Mostly Harmless Statistics Formula Packet
- Page ID
- 34990
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| Sample Mean: \(\bar{x} = \frac{\sum x}{n}\)le | Population Mean: \(\mu = \frac{\sum x}{N}\) |
| Weighted Mean: \(\bar{x} = \frac{\sum (xw)}{\sum w}\) | Range = \(\text{Max} - \text{Min}\) |
| Sample Standard Deviation: \(s = \sqrt{\frac{\sum \left(x - \bar{x}\right)^{2}}{n-1}}\) | Population Standard Deviation = \(\sigma\) |
| Sample Variance: \(s^{2} = \frac{\sum \left(x - \bar{x}\right)^{2}}{n-1}\) | Population Variance = \(\sigma^{2}\) |
| Coefficient of Variation: \(\text{CVar} = \left(\frac{s}{\bar{x}} \cdot 100\right) %\) | \(Z\)-Score: \(z = \frac{x - \bar{x}}{s}\) |
| Percentile Index: \(i = \frac{(n+1) \cdot p}{100}\) | Interquartile Range: \(\text{IQR} = Q_{3} - Q_{1}\) |
| Empirical Rule: \(z = 1, 2, 3 \Rightarrow 68%, 95%, 99.7%\) | Outlier Lower Limit: \(Q_{1} - (1.5 \cdot \text{IQR})\) |
| Chebyshev’s Inequality: \(\left(\left(1 - \frac{1}{(z)^{2}}\right) \cdot 100r \right) %\) | Outlier Upper Limit: \(Q_{3} + (1.5 \cdot \text{IQR})\) |
TI-84: Enter the data in a list and then press [STAT]. Use cursor keys to highlight CALC. Press 1 or [ENTER] to select 1:1-Var Stats. Press [2nd], then press the number key corresponding to your data list. Press [Enter] to calculate the statistics. Note: the calculator always defaults to L1 if you do not specify a data list.
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\(s_{x}\) is the sample standard deviation. You can arrow down and find more statistics. Use the min and max to calculate the range by hand. To find the variance simply square the standard deviation.
Chapter 4 Formulas
| Complement Rules: \(\begin{array}{l} \text{P}(A) + \text{P}(A^{C}) = 1 \\ \text{P}(A) = 1 - \text{P}(A^{C}) \\ \text{P}(A^{C}) = 1 - \text{P}(A) \end{array}\) | Mutually Exclusive Events: \(\text{P}(A \cup B) = 0\) |
| Union Rule: \(\text{P} (A \cup B) = \text{P}(A) + \text{P}(B) – \text{P}(A \cap B)\) | Independent Events: \(\text{P} (A \cup B) = \text{P}(A) \cdot \text{P}(B)\) |
| Intersection Rule: \(\text{P} (A \cap B) = \text{P}(A) \cdot \text{P} (A|B)\) | Conditional Probability Rule: \(\text{P} (A|B) = \frac{\text{P} (A \cap B)}{\text{P} (B)}\) |
| Fundamental Counting Rule: \(m_{1} \cdot m_{2} \cdots m_{n}\) | Factorial Rule: \(n! = n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1\) |
| Combination Rule: \({}_{n} C_{r} = \frac{n!}{(r! (n-r)!)}\) | Permutation Rule: \({}_{n} P_{r} = \frac{n!}{(n-r)!}\) |
