, SDs 6 and 8, and sizes 36 and 64. Test equal population means against a two-sided alternative at alpha=.05. Calculate the statistic and use a calculator's Welch output df=90 and p≈.0058. Explain the population parameter, conditions, result and design limits.
Review rubric and solution
- Define mu1−mu2 in context; H0: difference=0, Ha: difference!=0.
- Check independent random samples, small sampling fractions, and distribution conditions.
- Calculate SE=√2 and t=2.828.
- Compare .0058 with .05 and reject equality.
- Explain evidence of a population mean difference without assuming a causal treatment effect.
- SE=√(36/36+64/64)=√2; t=(18−14)/√2≈2.828.
- Welch df=(1+1)²/(1/35+1/63)=90; the two-sided p-value is about .0058.
- Reject equal population means at .05; the random-sampling design does not by itself establish causation.
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