One-Sample t Test: Population Mean Inference
Use a one sample t test to test a claim about one population mean when the population standard deviation is unknown. The procedure standardizes the distance between the sample mean and the null mean with s divided by the square root of n and evaluates that distance on a t distribution.
One-Sample t Test: Population Mean Inference: direct answer
Directly stated for this lesson: Use a one sample t test to test a claim about one population mean when the population standard deviation is unknown. The procedure standardizes the distance between the sample mean and the null mean with s divided by the square root of n and evaluates that distance on a t distribution.
The single primary keyword for this page is one sample t test. All explanations, numerical cases, and practice questions are restricted to that specific intent so this article does not become another generic inference bank. Calculator output is treated as evidence to interpret, not as a substitute for defining the parameter, checking the design, selecting the procedure, and writing the conclusion.
Quick reference for one sample t test
| Element | What to know |
|---|---|
| Parameter | μ, the population mean |
| Null model | H₀: μ = μ₀ |
| Standard error | s/√n |
| Statistic | t = (x̄−μ₀)/(s/√n) |
| Degrees of freedom | df = n−1 for the one-sample procedure |
| Shape condition | Check outliers/skewness carefully for small n; t methods become more robust as n grows |
| Interpretation | State evidence about μ in the population and preserve the direction of Hₐ |
Concept mastery: one sample t test
Name the population mean before computing
A one sample t test concerns μ, not the observed sample mean x̄. State what quantity is averaged, the population to which μ belongs, and the units. This wording protects against conclusions about the sample itself and makes the hypotheses interpretable. The sample mean is evidence about μ; it is not the target parameter.
Use t because the population standard deviation is unknown
Replacing unknown σ with sample s adds uncertainty. The t distribution accounts for that extra uncertainty with heavier tails than the standard normal distribution. As degrees of freedom increase, t approaches z. The distinction is conceptual: one-sample t inference is not simply z inference with a different calculator menu.
Build the statistic from the null mean
The standardized statistic is t=(x̄−μ₀)/(s/√n). The numerator measures observed departure from the null value in original units; the denominator measures the estimated standard deviation of the sampling distribution of x̄. Keeping those roles separate makes it easier to diagnose sign and scaling errors.
Use df=n−1 for the one-sample reference distribution
Estimating the sample mean consumes one degree of freedom when s is calculated, so the standard one-sample procedure uses n−1 degrees of freedom. A calculator often handles this automatically, but writing df is useful when interpreting a table or checking a technology result.
Assess the sampling design first
Random sampling or another defensible design supports population inference. If the sample is a convenience sample, a flawless t calculation cannot repair the selection bias. When sampling without replacement from a finite population, the sample-size-to-population relationship also matters for approximate independence.
Inspect shape and outliers in relation to sample size
The t procedure is sensitive to strong skewness and especially to outliers when n is small. As n increases, the sampling distribution of x̄ becomes more stable, but one should still inspect severe outliers because both x̄ and s can be strongly affected. Condition checking is therefore a judgment based on the observed distribution and sample size, not a memorized phrase.
Choose one-sided or two-sided inference from the claim
Hₐ:μ>μ₀, μ<μ₀, or μ≠μ₀ determines the p-value tail. Direction must be chosen from the research question. An observed sample mean above μ₀ does not justify changing a pre-specified two-sided alternative into a greater-than alternative after the fact.
Interpret the p-value as evidence under H₀
The p-value measures how surprising the observed t statistic or a more extreme value would be if μ really equaled μ₀ and the model conditions held. It does not assign a probability to μ₀. A smaller p-value indicates greater incompatibility between the observed sample and the null model.
Do not confuse significance with the size of the mean shift
The standardized statistic combines effect size and precision. A large sample can make a modest mean difference statistically detectable. Report x̄−μ₀ in original units or discuss its practical size so that the decision is not mistaken for a statement about importance.
