Statistical Procedure

One-Sample T Test for a Population Mean

A decision-and-workflow guide for a one-sample t test, covering method selection, conditions, mathematics, calculator evidence, and contextual reporting.

Course status: Revised 2026-27 course
Updated: July 18, 2026
Practice: Easy, Tough and Toughest

Method at a Glance: One Sample t-Test

A one-sample t test compares x-bar with a hypothesized mean using s over square root n and n-1 degrees of freedom, with robustness judged from sample size and shape.

Reader taskmean parameter, t statistic, degrees of freedom, conditions, and conclusion

Planned modules8

Mathematics2 expressions

Worked checks57

Boundary: Distinguish statistical significance from practical importance.

Procedure Workflow

Identify the data structure and parameter before selecting one sample t test; the name of a calculator menu is not method evidence.
State the hypotheses or estimation target for one sample t test using population notation and the order defined by the question.
Verify the design, independence, and approximation conditions that specifically justify one sample t test rather than reciting every condition learned in the course.
Compute the statistic, standard error, interval, or p-value for one sample t test with defined symbols, guard digits, and an independent arithmetic check.
Interpret one sample t test in the population and units named by the problem, then limit causation and generalization to what the collection design supports.

Procedure Formulas and Notation

One-sample t statistic

t=\frac{\bar x-\mu_0}{s/\sqrt n}

One-sample t statistic in One Sample t-Test: Keep the population mean, sample mean, sample standard deviation, and standard error distinct, including their original measurement units.

One-sample t degrees of freedom

df=n-1

One-sample t degrees of freedom in One Sample t-Test: This expression belongs specifically to a one-sample t test; define every symbol and apply the scope rule for mean parameter, t statistic, degrees of freedom, conditions, and conclusion before calculation.

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Step 1

Hypotheses

Decision

For Hypotheses in one sample t test, A constructed sample from a school library checkout study gives $n=25$ , $\bar x=52.70$ , and $s=7.20$ . Test $H_0:\mu=50$ two-sided.

Hypotheses result in one sample t test: The statistic is t=1.875 with df=24; p=0.0730.

t=\frac{52.70-50}{7.20/\sqrt{25}}=1.875.

Interpretation and validity

Hypotheses interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Hypotheses: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 2

Conditions

Decision

For Conditions in one sample t test, A constructed sample from a website response-time study gives $n=26$ , $\bar x=52.30$ , and $s=7.80$ . Test $H_0:\mu=50$ two-sided.

Conditions result in one sample t test: The statistic is t=1.504 with df=25; p=0.1452.

t=\frac{52.30-50}{7.80/\sqrt{26}}=1.504.

Interpretation and validity

Conditions interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Conditions: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 3

t statistic

Decision

For t statistic in one sample t test, A constructed sample from a greenhouse germination experiment gives $n=27$ , $\bar x=53.90$ , and $s=8.90$ . Test $H_0:\mu=50$ two-sided.

t statistic result in one sample t test: The statistic is t=2.277 with df=26; p=0.0313.

t=\frac{53.90-50}{8.90/\sqrt{27}}=2.277.

Interpretation and validity

t statistic interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and t statistic: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 4

Degrees of freedom

Decision

For Degrees of freedom in one sample t test, A constructed sample from a seedling-growth comparison gives $n=28$ , $\bar x=53.40$ , and $s=8.90$ . Test $H_0:\mu=50$ two-sided.

Degrees of freedom result in one sample t test: The statistic is t=2.021 with df=27; p=0.0532.

t=\frac{53.40-50}{8.90/\sqrt{28}}=2.021.

Interpretation and validity

Degrees of freedom interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Degrees of freedom: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 5

Calculator

Decision

For Calculator in one sample t test, A constructed sample from an online-course completion sample gives $n=29$ , $\bar x=53.60$ , and $s=8.60$ . Test $H_0:\mu=50$ two-sided.

Calculator result in one sample t test: The statistic is t=2.254 with df=28; p=0.0322.

t=\frac{53.60-50}{8.60/\sqrt{29}}=2.254.

Interpretation and validity

Calculator interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Calculator: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 6

P-value

Decision

For P-value in one sample t test, A constructed sample from a website response-time study gives $n=30$ , $\bar x=53.10$ , and $s=8.60$ . Test $H_0:\mu=50$ two-sided.

P-value result in one sample t test: The statistic is t=1.974 with df=29; p=0.0579.

t=\frac{53.10-50}{8.60/\sqrt{30}}=1.974.

Interpretation and validity

P-value interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and P-value: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 7

Conclusion

Decision

For Conclusion in one sample t test, A constructed sample from a recycling-behavior survey gives $n=31$ , $\bar x=53.50$ , and $s=7.50$ . Test $H_0:\mu=50$ two-sided.

Conclusion result in one sample t test: The statistic is t=2.598 with df=30; p=0.0144.

t=\frac{53.50-50}{7.50/\sqrt{31}}=2.598.

Interpretation and validity

Conclusion interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Conclusion: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Step 8

Worked FRQ

Decision

For Worked FRQ in one sample t test, A constructed sample from a water-filtration experiment gives $n=32$ , $\bar x=53.90$ , and $s=7.40$ . Test $H_0:\mu=50$ two-sided.

Worked FRQ result in one sample t test: The statistic is t=2.981 with df=31; p=0.0055.

t=\frac{53.90-50}{7.40/\sqrt{32}}=2.981.

Interpretation and validity

Worked FRQ interpretation for one sample t test: The p-value quantifies evidence against the null mean; the design determines whether the result generalizes or supports causation.

Condition evidence for one sample t test and Worked FRQ: Use a random sample or randomized design, independence, and a distribution without problematic skew or outliers for the sample size.

Procedure error: The t procedure accounts for estimating sigma with s; substituting a z distribution ignores that uncertainty.

Procedure Practice and Full Solutions

Every question in One-Sample T Test for a Population Mean is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.

Easy Practice

Easy 1: t statistic

Question P66-Easy-1. A constructed sample from a greenhouse germination experiment gives $n=25$ , $\bar x=54.40$ , and $s=8.90$ . Test $H_0:\mu=50$ two-sided.