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Chapter 5 Formulas
| Discrete Distribution Table: \(0 \leq \text{P} (x_{i}) \leq 1 \quad\quad\quad \sum \text{P} (x_{i}) = 1\) |
Discrete Distribution Mean: \(\mu = \sum \left(x_{i} \cdot \text{P} \left(x_{i}\right) \right)\) |
| Discrete Distribution Variance: \(\sigma^{2} = \sum \left(x_{i}^{2} \cdot \text{P} \left(x_{i}\right)\right) - \mu^{2}\) |
Discrete Distribution Standard Deviation: \(\sigma = \sqrt{\sigma^{2}}\) |
| Geometric Distribution: \(\text{P} (X=x) = p \cdot q^{x-1}, x = 1,2,3, \ldots\) |
Geometric Distribution Mean: \(\mu = \frac{1}{p}\) Variance: \(\sigma^{2} = \frac{1-p}{p^{2}}\) Standard Deviation: \(\sigma = \sqrt{\frac{1 - p}{p^{2}}}\) |
| Binomial Distribution: \(\text{P} (X=x) = {}_{n} C_{x} p^{x} \cdot q^{(n-x)}, x=0, 1, 2, \ldots, n\) |
Binomial Distribution Mean: \(\mu = n \cdot p\) Variance: \(sigma^{2} = n \cdot p \cdot q\) Standard Deviation: \(\sigma = \sqrt{n \cdot p \cdot q}\) |
| Hypergeometric Distribution: \(\text{P} (X=x) = \frac{\,_{a} C_{x} \cdot \,_{b} C_{n-x}}{\,_{N} C_{n}}\) |
\(p = \text{P(success)} \quad\quad\quad p = \text{P(failure)} = 1 - p\) \(n = \text{sample size} \quad\quad\quad N = \text{population size}\) |
| Unit Change for Poisson Distribution: \(\text{New } \mu = \text{old } \mu \left(\frac{\text{new units}}{\text{old units}}\right)\) |
Poisson Distribution: \(\text{P} (X=x) = \frac{e^{- \mu} \mu^{x}}{x!}\) |
| \(\text{P} (X=x)\) | \(\text{P} (X \leq x)\) | \(\text{P} (X \geq x)\) |
|---|---|---|
| Is the same as | Is less than or equal to | Is greater than or equal to |
| Is equal to | Is at most | Is at least |
| Is exactly the same as | Is not greater than | Is not less than |
| Has not changed from | Within | Is more than or equal to |
| Excel \(=\text{binom.dist}(x,n,p,0)\) \(=\text{HYPGEOM.DIST}(x,n,a,N,0)\) \(=\text{POISSON.DIST}(x,\mu,0)\) |
Excel \(=\text{binom.dist}(x,n,p,1)\) \(= \text{HYPGEOM.DIST}(x,n,a,N,1)\) \(=\text{POISSON.DIST}(x,\mu,1)\) |
Excel \(=1-\text{binom.dist}(x-1,n,p,1)\) \(=1- \text{HYPGEOM.DIST}(x-1,n,a,N,1)\) \(=1-\text{POISSON.DIST}(x-1,\mu,1)\) |
| TI Calculator \(\text{geometpdf}(p,x)\) \(\text{binompdf}(n,p,x)\) \(\text{poissonpdf}(\mu,x)\) |
TI Calculator \(\text{binomcdf}(n,p,x)\) \(\text{poissoncdf}(\mu,x)\) |
TI Calculator \(1-\text{binomcdf}(n,p,x-1)\) \(1-\text{poissoncdf}(\mu,x-1)\) |
| \(\text{P} (X>x)\) | \(\text{P} (X<x)\) | |
|---|---|---|
|
How do you tell them apart?