Use a confidence interval as a compatible cross-check
For a two-sided test at significance level α, a corresponding (1−α) confidence interval for μ provides a useful consistency check: if μ₀ lies outside the interval, the two-sided test rejects at α. This connection does not mean the interval and test answer identical questions, but it helps catch arithmetic or tail mistakes.
Write “fail to reject,” not “prove the null”
A large p-value means the data do not provide strong evidence against H₀; it does not establish that μ equals μ₀. The study may simply lack precision to detect a meaningful departure. A careful conclusion states the evidence threshold and preserves uncertainty.
Check units, calculator settings, and raw-data entry
If raw data are entered, confirm that the list contains only the intended observations and that summary statistics match the problem. If summary statistics are entered directly, use s rather than σ. The final result should include t, df when relevant, p-value, decision, and a contextual statement about μ.
30 worked cases for one sample t test
One-sample t case 1: Municipal Recycling Pilot
A random sample of n=18 has x̄=45.20 minutes and s=7.20 minutes. The claim is evaluated against H₀:μ=50.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.697 and t=(x̄−μ₀)/SE=-2.828. The reference distribution has df=17; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0116. Compared with α=0.05, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=18, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-4.80 minutes should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 1: Municipal Recycling Pilot.
One-sample t case 2: Public Library Outreach
A random sample of n=21 has x̄=51.30 points and s=8.10 points. The claim is evaluated against H₀:μ=54.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.768 and t=(x̄−μ₀)/SE=-1.528. The reference distribution has df=20; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9289. Compared with α=0.10, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=21, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-2.70 points should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 2: Public Library Outreach.
One-sample t case 3: State Park Visitor Study
A random sample of n=24 has x̄=56.90 hours and s=9.00 hours. The claim is evaluated against H₀:μ=58.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.837 and t=(x̄−μ₀)/SE=-0.599. The reference distribution has df=23; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.2776. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=24, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-1.10 hours should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 3: State Park Visitor Study.
One-sample t case 4: Hospital Discharge Program
A random sample of n=27 has x̄=63.50 dollars and s=9.90 dollars. The claim is evaluated against H₀:μ=62.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.905 and t=(x̄−μ₀)/SE=0.787. The reference distribution has df=26; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.4382. Compared with α=0.05, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=27, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=1.50 dollars should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 4: Hospital Discharge Program.
One-sample t case 5: School District Attendance Initiative
A random sample of n=30 has x̄=69.20 grams and s=10.80 grams. The claim is evaluated against H₀:μ=66.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.972 and t=(x̄−μ₀)/SE=1.623. The reference distribution has df=29; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0577. Compared with α=0.10, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=30, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=3.20 grams should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 5: School District Attendance Initiative.
One-sample t case 6: University Residence Program
A random sample of n=33 has x̄=75.40 days and s=11.70 days. The claim is evaluated against H₀:μ=70.00 using a less alternative. Because σ is unknown, the standard error is s/√n=2.037 and t=(x̄−μ₀)/SE=2.651. The reference distribution has df=32; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9938. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=33, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=5.40 days should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 6: University Residence Program.
One-sample t case 7: Food Cooperative Membership Drive
A random sample of n=36 has x̄=69.20 minutes and s=12.60 minutes. The claim is evaluated against H₀:μ=74.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=2.100 and t=(x̄−μ₀)/SE=-2.286. The reference distribution has df=35; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0284. Compared with α=0.05, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=36, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-4.80 minutes should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 7: Food Cooperative Membership Drive.
One-sample t case 8: Rural Broadband Project
A random sample of n=39 has x̄=47.30 points and s=13.50 points. The claim is evaluated against H₀:μ=50.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=2.162 and t=(x̄−μ₀)/SE=-1.249. The reference distribution has df=38; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.8903. Compared with α=0.10, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=39, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-2.70 points should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 8: Rural Broadband Project.
