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.
Understanding Statistics: The One-Sample t-Test
The most important idea is the standard error. A sample mean changes from sample to sample, even when every sample comes from the same population. The standard error estimates the usual size of that change.
It becomes smaller when the sample contains more independent observations. It becomes larger when individual values are spread out.
This explains why a difference of two units can be convincing in one study but unconvincing in another. A class average based on two hundred test scores is usually more stable than an average based on eight scores.
The method depends on how the data were collected. Random sampling gives the strongest basis for making a statement about a wider population. Random assignment is different.
It supports conclusions about the effect of a treatment in an experiment, but it does not automatically make the participants representative of everyone. Independence matters too.
Measurements from the same person, siblings, or nearby houses may be related. Treating related observations as separate independent data can make the evidence look stronger than it really is.
Before using the test, inspect the distribution of the measurements. For a small sample, serious skewness or a single extreme value can strongly affect the mean and sample standard deviation. A dot plot, histogram, or box plot can reveal these problems.
If the data have a clear outlier, check whether it is a recording error or a genuine observation. Deleting a real value only because it is inconvenient is not good practice. For large samples, the mean is often more reliable even when the individual data are not perfectly normal, but extreme skewness still deserves attention.
Choose the direction of the alternative claim before viewing the result. A two-sided test is appropriate when departures in either direction matter. A one-sided test is appropriate only when a direction was justified in advance and the opposite direction would not count as evidence for the claim.
Choosing a one-sided test after seeing that the sample mean moved in a convenient direction is misleading. In real settings, this issue appears when a factory checks whether average fill is below a required amount, when a school compares an average score with a benchmark, or when a health study checks whether mean recovery time has changed.
A statistically significant result does not tell how large or important the difference is. With a very large sample, even a tiny departure from a reference value can produce strong evidence. Report the sample mean and a confidence interval alongside the test result.
The interval shows a range of population mean values that fit the data reasonably well. Compare that range with a practical standard, not only with a significance level.
A result that fails to reject the null hypothesis does not prove the claimed mean is exactly correct. It may simply mean the sample was too small or too variable to detect a meaningful difference.
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.