|
x)\)">More than | Less than |
| x)\)">Greater than | Below | |
| x)\)">Above | Lower than | |
| x)\)">Higher than | Shorter than | |
| x)\)">Longer than | Smaller than | |
| x)\)">Bigger than | Decreased | |
| x)\)">Increased | Reduced | |
| x)\)"> | ||
| x)\)">Excel \(=1-\text{binom.dist}(x,n,p,1)\) \(=1- \text{HYPGEOM.DIST}(x,n,a,N,1)\) \(=1-\text{POISSON.DIST}(x,\mu,1)\) |
Excel \(=\text{binom.dist}(x-1,n,p,1)\) \(=\text{HYPGEOM.DIST}(x-1,n,a,N,1)\) \(=\text{POISSON.DIST}(x-1,\mu,1)\) |
|
| x)\)">TI Calculator \(1-\text{binomcdf}(n,p,x)\) \(1-\text{poissoncdf}(\mu,x)\) |
TI Calculator \(\text{binomcdf}(n,p,x-1)\) \(\text{poissoncdf}(\mu,x-1)\) |
Chapter 6 Formulas
| Uniform Distribution \(f(x) = \frac{1}{b-a}, \text{ for } a \leq x \leq b\) \(\text{P}(X \geq x) = \text{P} (X>x) = \left(\frac{1}{b-a}\right) \cdot (b-x)\) \(\text{P}(X \leq x) = \text{P} (X<x) = \left(\frac{1}{b-a}\right) \cdot (x-a)\) \(\text{P}\left(x_{1} \leq X \leq x_{2}\right) = \text{P} \left(x_{1} < X < x_{2}\right) = \left(\frac{1}{b-a}\right) \cdot \left(x_{2}-x_{1}\right)\) |
Exponential Distribution \(f(x) = \frac{1}{\mu} e^{(-x / \mu)}, \text{ for } x \geq 0\) \(\text{P}(X \geq x) = \text{P} (X > x) = e^{-x / \mu}\) \(\text{P}(X \leq x) = \text{P} (X < x) = 1 - e^{-x / \mu}\) \(\text{P}\left(x_{1} \leq X \leq x_{2}\right) = \text{P} \left(x_{1} < X < x_{2}\right) = e^{(-x_{1} / \mu)} - e^{(-x_{2} / \mu)}\) |
| Standard Normal Distribution \(\mu = 0, \sigma = 1\) \(z\)-score: \(z = \frac{x - \mu}{\sigma}\) \(x = z \sigma + \mu\) |
Central Limit Theorem Z-score: \(z = \frac{\bar{x} - \mu}{\left( \frac{\sigma}{\sqrt{n}} \right)}\) |
In the table below, note that when \(\mu = 0\) and \(\sigma = 1\) use the \(\text{NORM.S. DIST}\) or \(\text{NORM.S.INV}\) function in Excel for a standard normal distribution.
| \(\text{P} (X \leq x)\) or \(\text{P} (X < x)\) | \(\text{P} \left(x_{1} < X < x_{2}\right)\) or \(\text{P} \left(x_{1} \leq X \leq x_{2}\right)\) | \(\text{P} (X \geq x)\) or \(\text{P} (X > x)\) |
|---|---|---|
| Is less than or equal to | Between | x)\)">Is greater than or equal to |
| Is at most | x)\)">Is at least | |
| Is not greater than | x)\)">Is not less than | |
| Within | x)\)">More than | |
| Less than | x)\)">Greater than | |
| Below | x)\)">Above | |
| Lower than | x)\)">Higher than | |
| Shorter than | x)\)">Longer than | |
| Smaller than | x)\)">Bigger than | |
| Decreased | x)\)">Increased | |
| Reduced | x)\)">Larger | |
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x)\)">.png?revision=1) |
| Excel Finding a Probability: \(=\text{NORM.DIST}(x, \mu, \sigma, \text{true})\) Finding a Percentile: \(=\text{NORM.INV}(\text{area}, \mu, \sigma)\) |
Excel Finding a Probability: \(=\text{NORM.DIST}(x_{2},\mu,\sigma,\text{true}) - \text{NORM.DIST}(x_{1},\mu,\sigma,\text{true})\) Finding a Percentile: \(x_{1} = \text{NORM.INV}((1-\text{area})/2,\mu, \sigma)\) \(x_{2} = \text{NORM.INV}(1-((1-\text{area})/2),\mu,\sigma)\) |
