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Mostly Harmless Statistics Formula Packet

Last updated

Mar 12, 2023
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Chapter 3 Formulas

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.

Screenshots of a TI-84 calculator to enter data in a list, select 1-Var Stats, select the list, and calculate statistics for the list.

$s_{x}$ 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: ${}_{n} P_{r} = \frac{n!}{(n-r)!}$

All 52 playing card values from a standard pack.

Table of sums of the rolls of two 6-sided dice. Logic tree for determining whether to apply the Fundamental Counting Rule, factorials, permutations, or combinations to a situation.

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? Geometric – A percent or proportion is given. There is no set sample size until a success is achieved. Binomial – A percent or proportion is given. A sample size is given. Hypergeometric – Usually frequencies of successes are given instead of percentages. A sample size is given. Poisson – An average or mean is given. There is no set sample size until a success is achieved.	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
		x)\)">
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 $H_{0}: p < p_{0}$

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.

Flowchart for deciding which type of test to use, based on what information is given: proportions or means, the number of samples, etc.

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.
One-way ANOVA table showing the formulas for sum of squares, degrees of freedom, mean squares, and F-values for factor and error.

One-way ANOVA table showing the formulas for sum of squares, degrees of freedom, mean squares, and F-values for factor and error.

$\bar{x}_{i}$ = 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 ______________.

Two-way ANOVA table showing equations for SS, df, MS, and F-values for the row factor, column factor, interaction, and error.

Two-way ANOVA table showing equations for SS, df, MS, and F-values for the row factor, column factor, interaction, and error.

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 $H_{0}: \beta_{1} = 0; \ H_{1}: \beta_{1} \neq 0$
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

Ranking Data

Order the data from smallest to largest.
The smallest value gets a rank of 1.
The next smallest gets a rank of 2, etc.
If there are any values that tie, then each of the tied values gets the average of the corresponding ranks.

Sign Test

$H_{0}:$ Median $= MD_{0}$
$H_{1}:$ Median $\neq MD_{0}$
p-value uses binomial distribution with $p = 0.5$ and $n$ is the sample size not including ties with the median or differences of 0.

For a 2-tailed test, the test statistic, $x$ , is the smaller of the plus or minus signs. If $x$ is the test statistic, the p-value for a two-tailed test is $2 \cdot \text{P}(X \leq x)$ .
For a right-tailed test, the test statistic, $x$ , is the number of plus signs. For a left-tailed test, the test statistic, $x$ , is the number of minus signs. The p-value for a one-tailed test is $\text{P}(X \geq x)$ or $\text{P}(X \leq x)$ .

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

Table of Wilcoxon signed-rank critical values for both 1-tailed and 2-tailed tests, with alpha values of 0.01, 0.05, and 0.10.

Mann-Whitney U Critical Values

Table of critical values for 2-tailed Mann-Whitney U Test for alpha = 0.05.

Table of critical values for 2-tailed Mann-Whitney U Test for alpha = 0.01.

Chapter 3 Formulas

Chapter 4 Formulas

Chapter 5 Formulas

Chapter 6 Formulas

Chapter 7 Formulas

Chapter 8 Formulas

Chapter 9 Formulas

Chapter 10 Formulas

Chapter 11 Formulas

Chapter 12 Formulas

Chapter 13 Formulas

Wilcoxon Signed-Rank Critical Values

Mann-Whitney U Critical Values

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