Ultimate Inference Guide

Ultimate Inference Guide – AP Statistics

Confidence Intervals

Proportions

One-sample z-interval for a proportion

• Statistic: p̂

• Parameter: p

• Conditions:

  • Random sample

  • n ≤ 10%N

  • np̂ ≥ 10 and n(1 − p̂) ≥ 10

• Formula:
  p^±z∗p^(1−p^)n\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}p^​±z∗np^​(1−p^​)​​

• Calculator: 1-PropZInt

Two-sample z-interval for difference in proportions

• Statistic: p̂₁ − p̂₂

• Parameter: p₁ − p₂

• Conditions:

  • Independent random samples or randomized experiment

  • n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂

  • n₁p̂₁ ≥ 10, n₁(1 − p̂₁) ≥ 10

  • n₂p̂₂ ≥ 10, n₂(1 − p̂₂) ≥ 10

• Formula:
  (p^1−p^2)±z∗p^1(1−p^1)n1+p^2(1−p^2)n2(\hat{p}_1-\hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}(p^​1​−p^​2​)±z∗n1​p^​1​(1−p^​1​)​+n2​p^​2​(1−p^​2​)​​

• Calculator: 2-PropZInt

Means

One-sample t-interval (or paired t-interval)

• Statistic: x̄

• Parameter: μ

• Conditions:

  • Random sample or experiment

  • n ≤ 10%N

  • Population approx normal or n ≥ 30

• Formula:
  xˉ±t∗sn\bar{x} \pm t^* \frac{s}{\sqrt{n}}xˉ±t∗n​s​

• df = n − 1

• Calculator: TInterval

Two-sample t-interval for difference in means

• Statistic: x̄₁ − x̄₂

• Parameter: μ₁ − μ₂

• Conditions:

  • Independent samples

  • n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂

  • Each population approx normal or n ≥ 30

• Formula:
  (xˉ1−xˉ2)±t∗s12n1+s22n2(\bar{x}_1-\bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}(xˉ1​−xˉ2​)±t∗n1​s12​​+n2​s22​​​

• df = smaller of n₁ − 1 and n₂ − 1 (or use tech)

• Calculator: 2-SampTInt

Slope

t-interval for slope

• Statistic: b

• Parameter: β

• Conditions:

  • Linear relationship

  • n ≤ 10%N

  • Normal residuals

  • Equal variance

  • Random sample or experiment

• Formula:
  b±t∗SEbb \pm t^* SE_bb±t∗SEb​

• df = n − 2

• Calculator: LinRegTInt

Significance Tests

Proportions

One-sample z-test for a proportion

• H₀: p = p₀

• Conditions:

  • Random sample

  • n ≤ 10%N

  • np₀ ≥ 10 and n(1 − p₀) ≥ 10

• Formula:
  z=p^−p0p0(1−p0)nz = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}z=np0​(1−p0​)​​p^​−p0​​

• Calculator: 1-PropZTest

Two-sample z-test for difference in proportions

• H₀: p₁ − p₂ = 0

• Conditions:

  • Independent samples

  • n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂

  • Use pooled p̂

• Formula:
  z=(p^1−p^2)p^(1−p^)(1n1+1n2)z = \frac{(\hat{p}_1-\hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}z=p^​(1−p^​)(n1​1​+n2​1​)​(p^​1​−p^​2​)​

• Calculator: 2-PropZTest

Means

One-sample t-test (or paired t-test)

• H₀: μ = μ₀

• Conditions:

  • Random sample

  • n ≤ 10%N

  • Normal or n ≥ 30

• Formula:
  t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}t=s/n​xˉ−μ0​​

• df = n − 1

• Calculator: T-Test

Two-sample t-test for difference in means

• H₀: μ₁ − μ₂ = 0

• Conditions:

  • Independent samples

  • n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂

  • Normal or n ≥ 30

• Formula:
  t=(xˉ1−xˉ2)s12n1+s22n2t = \frac{(\bar{x}_1-\bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}t=n1​s12​​+n2​s22​​​(xˉ1​−xˉ2​)​

• df = smaller of n₁ − 1 and n₂ − 1

• Calculator: 2-SampTTest

Slope

t-test for slope

• H₀: β = β₀

• Conditions:

  • Linear relationship

  • Normal residuals

  • Equal variance

  • Random sample

• Formula:
  t=b−β0SEbt = \frac{b - \beta_0}{SE_b}t=SEb​b−β0​​

• df = n − 2

• Calculator: LinRegTTest

Chi-Square Tests

Goodness-of-fit

• H₀: Distribution is correct

• Hₐ: Distribution is not correct

• Conditions:

  • Random sample

  • n ≤ 10%N

  • All expected counts > 5

• Formula:
  χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}χ2=∑E(O−E)2​

• df = categories − 1

• Calculator: χ²GOF-Test

Homogeneity

• H₀: No difference across groups

• Hₐ: Difference exists

• Conditions:

  • Random samples or experiment

  • All expected counts > 5

• Formula:
  χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}χ2=∑E(O−E)2​

• df = (rows − 1)(columns − 1)

• Calculator: χ²-Test

Independence

• H₀: Variables are independent

• Hₐ: Variables are associated

• Conditions:

  • Random sample

  • All expected counts > 5

• Formula:
  χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}χ2=∑E(O−E)2​

• df = (rows − 1)(columns − 1)

• Calculator: χ²-Test