x)\)">Excel Finding a Probability: \(= 1-\text{NORM.DIST}(x, \mu, \sigma, \text{true})\) Finding a Percentile: \(= \text{NORM.INV}(1-\text{area}, \mu, \sigma)\) |
| TI Calculator Finding a Probability: \(=\text{normalcdf}(-1\text{E}99,x,\mu,\sigma)\) Finding a Percentile: \(=\text{invNorm}(\text{area},\mu,\sigma)\) |
TI Calculator Finding a Probability: \(=\text{normalcdf}(x_{1}, x_{2}, \mu, \sigma)\) Finding a Percentile: \(x_{1} = \text{invNorm}((1-\text{area})/2, \mu, \sigma)\) \(x_{2} = \text{invNorm}(1-((1-\text{area})/2), \mu, \sigma)\) |
x)\)">TI Calculator Finding a Probability: \(=\text{normalcdf}(x, 1\text{E}99, \mu, \sigma)\) Finding a Percentile: \(=\text{invNorm}(1-\text{area}, \mu, \sigma)\) |
Chapter 7 Formulas
| Confidence Interval for One Proportion \(\hat{p} \pm z_{\alpha/2} \sqrt{\left(\frac{\hat{p} \hat{q}}{n}\right)}\) \(\hat{p} = \frac{x}{n}\) \(\hat{q} = 1 - \hat{p}\) TI-84: \(1 - \text{PropZInt}\) |
Sample Size for Proportion \(n = p^{*} \cdot q^{*} \left(\frac{z_{\alpha/2}}{E}\right)^{2}\) Always round up to whole number. If \(p\) is not given use \(p^{*} = 0.5\). \(E\) = Margin of Error |
| Confidence Interval for One Mean Use z-interval when \(\sigma\) is given. Use t-interval when \(s\) is given. If \(n < 30\), population needs to be normal. |
Z-Confidence Interval \(\bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)\) TI-84: \(\text{ZInterval}\) |
| Z-Critical Values Excel: \(z_{\alpha/2} = \text{NORM.INV}(1-\text{area}/2, 0, 1)\) TI-84: \(z_{\alpha/2} = \text{invNorm}(1-\text{area}/2, 0, 1)\) |
t-Critical Values Excel: \(t_{\alpha/2} = \text{T.INV}(1-\text{area}/2, df)\) TI-84: \(t_{\alpha/2} = \text{invT}(1-\text{area}/2, df)\) |
| t-Confidence Interval \(\bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)\) \(df = n-1\) TI-84: \(\text{TInterval}\) |
Sample Size for Mean \(n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^{2}\) Always round up to whole number. \(E\) = Margin of Error |
Chapter 8 Formulas
| Hypothesis Test for One Mean Use z-test when \(\sigma\) is given. Use t-test when \(s\) is given. If \(n < 30\), population needs to be normal. |
Type I Error - Reject \(H_{0}\) when \(H_{0}\) is true. Type II Error - Fail to reject \(H_{0}\) when \(H_{0}\) is false. |
| Z-Test: \(H_{0}: \mu = \mu_{0}\) \(H_{1}: \mu \neq \mu_{0}\) \(z = \frac{\bar{x} - \mu_{0}}{\left(\frac{\sigma}{\sqrt{n}}\right)}\) TI-84: \(\text{Z-Test}\) |
t-Test: \(H_{0}: \mu = \mu_{0}\) \(H_{1}: \mu \neq \mu_{0}\) \(t = \frac{\bar{x} - \mu_{0}}{\left(\frac{s}{\sqrt{n}}\right)}\) TI-84: \(\text{T-Test}\) |
| z-Critical Values Excel: Two-tail: \(z_{\alpha/2} = \text{NORM.INV}(1-\alpha/2, 0, 1)\) Right-tail: \(z_{1 - \alpha} = \text{NORM.INV}(1-\alpha, 0, 1)\) Left-tail: \(z_{\alpha} = NORM.INV(\alpha, 0, 1)\) TI-84: Two-tail: \(z_{\alpha/2} = \text{invNorm}(1-\alpha/2, 0, 1)\) Right-tail: \(z_{1-\alpha} = \text{invNorm}(1-\alpha, 0, 1)\) Left-tail: \(z_{\alpha} = \text{invNorm}(\alpha, 0, 1)\) |
