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Exact Binomial and Known-Sigma z Test worksheets

Run exact one- and two-sided binomial tests and known-sigma z tests from raw data or summaries.

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For exact two-sided tests use the printed probability-ordering rule. For z tests, verify that sigma is known.

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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

Compute p-values and critical regions with an explicit exact extremeness rule or known-sigma normal model.

Worked example

For X~Binomial(4,0.5), what is the lower-tail p-value at x=0?

P(X<=0)=1/16=0.0625.

The observed endpoint is included in the directed exact tail.

Teaching notes

Make the practice count

  • Include the observed outcome in the tail.
  • Name the two-sided extremeness rule.
  • Use sigma/sqrt(n) for a sample mean.

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Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.1.Under H0, X is Binomial(n=20, p=0.4); observe x=10. Test HA: p is not 0.4 at alpha=0.1. Compute the exact p-value, critical region, and conclusion.Assumptions: There are 20 independent binary trials with one common success probability.Reference: Exact null table P0(X=x), x=0,...,20: 0.0000365616, 0.0004874878, 0.0030874227, 0.0123496907, 0.0349907904, 0.0746470195, 0.1244116992, 0.1658822656, 0.1797057878, 0.1597384780,0.1171415505, 0.0709948791, 0.0354974396, 0.0145630521, 0.0048543507, 0.0012944935, 0.0002696862, 0.0000423037, 0.0000047004, 0.0000003299, 0.0000000110. Two-sided extremeness includesexactly outcomes with P0(X=x)<=P0(X=observed), using unrounded probabilities for ordering.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 1 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.2.Data are 22, 23, 24, 25, 27. With known sigma=4, test H0: mu=21 against HA: mu>mu0 at alpha=0.01.Assumptions: The observations are independent and normal, or n is adequate for the mean; population sigma is known.Reference: Normal reference at printed z=1.788854382: P(Z>=|z|)=0.0368190849. Critical z=2.326347015.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 2 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.3.Under H0, X is Binomial(n=11, p=0.6); observe x=4. Test HA: p<0.6 at alpha=0.05. Compute the exact p-value, critical region, and conclusion.Assumptions: There are 11 independent binary trials with one common success probability.Reference: Exact null table P0(X=x), x=0,...,11: 0.0000419430, 0.0006920602, 0.0051904512, 0.0233570304, 0.0700710912, 0.1471492915, 0.2207239373, 0.2364899328, 0.1773674496, 0.0886837248,0.0266051174, 0.0036279706. Use the directed tail including the observed count.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 3 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.4.A sample has x-bar=25.1, n=16. With known sigma=6, test H0: mu=23 against HA: mu<mu0 at alpha=0.1.Assumptions: The observations are independent and normal, or n is adequate for the mean; population sigma is known.Reference: Normal reference at printed z=1.400000000: Phi(z)=0.9192432862. Critical z=-1.281551956.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 4 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.5.Under H0, X is Binomial(n=13, p=0.4); observe x=8. Test HA: p>0.4 at alpha=0.01. Compute the exact p-value, critical region, and conclusion.Assumptions: There are 13 independent binary trials with one common success probability.Reference: Exact null table P0(X=x), x=0,...,13: 0.0013060694, 0.0113192681, 0.0452770726, 0.1106772886, 0.1844621476, 0.2213545771, 0.1967596241, 0.1311730827, 0.0655865414, 0.0242913116,0.0064776831, 0.0011777606, 0.0001308623, 0.0000067109. Use the directed tail including the observed count.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 5 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.6.Under H0, X is Binomial(n=14, p=0.5); observe x=10. Test HA: p>0.5 at alpha=0.05. Compute the exact p-value, critical region, and conclusion.Assumptions: There are 14 independent binary trials with one common success probability.Reference: Exact null table P0(X=x), x=0,...,14: 0.0000610352, 0.0008544922, 0.0055541992, 0.0222167969, 0.0610961914, 0.1221923828, 0.1832885742, 0.2094726563, 0.1832885742, 0.1221923828,0.0610961914, 0.0222167969, 0.0055541992, 0.0008544922, 0.0000610352. Use the directed tail including the observed count.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 6 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.7.A sample has x-bar=23.7, n=16. With known sigma=9, test H0: mu=26 against HA: mu<mu0 at alpha=0.1.Assumptions: The observations are independent and normal, or n is adequate for the mean; population sigma is known.Reference: Normal reference at printed z=-1.022222222: Phi(z)=0.1533378861. Critical z=-1.281551956.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 7 of 8
Math SheetNameScoreTeacherDateExact Binomial and Known-Sigma z TestsFor exact two-sided tests, include outcomes whose null probability is no larger than the observed outcome. For z tests, sigma is known.8.Under H0, X is Binomial(n=16, p=0.3); observe x=7. Test HA: p is not 0.3 at alpha=0.01. Compute the exact p-value, critical region, and conclusion.Assumptions: There are 16 independent binary trials with one common success probability.Reference: Exact null table P0(X=x), x=0,...,16: 0.0033232931, 0.0227882952, 0.0732480919, 0.1464961837, 0.2040482559, 0.2098782061, 0.1649043048, 0.1009618193, 0.0486780200, 0.0185440076,0.0055632023, 0.0013004888, 0.0002322302, 0.0000306238, 0.0000028124, 0.0000001607, 0.0000000043. Two-sided extremeness includes exactly outcomes with P0(X=x)<=P0(X=observed), usingunrounded probabilities for ordering.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 8 of 8

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