Traditional Statistical Inference

Read It: Summary of Traditional Statistical Inference

Recall from earlier in the lesson, you used the z* values and the z-statistics to calculate confidence intervals and p-values. You can also use those values to make conclusions from each of the hypothesis tests that we covered in Lesson 5. Below is a table that outlines how to fond the standard error, confidence interverals, and p-value for each of the hypothesis tests using the z-related formlas.

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Table showing the forumlae needed to calculate the standard error, confidence interval, and p-value for each hypothesis test discussed in Lesson 5.

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Single Proportion

Sample Statistic: $\hat{p}$
Standard Error: $\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$
Confidence Interval: $\hat{p} \pm z^{*} \times \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$
Hypothesis Test Statistic: $\frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$

Single Mean

Sample Statistic: $\bar{x}$
Standard Error: $\frac{s}{\sqrt{n}}$
Confidence Interval: $\bar{x} \pm t^{*} \times \frac{s}{\sqrt{n}}$
Hypothesis Test Statistic: $\frac{\bar{x} - μ_{0}}{s / \sqrt{n}}$

Difference in Proportions

Sample Statistic: ${\hat{p}}_{1} - {\hat{p}}_{2}$
Standard Error: $\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$
Confidence Interval: $({\hat{p}}_{1} - {\hat{p}}_{2}) \pm z^{*} \times S E$
Hypothesis Test Statistic: $\frac{({\hat{p}}_{1} - {\hat{p}}_{2}) - 0}{\sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}}$

Difference in Means

Sample Statistic: ${\bar{x}}_{1} - {\bar{x}}_{2}$
Standard Error: $\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$
Confidence Interval: $({\bar{x}}_{1} - {\bar{x}}_{2}) \pm t^{*} \times S E$
Hypothesis Test Statistic: $\frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - 0}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$

Paired Difference in Means

Sample Statistic: ${\bar{x}}_{d}$
Standard Error: $\frac{s_{d}}{\sqrt{n_{d}}}$
Confidence Interval: ${\bar{x}}_{d} \pm t^{*} \times \frac{s_{d}}{\sqrt{n_{d}}}$
Hypothesis Test Statistic: $\frac{{\bar{x}}_{d} - 0}{s_{d} / \sqrt{n_{d}}}$

Download pdf: Summary of Types of Inference [2]