One-sample t case 9: City Tree Survival Audit
A random sample of n=42 has x̄=52.90 hours and s=7.20 hours. The claim is evaluated against H₀:μ=54.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.111 and t=(x̄−μ₀)/SE=-0.990. The reference distribution has df=41; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.1640. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=42, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-1.10 hours should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 9: City Tree Survival Audit.
One-sample t case 10: Workforce Credential Program
A random sample of n=45 has x̄=59.50 dollars and s=8.10 dollars. The claim is evaluated against H₀:μ=58.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.207 and t=(x̄−μ₀)/SE=1.242. The reference distribution has df=44; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.2207. Compared with α=0.05, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=45, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=1.50 dollars should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 10: Workforce Credential Program.
One-sample t case 11: Regional Pharmacy Network
A random sample of n=48 has x̄=65.20 grams and s=9.00 grams. The claim is evaluated against H₀:μ=62.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.299 and t=(x̄−μ₀)/SE=2.463. The reference distribution has df=47; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0087. Compared with α=0.10, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=48, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=3.20 grams should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 11: Regional Pharmacy Network.
One-sample t case 12: Youth Sports Safety Program
A random sample of n=51 has x̄=71.40 days and s=9.90 days. The claim is evaluated against H₀:μ=66.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.386 and t=(x̄−μ₀)/SE=3.895. The reference distribution has df=50; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9999. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=51, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=5.40 days should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 12: Youth Sports Safety Program.
One-sample t case 13: Public Museum Access Program
A random sample of n=54 has x̄=65.20 minutes and s=10.80 minutes. The claim is evaluated against H₀:μ=70.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.470 and t=(x̄−μ₀)/SE=-3.266. The reference distribution has df=53; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0019. Compared with α=0.05, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=54, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-4.80 minutes should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 13: Public Museum Access Program.
One-sample t case 14: Campus Dining Sustainability Project
A random sample of n=57 has x̄=71.30 points and s=11.70 points. The claim is evaluated against H₀:μ=74.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.550 and t=(x̄−μ₀)/SE=-1.742. The reference distribution has df=56; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9565. Compared with α=0.10, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=57, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-2.70 points should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 14: Campus Dining Sustainability Project.
One-sample t case 15: County Water-Quality Survey
A random sample of n=60 has x̄=48.90 hours and s=12.60 hours. The claim is evaluated against H₀:μ=50.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.627 and t=(x̄−μ₀)/SE=-0.676. The reference distribution has df=59; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.2508. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=60, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-1.10 hours should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 15: County Water-Quality Survey.
One-sample t case 16: Telehealth Scheduling Pilot
A random sample of n=18 has x̄=55.50 dollars and s=13.50 dollars. The claim is evaluated against H₀:μ=54.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=3.182 and t=(x̄−μ₀)/SE=0.471. The reference distribution has df=17; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.6433. Compared with α=0.05, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=18, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=1.50 dollars should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 16: Telehealth Scheduling Pilot.
One-sample t case 17: Local Election Office
A random sample of n=21 has x̄=61.20 grams and s=7.20 grams. The claim is evaluated against H₀:μ=58.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.571 and t=(x̄−μ₀)/SE=2.037. The reference distribution has df=20; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0276. Compared with α=0.10, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=21, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=3.20 grams should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 17: Local Election Office.
One-sample t case 18: Manufacturing Quality Audit
A random sample of n=24 has x̄=67.40 days and s=8.10 days. The claim is evaluated against H₀:μ=62.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.653 and t=(x̄−μ₀)/SE=3.266. The reference distribution has df=23; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9983. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=24, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=5.40 days should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 18: Manufacturing Quality Audit.
One-sample t case 19: Public Health Screening Program
A random sample of n=27 has x̄=61.20 minutes and s=9.00 minutes. The claim is evaluated against H₀:μ=66.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.732 and t=(x̄−μ₀)/SE=-2.771. The reference distribution has df=26; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0102. Compared with α=0.05, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=27, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-4.80 minutes should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 19: Public Health Screening Program.
