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One-Sample, Paired, Pooled, and Welch t Test worksheets

Select and run one-sample, paired, equal-variance pooled, and unequal-variance Welch t tests.

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Identify the design before calculating. Keep paired differences and Welch noninteger degrees of freedom explicit.

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Student pages print the assumptions, tail convention, distribution reference, and workspace needed to complete each inference problem. Separate worked keys reproduce the decision from the same validated document.

Learning objective

What this activity builds

Match the study design to the correct t statistic, assumptions, degrees of freedom, and conclusion.

Worked example

Five paired differences are analyzed. What df is used?

df=5-1=4.

A paired test is a one-sample t test on the differences.

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  • Form paired differences first.
  • Pool only with an equal-variance assumption.
  • Retain Welch df as a decimal.

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Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.1.Independent samples give group 1: n=6, mean=14, s=2; group 2: n=9, mean=15, s=3. Use the equal-variance pooled test of H0: mu1-mu2=0 at alpha=0.1.Assumptions: Samples are independent and each population is approximately normal. Population variances are assumed equal.Reference: Pool the variance and use df=n1+n2-2. t reference: P(T_13>=0.713229032)=0.2441552908; two-sided p=2(0.2441552908)=0.4883105816. Critical |t|=1.770933396.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 1 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.2.Matched before values are 13,16,15,19,17; after values are 14,18,14,22,20. Test whether the mean paired change differs from 0 at alpha=0.01.Assumptions: Pairs are independent of other pairs; paired differences are approximately normal.Reference: Use d=after-before, t=d-bar/(s_d/sqrt(n)), df=n-1. t reference: P(T_4>=2.138089935)=0.0496503416; two-sided p=2(0.0496503416)=0.0993006832. Critical |t|=4.604094871.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 2 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.3.A sample has n=8, x-bar=16, and s=3. Test H0: mu=12 against HA: mu differs from 12 at alpha=0.05.Assumptions: The observations are independent and the population is approximately normal.Reference: One-sample t=(x-bar-mu0)/(s/sqrt(n)), df=n-1. t reference: P(T_7>=3.771236166)=0.0034858275; two-sided p=2(0.0034858275)=0.0069716550. Critical |t|=2.364624252.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 3 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.4.Independent samples give group 1: n=9, mean=17, s=3.5; group 2: n=7, mean=15, s=7. Use the unequal-variance Welch test of H0: mu1-mu2=0 atalpha=0.1.Assumptions: Samples are independent and each population is approximately normal. Equal variances are not assumed.Reference: Use SE=sqrt(s1^2/n1+s2^2/n2) and Welch-Satterthwaite df without rounding. t reference: P(T_8.324141812>=0.691668501)=0.2539874797; two-sided p=2(0.2539874797)=0.5079749594.Critical |t|=1.850204075.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 4 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.5.Matched before values are 16,19,18,22,20; after values are 17,21,17,25,22. Test whether the mean paired change differs from 0 at alpha=0.01.Assumptions: Pairs are independent of other pairs; paired differences are approximately normal.Reference: Use d=after-before, t=d-bar/(s_d/sqrt(n)), df=n-1. t reference: P(T_4>=2.064187386)=0.0539694112; two-sided p=2(0.0539694112)=0.1079388224. Critical |t|=4.604094871.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 5 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.6.A sample has n=6, x-bar=19, and s=2.5. Test H0: mu=10 against HA: mu differs from 10 at alpha=0.05.Assumptions: The observations are independent and the population is approximately normal.Reference: One-sample t=(x-bar-mu0)/(s/sqrt(n)), df=n-1. t reference: P(T_5>=8.818163074)=0.0001557151; two-sided p=2(0.0001557151)=0.0003114302. Critical |t|=2.570581836.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 6 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.7.Independent samples give group 1: n=7, mean=20, s=3; group 2: n=10, mean=15, s=3. Use the equal-variance pooled test of H0: mu1-mu2=0 atalpha=0.1.Assumptions: Samples are independent and each population is approximately normal. Population variances are assumed equal.Reference: Pool the variance and use df=n1+n2-2. t reference: P(T_15>=3.381997708)=0.0020533544; two-sided p=2(0.0020533544)=0.0041067088. Critical |t|=1.753050356.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 7 of 8
Math SheetNameScoreTeacherDateOne-Sample, Paired, Pooled, and Welch t TestsIdentify the design, statistic, degrees of freedom, assumptions, and conclusion. Keep Welch degrees of freedom unrounded during probability calculations.8.Independent samples give group 1: n=8, mean=14, s=3.5; group 2: n=11, mean=15, s=7. Use the unequal-variance Welch test of H0: mu1-mu2=0 atalpha=0.01.Assumptions: Samples are independent and each population is approximately normal. Equal variances are not assumed.Reference: Use SE=sqrt(s1^2/n1+s2^2/n2) and Welch-Satterthwaite df without rounding. t reference: P(T_15.448794462>=0.408732399)=0.3441729238; two-sided p=2(0.3441729238)=0.6883458476.Critical |t|=2.934610110.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 8 of 8

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