t-Critical Values Excel: Two-tail: \(t_{\alpha/2} = \text{T.INV}(1-\alpha/2, df)\) Right-tail: \(t_{1-\alpha} = \text{T.INV}(1-\alpha, df)\) Left-tail: \(t_{\alpha} = \text{T.INV}(\alpha, df)\) TI-84: Two-tail: \(t_{\alpha/2} = \text{invT}(1-\alpha/2, df)\) Right-tail: \(t_{1-\alpha} = \text{invT}(1-\alpha, df)\) Left-tail: \(t_{\alpha} = \text{invT}(\alpha, df)\) |
| Hypothesis Test for One Proportion \(H_{0}: p = p_{0}\) \(H_{1}: p \neq p_{0}\) \(z = \frac{\hat{p} - p_{0}}{\sqrt{\left(\frac{p_{0} q_{0}}{n}\right)}}\) TI-84: \(1\text{-PropZTest}\) |
Rejection Rules: P-value method: reject \(H_{0}\) when the p-value \(\leq \alpha\). Critical value method: reject \(H_{0}\) when the test statistic is in the critical region (shaded tails). |
| Two-tailed Test | Right-tailed Test | Left-tailed Test |
|---|---|---|
| \(H_{0}: \mu = \mu_{0}\) or \(H_{0}: p = p_{0}\) \(H_{1}: \mu \neq \mu_{0}\) or \(H_{0}: p \neq p_{0}\) |
\(H_{0}: \mu = \mu_{0}\) or \(H_{0}: p = p_{0}\) \(H_{1}: \mu > \mu_{0}\) or \(H_{0}: p > p_{0}\) |
\(H_{0}: \mu = \mu_{0}\) or \(H_{0}: p = p_{0}\) \(H_{1}: \mu < \mu_{0}\) or \(H_{0}: p < p_{0}\) |
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| Claim is in the Null Hypothesis | ||
|---|---|---|
| = | \(\leq\) | \(\geq\) |
| Is equal to | Is less than or equal to | Is greater than or equal to |
| Is exactly the same as | Is at most | Is at least |
| Has not changed from | Is not more than | Is not less than |
| Is the same as | Within | Is more than or equal to |
| Claim is in the Alternative Hypothesis | ||
|---|---|---|
| \(\neq\) | > | < |
| Is not | More than | Less than |
| Is not equal to | Greater than | Below |
| Is different from | Above | Lower than |
| Has changed from | Higher than | Shorter than |
| Is not the same as | Longer than | Smaller than |
| Bigger than | Decreased | |
| Increased | Reduced | |
Chapter 9 Formulas
| Hypothesis Test for Two Dependent Means \(H_{0}: \mu_{D} = 0\) \(H_{1}: \mu_{D} \neq 0\) \(t = \frac{\bar{D} - \mu_{D}}{\left(\frac{s_{D}}{\sqrt{n}}\right)}\) TI-84: \(\text{T-Test}\) |
Confidence Interval for Two Dependent Means \(\bar{D} \pm t_{\alpha/2} \left(\frac{s_{D}}{\sqrt{n}}\right)\) TI-84: \(\text{TInterval}\) |
| Hypothesis Test for Two Independent Means Z-Test: \(H_{0}: \mu_{1} = \mu_{2}\) \(H_{1}: \mu_{1} \neq \mu_{2}\) \(z = \frac{\left(\bar{x}_{1} - \bar{x}_{2}\right) - \left(\mu_{1} - \mu_{2}\right)_{0}}{\sqrt{\left( \frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}} \right)}}\) TI-84: \(2\text{-SampZTest}\) |
Confidence Interval for Two Independent Means Z-Interval \(\left(\bar{x}_{1} - \bar{x}_{2}\right) \pm z_{\alpha/2} \sqrt{\left( \frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}\right)}\) TI-84: \(2\text{-SampZInt}\) |