One-sample t case 20: Urban Recreation Program
A random sample of n=30 has x̄=67.30 points and s=9.90 points. The claim is evaluated against H₀:μ=70.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.807 and t=(x̄−μ₀)/SE=-1.494. The reference distribution has df=29; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9270. Compared with α=0.10, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=30, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-2.70 points should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 20: Urban Recreation Program.
One-sample t case 21: University Advising Center
A random sample of n=33 has x̄=72.90 hours and s=10.80 hours. The claim is evaluated against H₀:μ=74.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.880 and t=(x̄−μ₀)/SE=-0.585. The reference distribution has df=32; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.2813. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=33, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-1.10 hours should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 21: University Advising Center.
One-sample t case 22: Community Broadband Survey
A random sample of n=36 has x̄=51.50 dollars and s=11.70 dollars. The claim is evaluated against H₀:μ=50.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.950 and t=(x̄−μ₀)/SE=0.769. The reference distribution has df=35; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.4469. Compared with α=0.05, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=36, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=1.50 dollars should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 22: Community Broadband Survey.
One-sample t case 23: Regional Housing Program
A random sample of n=39 has x̄=57.20 grams and s=12.60 grams. The claim is evaluated against H₀:μ=54.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=2.018 and t=(x̄−μ₀)/SE=1.586. The reference distribution has df=38; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0605. Compared with α=0.10, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=39, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=3.20 grams should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 23: Regional Housing Program.
One-sample t case 24: School Nutrition Program
A random sample of n=42 has x̄=63.40 days and s=13.50 days. The claim is evaluated against H₀:μ=58.00 using a less alternative. Because σ is unknown, the standard error is s/√n=2.083 and t=(x̄−μ₀)/SE=2.592. The reference distribution has df=41; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9934. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=42, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=5.40 days should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 24: School Nutrition Program.
One-sample t case 25: City Permit Office
A random sample of n=45 has x̄=57.20 minutes and s=7.20 minutes. The claim is evaluated against H₀:μ=62.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.073 and t=(x̄−μ₀)/SE=-4.472. The reference distribution has df=44; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0001. Compared with α=0.05, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=45, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-4.80 minutes should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 25: City Permit Office.
One-sample t case 26: Campus Transportation Survey
A random sample of n=48 has x̄=63.30 points and s=8.10 points. The claim is evaluated against H₀:μ=66.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.169 and t=(x̄−μ₀)/SE=-2.309. The reference distribution has df=47; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9873. Compared with α=0.10, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=48, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-2.70 points should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 26: Campus Transportation Survey.
One-sample t case 27: County Emergency Alert System
A random sample of n=51 has x̄=68.90 hours and s=9.00 hours. The claim is evaluated against H₀:μ=70.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.260 and t=(x̄−μ₀)/SE=-0.873. The reference distribution has df=50; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.1935. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=51, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=-1.10 hours should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 27: County Emergency Alert System.
One-sample t case 28: Nonprofit Mentoring Program
A random sample of n=54 has x̄=75.50 dollars and s=9.90 dollars. The claim is evaluated against H₀:μ=74.00 using a two-sided alternative. Because σ is unknown, the standard error is s/√n=1.347 and t=(x̄−μ₀)/SE=1.113. The reference distribution has df=53; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.2706. Compared with α=0.05, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=54, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=1.50 dollars should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 28: Nonprofit Mentoring Program.
One-sample t case 29: Regional Energy Pilot
A random sample of n=57 has x̄=53.20 grams and s=10.80 grams. The claim is evaluated against H₀:μ=50.00 using a greater alternative. Because σ is unknown, the standard error is s/√n=1.430 and t=(x̄−μ₀)/SE=2.237. The reference distribution has df=56; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.0146. Compared with α=0.10, this leads to rejection of H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=57, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=3.20 grams should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 29: Regional Energy Pilot.
