Statistics: Means & Intervals

Cheatsheet Content

### One-Sample Formulas (Single Group) Use when you have one set of data ($n$, $\bar{x}$, $s$). | Goal | Formula | Table / Multiplier | |---|---|---| | Confidence Interval (CI) | $\bar{x} \pm t \cdot \left(\frac{s}{\sqrt{n}}\right)$ | Table IV: Row $df=n-1$, Two-tail $\alpha$ | | Prediction Interval (PI) | $\bar{x} \pm t \cdot s \sqrt{1 + \frac{1}{n}}$ | Table IV: Row $df=n-1$, Two-tail $\alpha$ | | Mean of $m$ Future Items | $\bar{x} \pm t \cdot s \sqrt{\frac{1}{m} + \frac{1}{n}}$ | Table IV: Row $df=n-1$, Two-tail $\alpha$ | | Tolerance Interval (TI) | $\bar{x} \pm k \cdot s$ | Table V: Use $n$ and % of population | ### Two-Sample Formulas (Comparing Two Groups) Use when comparing Group 1 vs. Group 2 (e.g., High Purity vs. Commercial). - **Standard Error (SE):** $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ - **Degrees of Freedom ($df$):** Use the smaller of $(n_1-1)$ or $(n_2-1)$. | Goal | Formula | Table / Multiplier | |---|---|---| | Two-Sided Interval | $(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, df} \cdot SE$ | Table IV: Two-tail column | | Lower Confidence Bound | $(\bar{x}_1 - \bar{x}_2) - t_{\alpha, df} \cdot SE$ | Table IV: One-tail column | | Upper Confidence Bound | $(\bar{x}_1 - \bar{x}_2) + t_{\alpha, df} \cdot SE$ | Table IV: One-tail column | ### Key Vocabulary & Logic - **Critical Value:** The number from the table ($t$, $z$, or $k$). - **Standard Error (SE):** The "spread" of the sample mean ($\frac{s}{\sqrt{n}}$). - **Margin of Error ($E$):** Everything after the $\pm$ sign ($E = \text{Critical Value} \cdot SE$). - **Confidence Level ($1-\alpha$):** Usually 95% ($\alpha=0.05$) or 99% ($\alpha=0.01$). - **Precision:** Refers to the width of the interval. A narrower interval is more precise. (Increase $n$ to increase precision). - **Reliability:** Refers to the confidence level. 99% is more reliable than 95%, but results in a wider (less precise) interval. - **$n$**: Sample size (number of observations in a group). - **$\bar{x}$**: Sample mean (average of observations in a group). - **$s$**: Sample standard deviation (measure of spread of observations in a group). - **$\alpha$**: Significance level (probability of making a Type I error, $1 - \text{Confidence Level}$). - **$df$**: Degrees of freedom (relates to sample size, used for $t$-distribution). - **$m$**: Number of future items for prediction. ### Table Strategy (Which one do I use?) - **Table IV ($t$-distribution):** - **Two-tail:** Use for any "Interval" ($\pm$). - **One-tail:** Use for "Lower Bound" or "Upper Bound" (only one side). - **$z$ row:** If $n$ is very large (usually $>40$), use the bottom row of the $t$-table. - **Table V (Tolerance Factors):** - Use ONLY when the question asks to capture a specific percentage of the population (e.g., "capture 99% of all cadences"). - **Two-sided:** Use for an "Interval." - **One-sided:** Use for a "Bound." ### Calculator Shortcuts - **Standard Deviation ($s$):** $\sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$ - **Standard Error of Mean (SE Mean):** $\frac{s}{\sqrt{n}}$ - **Point Estimate for Difference:** $\bar{x}_1 - \bar{x}_2$