| Hypothesis Test for Two Independent Means \(H_{0}: \mu_{1} = \mu_{2}\) \(H_{1}: \mu_{1} \neq \mu_{2}\) T-Test: Assume variances are unequal \(t = \dfrac{\left(\bar{x}_{1} - \bar{x}_{2}\right) - \left(\mu_{1} - \mu_{2}\right)_{0}}{\sqrt{\left( \frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}} \right)}}\) TI-84: \(2\text{-SampTTest}\) \(df = \dfrac{\left( \frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}} \right)^2}{\left( \left(\frac{s_{1}^{2}}{n_{1}}\right)^{2} \left(\frac{1}{n_{1}-1}\right) + \left(\frac{s_{2}^{2}}{n_{2}}\right)^{2} \left(\frac{1}{n_{2}-1}\right) \right)}\) T-Test: Assume variances are equal \(t = \dfrac{\left(\bar{x}_{1} - \bar{x}_{2}\right) - \left(\mu_{1} - \mu_{2}\right)}{\sqrt{ \left(\frac{\left(n_{1} - 1\right) s_{1}^{2} + \left(n_{2} - 1\right) s_{2}^{2}}{\left(n_{1} + n_{2} - 2\right)} \right) \left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right) }}\) \(df = n_{1} - n_{2} - 2\) |
Confidence Interval for Two Independent Means T-Interval: Assume variances are unequal \(\left(\bar{x}_{1} - \bar{x}_{2}\right) \pm t_{\alpha/2} \sqrt{\left(\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}\right)}\) TI-84: \(2\text{-SampTInt}\) \(df = \dfrac{\left( \frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}} \right)^2}{\left( \left(\frac{s_{1}^{2}}{n_{1}}\right)^{2} \left(\frac{1}{n_{1}-1}\right) + \left(\frac{s_{2}^{2}}{n_{2}}\right)^{2} \left(\frac{1}{n_{2}-1}\right) \right)}\) T-Interval: Assume variances are equal \(\left(\bar{x}_{1} - \bar{x}_{2}\right) \pm t_{\alpha/2} \sqrt{\left( \left(\frac{\left(n_{1} - 1\right) s_{1}^{2} + \left(n_{2} - 1\right) s_{2}^{2}}{\left(n_{1} - n_{2} - 2\right)}\right) \left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right) \right)}\) \(df = n_{1} - n_{2} - 2\) |
| Hypothesis Test for Two Proportions \(H_{0}: p_{1} = p_{2}\) \(H_{1}: p_{1} \neq p_{2}\) \(z = \dfrac{\left(\hat{p}_{1} - \hat{p}_{2}\right) - \left(p_{1} - p_{2}\right)}{\sqrt{ \left( \hat{p} \cdot \hat{q} \left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right) \right) }}\) \(\hat{p} = \frac{\left(x_{1} + x_{2}\right)}{\left(n_{1} + n_{2}\right)} = \frac{\left(\hat{p}_{1} \cdot n_{1} + \hat{p}_{2} \cdot n_{2}\right)}{\left(n_{1} + n_{2}\right)}\) \(\hat{q} = 1 - \hat{p}\) \(\hat{p}_{1} = \frac{x_{1}}{n_{1}}, \quad\quad \hat{p}_{2} = \frac{x_{2}}{n_{2}}\) TI-84: \(2\text{-PropZTest}\) |
Confidence Interval for Two Proportions \(\left(\hat{p}_{1} - \hat{p}_{2}\right) \pm z_{\alpha/2} \sqrt{\left( \frac{\hat{p}_{1} \hat{q}_{1}}{n_{1}} + \frac{\hat{p}_{2} \hat{q}_{2}}{n_{2}} \right)}\) \(\hat{p}_{1} = \frac{x_{1}}{n_{1}} \quad\quad\quad\. \hat{p}_{2} = \frac{x_{2}}{n_{2}}\) \(\hat{q}_{1} = 1 - \hat{p}_{1} \quad\quad \hat{q}_{2} = 1 - \hat{p}_{2}\) TI-84: \(2\text{-PropZInt}\) |