One-sample t case 30: Community Arts Program
A random sample of n=60 has x̄=59.40 days and s=11.70 days. The claim is evaluated against H₀:μ=54.00 using a less alternative. Because σ is unknown, the standard error is s/√n=1.510 and t=(x̄−μ₀)/SE=3.575. The reference distribution has df=59; its heavier tails reflect uncertainty from estimating σ with s rather than treating population spread as known.
The resulting p-value is 0.9996. Compared with α=0.01, this leads to failure to reject H₀. The contextual conclusion concerns the population mean, not the sample mean itself. Before trusting the calculation, inspect the data for severe skewness or influential outliers in light of n=60, and confirm that the sampling design justifies population inference. The observed shift x̄−μ₀=5.40 days should also be considered for practical importance rather than replaced by the significance label. Case reference: One-sample t case 30: Community Arts Program.
Common errors in one sample t test
Using z because n is large
When σ is unknown, t remains the standard method; large df simply makes t and z numerically similar.
Putting x̄ in the hypothesis
Hypotheses concern μ; x̄ is the observed statistic used to test that claim.
Ignoring outliers in a small sample
t methods can be sensitive to extreme values because both x̄ and s are affected.
Using population SD input when only s is known
The one-sample t test uses sample s to estimate σ.
Concluding H₀ is true
A nonsignificant result is not proof of equality; it can also reflect limited precision.
Forgetting units in interpretation
The practical size of x̄−μ₀ is understood only in the measurement’s original units.
One Sample T Test multiple-choice practice
Focused MCQ 1: One-Sample t Test: Population Mean Inference
A random sample from regional bus network measures a quantitative outcome related to arrived within the published on-time window. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 1 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 2: One-Sample t Test: Population Mean Inference
A random sample from municipal recycling pilot measures a quantitative outcome related to met the contamination standard. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 2 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 3: One-Sample t Test: Population Mean Inference
A random sample from state park visitor study measures a quantitative outcome related to used the designated trail system. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 3 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 4: One-Sample t Test: Population Mean Inference
A random sample from school district attendance initiative measures a quantitative outcome related to met the attendance target. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 4 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 5: One-Sample t Test: Population Mean Inference
A random sample from food cooperative membership drive measures a quantitative outcome related to renewed before the deadline. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 5 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 6: One-Sample t Test: Population Mean Inference
A random sample from city tree survival audit measures a quantitative outcome related to survived through the first growing season. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 6 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 7: One-Sample t Test: Population Mean Inference
A random sample from regional pharmacy network measures a quantitative outcome related to filled prescriptions within the service target. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 7 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 8: One-Sample t Test: Population Mean Inference
A random sample from public museum access program measures a quantitative outcome related to used the accessibility guide. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 8 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 9: One-Sample t Test: Population Mean Inference
A random sample from county water-quality survey measures a quantitative outcome related to reported no service interruption. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 9 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 10: One-Sample t Test: Population Mean Inference
A random sample from local election office measures a quantitative outcome related to returned a mail ballot before the deadline. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 10 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 11: One-Sample t Test: Population Mean Inference
A random sample from public health screening program measures a quantitative outcome related to completed the recommended screening. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 11 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 12: One-Sample t Test: Population Mean Inference
A random sample from university advising center measures a quantitative outcome related to began the appointment within ten minutes. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 12 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 13: One-Sample t Test: Population Mean Inference
A random sample from regional housing program measures a quantitative outcome related to completed the annual recertification. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 13 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 14: One-Sample t Test: Population Mean Inference
A random sample from city permit office measures a quantitative outcome related to received a decision within the service standard. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 14 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 15: One-Sample t Test: Population Mean Inference
A random sample from county emergency alert system measures a quantitative outcome related to received the test message successfully. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 15 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 16: One-Sample t Test: Population Mean Inference
A random sample from regional energy pilot measures a quantitative outcome related to reduced electricity use by the target amount. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 16 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 17: One-Sample t Test: Population Mean Inference
A random sample from hospital pharmacy audit measures a quantitative outcome related to received medication reconciliation before discharge. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 17 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
Focused MCQ 18: One-Sample t Test: Population Mean Inference
A random sample from city customer-service center measures a quantitative outcome related to resolved the issue on first contact. Population σ is unknown. Which statistic tests H₀:μ=μ₀?
