A one-sample t-test is used to decide whether a sample mean provides strong evidence that a population mean is different from a claimed value. It matters when the population standard deviation is unknown, which is common in real data collection. The test compares the observed sample mean to the hypothesized mean while accounting for sample size and sample variability.
The result helps researchers judge whether an observed difference is likely due to random sampling or a real effect.
The test statistic follows a t-distribution with n - 1 degrees of freedom when the data are approximately normal or the sample size is large enough. A larger absolute t-value means the sample mean is farther from the hypothesized mean in standard error units. The p-value gives the probability of getting a result at least as extreme as the observed one if the null hypothesis is true.
In a worked example, you compute t = (x̄ - μ0)/(s/√n), find the degrees of freedom, compare the p-value to α, and make a conclusion in context.
Key Facts
- Use a one-sample t-test when testing a population mean μ and the population standard deviation σ is unknown.
- Null hypothesis: H0: μ = μ0, where μ0 is the claimed or reference mean.
- Common alternative hypotheses are Ha: μ ≠ μ0, Ha: μ > μ0, or Ha: μ < μ0.
- Test statistic: t = (x̄ - μ0)/(s/√n).
- Degrees of freedom: df = n - 1.
- Decision rule: reject H0 if p-value ≤ α, and fail to reject H0 if p-value > α.
Vocabulary
- One-sample t-test
- A hypothesis test used to determine whether a sample mean differs significantly from a hypothesized population mean.
- Null hypothesis
- The statement being tested, usually that the population mean equals a specific value.
- Test statistic
- A standardized value that measures how far the sample result is from the null hypothesis value.
- Degrees of freedom
- The number of independent pieces of information used to estimate variability, equal to n - 1 for a one-sample t-test.
- p-value
- The probability, assuming the null hypothesis is true, of observing a test statistic as extreme as or more extreme than the one found.
Common Mistakes to Avoid
- Using z instead of t when σ is unknown, because the t-test is designed for estimating population variability with the sample standard deviation s.
- Forgetting to divide s by √n, because the denominator is the standard error of the mean, not the standard deviation of individual data values.
- Using df = n instead of df = n - 1, because estimating the sample mean uses up one degree of freedom.
- Rejecting H0 just because x̄ is not equal to μ0, because random samples almost never match the hypothesized mean exactly and the decision must use the p-value or critical value.
Practice Questions
- 1 A sample of n = 16 students has a mean score x̄ = 82 and sample standard deviation s = 8. Test H0: μ = 78 using the test statistic formula. What is the t-value and the degrees of freedom?
- 2 A company claims its batteries last 50 hours. A sample of n = 25 batteries has x̄ = 47.8 hours and s = 5 hours. For H0: μ = 50 and Ha: μ < 50, compute the t-statistic.
- 3 A two-sided one-sample t-test gives p-value = 0.032 at α = 0.05. Explain the decision and write a conclusion in the context of testing whether a population mean differs from a claimed value.