| Hypothesis Test for Two Variances \(H_{0}: \sigma_{1}^{2} = \sigma_{2}^{2}\) \(H_{1}: \sigma_{1}^{2} \neq \sigma_{2}^{2}\) \(F = \frac{s_{1}^{2}}{s_{2}^{2}}\) \(df_{\text{N}} = n_{1} - 1, \quad\quad df_{\text{D}} = n_{2} - 1\) TI-84: \(2\text{-SampFTest}}\) |
Hypothesis Test for Two Standard Deviations \(H_{0}: \sigma_{1} = \sigma_{2}\) \(H_{1}: \sigma_{1} \neq \sigma_{2}\) \(F = \frac{s_{1}^{2}}{s_{2}^{2}}\) \(df_{\text{N}} = n_{1} - 1, \quad\quad df_{\text{D}} = n_{2} - 1\) TI-84: \(2\text{-SampFTest}}\) |
| F-Critical Values Excel: Two-tail: \(F_{\alpha/2} = \text{F.INV}(1 - \alpha/2, 0, 1)\) Right-tail: \(F_{1-\alpha} = \text{F.INV}(1 - \alpha, 0, 1)\) Left-tail: \(F_{\alpha} = \text{F.INV}(\alpha, 0, 1)\) |
For z and t-Critical Values refer back to Chapter 8 TI-84: invF program can be downloaded at http://www.MostlyHarmlessStatistics.com. |
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Chapter 10 Formulas
| Goodness of Fit Test \(H_{0}: p_{1} = p_{0}, p_{2} = p_{0}, \ldots, p_{k} = p_{0}\) \(H_{1}:\) At least one proportion is different. \(\chi^{2} = \sum \frac{(O-E)^{2}}{E}\) \(df = k-1, p_{0} = 1/k \text{ or given %}\) TI-84: \(\chi^{2} \text{ GOF-Test}\) |
Test for Independence \(H_{0}:\) Variable 1 and Variable 2 are independent. \(H_{1}:\) Variable 1 and Variable 2 are dependent. \(\chi^{2} = \sum \frac{(O-E)^{2}}{E}\) \(df = (R-1)(C-1)\) TI-84: \(\chi^{2} \text{-Test}\) |
Chapter 11 Formulas
| One-Way ANOVA: \(H_{0}: \mu_{1} = \mu_{2} = \mu_{3} = \ldots = \mu_{k} \quad\quad k = \text{number of groups}\) \(H_{1}:\) At least one mean is different. .png?revision=1) \(\bar{x}_{i}\) = sample mean from the \(i^{th}\) group \(n_{i}\) = sample size of the \(i^{th}\) group \(s_{i}^{2}\) = sample variance from the \(i^{th}\) group \(N = n_{1} + n_{2} + \cdots + n_{k}\) \(\bar{x}_{GM} = \frac{\sum x_{i}}{N}\) |
| Bonferroni test statistic: \(t = \dfrac{\bar{x}_{i} - \bar{x}_{j}}{\sqrt{\left( MSW \left(\frac{1}{n_{i}} + \frac{1}{n_{j}}\right) \right)}}\) \(H_{0}: \mu_{i} = \mu_{j}\) \(H_{1}: \mu_{i} \neq \mu_{j}\) Multiply p-value by \(m = {}_{k} C_{2}\), divide area for critical value by \(m = {}_{k} C_{2}\) |
| Two-Way ANOVA: Row Effect (Factor A): \(H_{0}:\) The row variable has no effect on the average ______________. \(H_{1}:\) The row variable has an effect on the average ______________. Column Effect (Factor B): \(H_{0}:\) The column variable has no effect on the average ______________. \(H_{1}:\) The column variable has an effect on the average ______________. Interaction Effect (A \(\times\) B\): \(H_{0}:\) There is no interaction effect between row variable and column variable on the average ______________. \(H_{1}:\) There is an interaction effect between row variable and column variable on the average ______________. .png?revision=1) |
Chapter 12 Formulas
| \(SS_{xx} = (n-1) s_{x}^{2}\) \(SS_{yy} = (n-1) s_{y}^{2}\) \(SS_{xy} = \sum (xy) - n \cdot \bar{x} \cdot \bar{y}\) |
Correlation Coefficient \(r = \frac{SS_{xy}}{\sqrt{\left( SS_{xx} \cdot SS_{yy} \right)}}\) |
