AP Statistics Exam Notes

Confidence Intervals for Proportions

• One-sample z-interval:
  • Parameter: $p$
  • Conditions: Random sample, $n ≤ 10\%N$, $n\hat{p}≥ 10$, $n(1 - \hat{p}) ≥ 10$
  • Formula: $\hat{p} ± z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
  • Calculator: 1-PropZInt
• Two-sample z-interval:
  • Parameter: $p<em>1 - p</em>2$
  • Conditions: Independent random samples/randomized experiment, $n<em>1 ≤ 10\%N</em>1$, $n<em>2 ≤ 10\%N</em>2$, $n<em>1\hat{p}</em>1 ≥ 10$, $n<em>1(1 - \hat{p}</em>1) ≥ 10$, $n<em>2\hat{p}</em>2 ≥ 10$, $n<em>2(1 - \hat{p}</em>2) ≥ 10$
  • Formula: $(\hat{p}<em>1 - \hat{p}</em>2) ± z^* \sqrt{\frac{\hat{p}<em>1(1 - \hat{p}</em>1)}{n<em>1} + \frac{\hat{p}</em>2(1 - \hat{p}<em>2)}{n</em>2}}$
  • Calculator: 2-PropZInt

Confidence Intervals for Means

• One-sample t-interval:
  • Parameter: $\mu$
  • Conditions: Random sample/randomized experiment, $n ≤ 10\%N$, population distribution ≈ normal or $n ≥ 30$
  • Formula: $\bar{x} ± t^* \frac{s}{\sqrt{n}}$
  • Calculator: TInterval, df = n – 1
• Two-sample t-interval:
  • Parameter: $\mu<em>1 - \mu</em>2$
  • Conditions: Independent random samples/randomized experiment, $n<em>1 ≤ 10\%N</em>1$, $n<em>2 ≤ 10\%N</em>2$, population distributions ≈ normal or $n ≥ 30$
  • Formula: $(\bar{x}<em>1 - \bar{x}</em>2) ± t^* \sqrt{\frac{(s<em>1)^2}{n</em>1} + \frac{(s<em>2)^2}{n</em>2}}$
  • Calculator: 2-SampTInt, df = smaller of n1 – 1 and n2 – 1

Confidence Intervals for Slope

• t-interval for slope:
  • Parameter: $\beta$
  • Conditions: Linear relationship, $n ≤ 10\%N$, y is ≈ normal for each x, y has same standard deviation for each x, random sample/randomized experiment
  • Formula: $b ± t^*SE_b$
  • Calculator: LinRegTInt, df = n – 2

Significance Tests for Proportions

• One-sample z-test:
  • Null Hypothesis: $H<em>0: p = p</em>0$
  • Conditions: Random sample, $n ≤ 10\%N$, $np<em>0 ≥ 10$, $n(1 - p</em>0) ≥ 10$
  • Formula: $z = \frac{\hat{p} - p<em>0}{\sqrt{\frac{p</em>0(1 - p_0)}{n}}}$
  • Calculator: 1-PropZTest
• Two-sample z-test:
  • Null Hypothesis: $H<em>0: p</em>1 – p_2 = 0$
  • Conditions: Independent random samples/randomized experiment, $n<em>1 ≤ 10\%N</em>1$, $n<em>2 ≤ 10\%N</em>2$, $n<em>1\hat{p}</em>c ≥ 10$, $n<em>1(1 - \hat{p}</em>c) ≥ 10$, $n<em>2\hat{p}</em>c ≥ 10$, $n<em>2(1 - \hat{p}</em>c) ≥ 10$, where $\hat{p}<em>c = \frac{x</em>1 + x<em>2}{n</em>1 + n_2}$
  • Formula: $z = \frac{(\hat{p}<em>1 - \hat{p}</em>2) - 0}{\sqrt{\frac{\hat{p}<em>c(1 - \hat{p}</em>c)}{n<em>1} + \frac{\hat{p}</em>c(1 - \hat{p}<em>c)}{n</em>2}}}$
  • Calculator: 2-PropZTest

Significance Tests for Means

• One-sample t-test:
  • Null Hypothesis: $H<em>0: \mu = \mu</em>0$
  • Conditions: Random sample/randomized experiment, $n ≤ 10\%N$, population distribution ≈ normal or $n ≥ 30$
  • Formula: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$
  • Calculator: T-Test, df = n – 1
• Two-sample t-test:
  • Null Hypothesis: $H<em>0: \mu</em>1 - \mu_2 = 0$
  • Conditions: Independent random samples/randomized experiment, $n<em>1 ≤ 10\%N</em>1$, $n<em>2 ≤ 10\%N</em>2$, population distributions ≈ normal or $n ≥ 30$
  • Formula: $t = \frac{(\bar{x}<em>1 - \bar{x}</em>2) - (\mu<em>1 - \mu</em>2)}{\sqrt{\frac{(s<em>1)^2}{n</em>1} + \frac{(s<em>2)^2}{n</em>2}}}$
  • Calculator: 2-SampTTest, df = smaller of n1 – 1 and n2 – 1

Significance Tests for Slope

• t-test for slope:
  • Null Hypothesis: $H<em>0: \beta = \beta</em>0$
  • Conditions: Linear relationship, $n ≤ 10\%N$, y is ≈ normal for each x, y has same standard deviation for each x, random sample/randomized experiment
  • Formula: $t = \frac{b - \beta<em>0}{SE</em>b}$
  • Calculator: LinRegTTest, df = n – 2

Chi-Square Tests

• Goodness-of-fit:
  • Hypotheses: H0: Claimed distribution is correct. Ha: Claimed distribution is incorrect.
  • Conditions: Random sample/randomized experiment, $n ≤ 10\%N$, all expected counts > 5
  • Formula: $\chi^2 = \sum \frac{(observed - expected)^2}{expected}$
  • Calculator: $\chi^2$GOF-Test, df = # of categories – 1
• Homogeneity:
  • Hypotheses: H0: No difference in distribution across populations. Ha: Difference in distribution across populations.
  • Conditions: Random samples/randomized experiment, $n ≤ 10\%N$, all expected counts > 5
  • Formula: $\chi^2 = \sum \frac{(observed - expected)^2}{expected}$
  • Calculator: $\chi^2$-Test, df = (# of rows – 1) (# of columns – 1)
• Independence:
  • Hypotheses: H0: No association between variables. Ha: Variables are associated.
  • Conditions: Random sample/randomized experiment, $n ≤ 10\%N$, all expected counts > 5
  • Formula: $\chi^2 = \sum \frac{(observed - expected)^2}{expected}$
  • Calculator: $\chi^2$-Test, df = (# of rows – 1) (# of columns – 1)