Hypothesis test and confidence interval

Procedures for Hypothesis Test¶

1. Define context
   • $H_0: parameter =\ ...$
   • $H_a: parameter\ condition\ ...$
   • $parameter$ is the ...
   • $\alpha =\ ...$
2. Verify inference conditions
3. Find p-value
4. Conclusion
   • $p\ condition\ \alpha$
   • $H_0$ should/should not be rejected
   • The data provides sufficient evidence that ...

Procedures for Confidence Interval¶

1. Verify confidence interval conditions
2. Find confidence interval
   • $CI = statistic \pm (critical\ value) \times (standard\ error\ of\ statistic)$
3. We can be $C\%$ confident that the interval from $lower\ limit$ to $upper\ limit$ captures the actual value of the $parameter$

Population Proportions¶

One-sample z-test.

Conditions:

• Independence condition
  • Random sampling/assignment
  • If sampling without replacement, $n < 0.1 N$
• Approximately normally distributed
  • $\left\{\begin{aligned} np_0 &\geq 10 \\ n(1-p_0) &\geq 10 \end{aligned}\right.$

Test statistic:

• $z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0 (1-p_0)}{n}}}$
• standard error = $\sqrt{\dfrac{p_0 (1-p_0)}{n}}$

Differences in Population Proportions¶

Two-sample z-test .

Conditions:

• Independence condition
  • Random sampling/assignment
  • If sampling without replacement, $n < 0.1 N$
• Approximately normally distributed
  • Combined proportion/pooled proportion $\hat{p}_c = \dfrac{X_1 + X_2}{n_1 + n_2}$, $X = n \hat{p}$
  • $\left\{\begin{aligned} n_1\hat{p}_c &\geq 10 \\ n_1(1-\hat{p}_c) &\geq 10 \\ n_2\hat{p}_c &\geq 10 \\ n_2(1-\hat{p}_c) &\geq 10 \end{aligned}\right.$

Test statistic:

• $z = \dfrac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}$
• standard error = $\sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Population Means¶

One-sample t-test.

Conditions:

• Independence condition
  • Random sampling/assignment
  • If sampling without replacement, $n < 0.1 N$
• Approximately normally distributed
  • Approximately symmetric
  • No outliers
• If very skewed, $n \geq 30$

Test statistic:

• $t = \dfrac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}$
• standard error = $\dfrac{s}{\sqrt{n}}$

Differences in Population Means¶

Two-sample t-test.

Conditions:

• Independence condition
  • Random sampling/assignment
  • If sampling without replacement, $n < 0.1 N$
• Approximately normally distributed
  • Approximately symmetric
  • No outliers
• If very skewed, $n \geq 30$

Test statistic:

• $t = \dfrac{\overline{x}_1 - \overline{x}_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$
• standard error = $\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$

Differences in Matched Pairs¶

One-sample t-test.

Conditions:

• Two measures come from the same items within the population
• Independence condition
  • Random sampling/assignment
  • If sampling without replacement, $n < 0.1 N$
• Approximately normally distributed
  • Approximately symmetric
  • No outliers
• If very skewed, $n \geq 30$

Test statistic:

• $t = \dfrac{\overline{x}_d - \mu_d}{\frac{s_d}{\sqrt{n}}}$
• standard error = $\dfrac{s_d}{\sqrt{n}}$

Goodness of Fit¶

$\chi^2$-test.

Conditions:

• Independence condition
  • Random sampling
  • If sampling without replacement, $n < 0.1 N$
• Large counts condition
  • Each expected value ≥ 5

Test statistic:

• $\chi^2 = \sum \dfrac{(observed - expected)^2}{expected}$

Independence¶

One-sample $\chi^2$-test.

Conditions:

• Independence condition
  • Random sampling
  • If sampling without replacement, $n < 0.1 N$
• Large counts condition
  • Each expected value ≥ 5
  • or ≥ 80\% expected values > 5 and all are ≥ 1

Test statistic:

• $\chi^2 = \sum \dfrac{(observed - expected)^2}{expected}$
• $Expected\ value$ = $\dfrac{Row\ total \times Column\ total}{Total\ number\ in\ sample}$
• $dof = (n_{row}-1)(n_{col}-1)$

Homogeneity¶

Multi-sample $\chi^2$-test.

Conditions:

• Independence condition
  • Random sampling
  • If sampling without replacement, $n < 0.1 N$
• Large counts condition
  • Each expected value ≥ 5

Test statistic:

• $\chi^2 = \sum \dfrac{(observed - expected)^2}{expected}$
• $dof = (n_{row}-1)(n_{col}-1)$

Regression Line¶

One-sample t-test.

Conditions:

• Relationship between $x$ and $y$ is linear
• $\sigma_y$ cannot vary with $x$
• Independence condition
  • Random sampling
  • If sampling without replacement, $n < 0.1 N$
• For a given value of $x$, $y$-values follow an approximate normal distribution
• If $n<30$, $y$-values distribution have no strong skew and no outliers

Test statistic:

• $t = \dfrac{b - \beta}{s_b}$
• standard error $s_b = \dfrac{s}{s_x \sqrt{n-1}}$
  • $s = \sqrt{\dfrac{\sum (y_i - \hat{y_i})^2}{n-2}}$
  • $s_x = \sqrt{\dfrac{\sum (x_i - \overline{x})^2}{n-1}}$
• $t$-distribution with $dof=n-2$