| Slope = \(b_{1} = \frac{SS_{xy}}{SS_{xx}}\) y-intercept = \(b_{0} = \bar{y} - b_{1} \bar{x}\) Regression Equation (Line of Best Fit): \(\hat{y} = b_{0} + b_{1} x\) |
Correlation t-test \(H_{0}: \rho = 0; \ H_{1}: \rho \neq 0 \quad\quad\quad t = r \sqrt{\left(\frac{n-2}{1-r^{2}}\right)} \quad df = n-2\) Slope t-test \(H_{0}: \beta_{1} = 0; \ H_{1}: \beta_{1} \neq 0 \quad\quad\quad t = \frac{b_{1}}{\sqrt{\left( \frac{MSE}{SS_{xx}} \right)}} \quad df = n - p - 1 = n-2\) |
| Residual \(e_{i} = y_{i} - \hat{y}_{i}\) (Residual plots should have no patterns.) Standard Error of Estimate \(s_{est} = \sqrt{\frac{\sum \left(y_{i} - \hat{y}_{i}\right)^{2}}{n - 2}} = \sqrt{MSE}\) Prediction Interval \(\hat{y} = t_{\alpha/2} \cdot s_{est} \sqrt{\left(1 + \frac{1}{n} + \frac{\left(x - \bar{x}\right)^{2}}{SS_{xx}}\right)}\) |
Slope/Model F-test \(H_{0}: \beta_{1} = 0; \ H_{1}: \beta_{1} \neq 0\) .png?revision=1) |
| Multiple Linear Regression Equation \(\hat{y} = b_{0} + b_{1} x_{1} + b_{2} x_{2} + \cdots + b_{p} x_{p}\) |
Coefficient of Determination \(R^{2} = (r)^{2} = \frac{SSR}{SST}\) |
| Model F-Test for Multiple Regression \(H_{0}: \beta_{1} = \beta_{2} = \cdots \beta_{p} = 0\) \(H_{1}:\) At least one slope is not zero. |
Adjusted Coefficient of Determination \(R_{adj}^{2} = 1 - \left(\frac{\left(1 - R^{2}\right) (n-1)}{(n - p - 1)}\right)\) |
Chapter 13 Formulas
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Ranking Data
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Sign Test \(H_{0}:\) Median \(= MD_{0}\)
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| Wilcoxon Signed-Rank Test \(n\) is the sample size not including a difference of 0. When \(n < 30\), use test statistic \(w_{s}\), which is the absolute value of the smaller of the sum of ranks. CV uses table below. If critical value is not in table then use an online calculator: http://www.socscistatistics.com/tests/signedranks When \(n \geq 30\), use z-test statistic: \(z = \frac{\left(w_{s} - \left(\frac{n (n+1)}{4}\right) \right)}{\sqrt{\left( \frac{n(n+1)(2n+1)}{24} \right)}}\) |
Mann-Whitney U Test When \(n_{1} \leq 20\) and \(n_{2} \leq 20\) \(U_{1} = R_{1} - \frac{n_{1} \left(n_{1}+1\right)}{2}, \ U_{2} = R_{2} - \frac{n_{2} \left(n_{2}+1\right)}{2}\). \(U = \text{Min} \left(U_{1}, U_{2}\right)\) CV uses tables below. If critical value is not in tables then use an online calculator: https://www.socscistatistics.com/tests/mannwhitney/default.aspx When \(n_{1} > 20\) and \(n_{2} > 20\), use z-test statistic: \(z = \frac{\left( U - \left(\frac{n_{1} \cdot n_{2}}{2}\right) \right)}{\sqrt{\left( \frac{n_{1} \cdot n_{2} \left(n_{1} + n_{2} + 1\right)}{12} \right)}}\) |
Wilcoxon Signed-Rank Critical Values
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Mann-Whitney U Critical Values
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