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 \(dof=n-2\)