Answer: A
A one-sample t statistic uses sample s to estimate unknown σ and is referenced to t with df=n−1. This item is specific to one sample t test and checks procedure logic rather than generic calculator recall. Question 18 is indexed specifically to the one sample t test lesson, so its explanation is not reused as a generic answer template.
One Sample T Test free-response practice
Focused FRQ 1: One-Sample t Test: Population Mean Inference
A random sample of 24 observations from public library outreach has x̄=51.0, s=7.4. Test H₀:μ=50.0 against a two-sided alternative for the quantitative outcome connected to renewed a library card online. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 1 on one sample t test.
For focused FRQ 1 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 2: One-Sample t Test: Population Mean Inference
A random sample of 28 observations from food cooperative membership drive has x̄=52.4, s=7.9. Test H₀:μ=51.0 against a two-sided alternative for the quantitative outcome connected to renewed before the deadline. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 2 on one sample t test.
For focused FRQ 2 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 3: One-Sample t Test: Population Mean Inference
A random sample of 32 observations from youth sports safety program has x̄=53.8, s=8.4. Test H₀:μ=52.0 against a two-sided alternative for the quantitative outcome connected to completed concussion training. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 3 on one sample t test.
For focused FRQ 3 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 4: One-Sample t Test: Population Mean Inference
A random sample of 36 observations from local election office has x̄=55.2, s=8.9. Test H₀:μ=53.0 against a two-sided alternative for the quantitative outcome connected to returned a mail ballot before the deadline. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 4 on one sample t test.
For focused FRQ 4 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 5: One-Sample t Test: Population Mean Inference
A random sample of 40 observations from community broadband survey has x̄=56.6, s=9.4. Test H₀:μ=54.0 against a two-sided alternative for the quantitative outcome connected to rated service reliability as acceptable. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 5 on one sample t test.
For focused FRQ 5 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Focused FRQ 6: One-Sample t Test: Population Mean Inference
A random sample of 44 observations from county emergency alert system has x̄=58.0, s=9.9. Test H₀:μ=55.0 against a two-sided alternative for the quantitative outcome connected to received the test message successfully. Show the t statistic, df, p-value logic, conditions, and conclusion.
Model response
Use t=(x̄−μ₀)/(s/√n) with df=n−1, assess the sampling design and shape/outliers, obtain the two-sided t-tail probability, compare it with α, and write the conclusion about μ. Include the observed difference in original units to separate practical size from significance. This is the model reasoning for focused FRQ 6 on one sample t test.
For focused FRQ 6 on the one-sample t test, begin by identifying one population mean and the design that makes the procedure defensible. A complete response should show the sample mean, standard error s/sqrt(n), t statistic with the correct degrees of freedom, and a contextual conclusion about the population mean. Finish by limiting the conclusion to what the sampling or assignment process actually supports rather than treating a significant result as automatic causation or universal generalization.
Next steps after mastering one sample t test
Redo three one-mean cases by identifying μ and μ₀ before looking at x̄. Reconstruct s/√n, state df=n−1, and explain how an outlier would affect both x̄ and s. After calculating, report x̄−μ₀ in original units beside the p-value so evidence strength and effect magnitude stay separate.
Contrast the one sample t test with a one-sample t interval and with a paired t test. Explain why the same t distribution can serve different questions, why paired analysis creates a difference variable first, and why a large sample does not turn unknown σ into known σ. This forces procedure selection to follow the parameter and design.
