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Chapter 3 – Descriptive Statistics
Mean
\[ \bar{x} = \frac{\sum x}{n} \]
Mean (Frequency Table)
\[ \bar{x} = \frac{\sum (f \cdot x)}{\sum f} \]
Sample Standard Deviation
\[ s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}} \]
Shortcut Standard Deviation
\[ s = \sqrt{\frac{n\sum x^2-(\sum x)^2}{n(n-1)}} \]
Standard Deviation (frequency table)
\[ s = \sqrt{\frac{n[\sum (f \cdot x^2)] - [\sum (f \cdot x)]^2}{n(n - 1)}} \]
Variance
\[ s^2 \]
Chapter 4 – Probability
Mutually Exclusive events
\[ P(A \text{ or } B)=P(A)+P(B) \]
Not Mutually Exclusive events
\[ P(A \text{ or } B)=P(A)+P(B) - P(A \text{ and } B) \]
Independent Events
\[ P(A \text{ and } B) = P(A) \cdot P(B) \]
Dependent Events
\[ P(A \text{ and } B) = P(A) \cdot P(B | A) \]
Rule of Complements
\[ P(\overline{A}) = 1 - P(A) \]
Permutations (no elements alike)
\[ _nP_r=\frac{n!}{(n-r)!} \]
Permutations (\(n_1\) alike, ...)
\[ \frac{n!}{n_1! \; n_2! \; \cdots \; n_k!} \]
Combinations
\[ _nC_r=\frac{n!}{(n-r)!r!} \]
Chapter 5 – Probability Distributions
Mean (Probability Distribution)
\[ \mu = \sum [ x \cdot P(x) ] \]
Standard Deviation (Probability Distribution)
\[ \sigma = \sqrt{\sum [x^2 \cdot P(x)]-\mu^2} \]
Binomial Probability
\[ P(x)=\frac{n!}{(n-x)! \; x!} \cdot p^x \cdot q^{n-x} \]
Mean (binomial)
\[ \mu=n \cdot p \]
Variance (binomial)
\[ \sigma^2=n \cdot p \cdot q \]
Standard Deviation (binomial)
\[ \sigma = \sqrt{n \cdot p \cdot q} \]
Poisson Distribution
\[ P(x)=\frac{\mu^x \cdot e^{-\mu}}{x!} \;\; \text{ where } \;\; e = 2.71828 \]
Chapter 7 – Confidence Intervals (One Population)
Proportion
\[ \hat{p} - E < p < \hat{p} + E \]
Margin of Error (Proportion)
\[ E = z_{\alpha/2}\sqrt{\frac{\hat{p}\hat{q}}{n}} \]
Mean
\[ \bar{x} - E < \mu < \bar{x} + E \]
Margin of Error (\(\sigma\) unknown)
\[ E = t_{\alpha/2}\frac{s}{\sqrt{n}} \]
Margin of Error (\(\sigma\) known)
\[ E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \]
Variance
\[ \frac{(n - 1)s^2}{\chi_R^2} < \sigma^2 < \frac{(n - 1)s^2}{\chi_L^2} \]
Chapter 7 – Sample Size Determination
Proportion (no prior estimate so \( \hat{p} \) and \( \hat{q} \) are unknown)
\[ n = \frac{[z_{\alpha/2}]^2 \cdot 0.25}{E^2} \]
Proportion (with prior estimate so \( \hat{p} \) and \( \hat{q} \) are known)
\[ n = \frac{[z_{\alpha/2}]^2 \cdot \hat{p}\hat{q}}{E^2} \]
Mean
\[ n = \left[\frac{z_{\alpha/2}\sigma}{E}\right]^2 \]
Chapter 8 – Test Statistics (one population)
Proportion - one population
\[ z = \frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \]
Mean - one population (σ unknown)
\[ t = \frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}} \]
Mean - one population (σ known)
\[ z = \frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}} \]
Standard deviation or variance - one population
\[ \chi = \frac{(n - 1)s^2}{\sigma^2} \]