NotesSheetFinal.tex

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\begin{center}
STAT 4033
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Final Notes \hfill Name: \uline{Cooper Morris}
\begin{multicols}{2}
\textbf{\uline{C1S3:}}\\
\textbf{Histogram:} Number of classes should be smallest whole number K that makes $2^K \geq$ number of measurements. For large data sets either $\log_2(n)$ or $2n^{1/3}$\\
\textbf{Unimodal:} One major peak\\
\textbf{Bimodal:} Two major peaks\\
\textbf{Symmetric:} Symmetric\\
\textbf{Right Skewed:} Long right tail, short left tail\\
\textbf{Boxplots:} Outliers are outside 1.5$\times$IQR. Box goes from $Q_1$ to $Q_3$, horizontal line at median, whiskers to largest data point inside 1.5$\times$IQR, X's for outliers\\
\textbf{Five Number Summary:} Min, Max, Median, Q1, and Q3\\
\textbf{\uline{C2S6:}}\\
\textbf{Jointly Distributed Random Variable:} Two or more random variables that are related when considering ``individuals" in a population.\\
\textbf{Joint Probability Mass Function:} \\
\(P(X=x, Y=y) = p(x,y)\) \(P(X=x \cap Y=y)\) \\
\textbf{Marginal Probability Mass Function:} \(P_x(x) = \sum_y p(x,y)\)\\
\(f_x(x) = \int_{-\infty}^\infty f(x,y)\,dy\)\\
Summing or integrating out the opposite variable gives you the marginal probability mass function for the variable you desire.\\
\textbf{\uline{C6S1: Large Sample Tests for Population Mean}}\\
\[z_{test} = \frac{\bar{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}\]
\textbf{Alternative Hypothesis:} The claim about the population that we are trying to find evidence for.\\
\textbf{Null Hypothesis:} What is assumed to be true.\\
Reject the null hypothesis if P-Value is less than \(\alpha\)\\
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\textbf{\uline{C6S3: Tests for a Population Proportion}}\\
\[z_{test}=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\cdot(1-p_0)}{n}}}\]
\textbf{\uline{C6S4: Small Sample Tests for \(\mu\)}}\\
\[t_{test}=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}\]
Use Table 2\\
\textbf{\uline{C6S7: Small Sample Tests for Difference Between Two Means}}\\
If \(\sigma_1 = \sigma_2\)\\
\[s_p^2 = \frac{(n_1-1)s_1^2+(n_2-1)s^2_2}{n_1+n_2+2}\]\\
\[t_{test} = \frac{(\bar{x_1}-\bar{x_2})-D_0}{s_p\cdot\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\]\\
If \(\sigma_1 \neq \sigma_2\)\\
\[t_{test} = \frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]\\
\[\nu = \frac{(\frac{s^2_1}{n^1}+\frac{s^2_2}{n_2})^2}{\frac{(\frac{s^2_1}{n_1})^2}{n_1-1}+\frac{(\frac{s^2_2}{n_2})^2}{n_2-1}}\]\\
\textbf{\uline{C6S8: Tests with Paired Data}}\\
\[t_{test} = \frac{\bar{d}-D_0}{\frac{s_d}{\sqrt{n}}}\]
Where \(\bar{d}\) is the difference between sample means and \(s_d\) is the difference between standard deviations.

\end{multicols}
\begin{tabular}{c|c|c}
                                    		& \multicolumn{2}{c}{\textbf{Null Hypothesis}}\\
Our Decision & True & False\\ \hline
Reject H\textsubscript{0} & \textbf{X} Type 1 error (false positive) & \checkmark \\ \hline
Fail to reject H\textsubscript{0} & \checkmark & \textbf{X} Type 2 error (false negative)
\end{tabular}
\newpage
\begin{multicols}{2}
\textbf{\uline{C7S1: Linear Correlation}}\\
\textbf{Correlation Coefficient:} The direction and strength of a linear relationship.
\[r=\frac{1}{n-1}\sum_{i=1}^n \biggr(\frac{x_i-\bar{x}}{s_x}\biggr)\biggr(\frac{y_i-\bar{y}}{s_y}\biggr)\]\\
\(-1\leq r \leq 1\)\\
\textbf{\uline{C7S2: Least Squares Line}}\\
\(y=\beta_0+\beta_1x+\epsilon\)\\
\[b_1=\frac{SS_{xy}}{SS_{xy}}\]\\
\[b_0=\bar{y}-b_1\bar{x}\]\\
Sum of Squared Error (SSE):\\
\[SSE = \sum_{i=1}^n(y_i-\hat{y_i})^2\]\\
Estimate Standard Deviation:\\
\(\hat{\sigma_\epsilon = s = \sqrt{\frac{SSE}{n-2}}}\)\\
\textbf{\uline{C7S3: Uncertainties in the Least-Squares Coefficients}}\\
\[s_{b0} = s\cdot\sqrt{\frac{1}{n}+\frac{\bar{x}^2}{SS_{xx}}}\]
\[s_{b1} = \frac{s}{\sqrt{SS_{xx}}}\]
\[b_i \pm t_{\alpha/2, n-2} \cdot s_{bi}\]
\[t_{test} = \frac{b_i}{s_{bi}}\]
S is residual standard error if given R output.\\
Use a t-distribution with n-2 degrees of freedom.\\
Confidence Intervals for the Mean Value of y:
\[D = \frac{1}{n}+\frac{(x-\bar{x})^2}{SS_{xx}}\]
\[s_{\hat{y}} = s\sqrt{\frac{1}{n}+\frac{(x-\bar{x})^2}{SS_{xx}}} = s\cdot\sqrt{D}\]
Confidence for mean value of y at given value x:
\[\hat{y} \pm t_{\alpha/2, \nu} \cdot s_{\hat{y}}\]
\(\nu = n-2\)\\
Prediction for mean values of y:
\[s_{pred} = s\cdot\sqrt{1+D}\]
\[\hat{y} \pm t_{\alpha/2, \nu} \cdot s_{pred}\]
\[SSTotal = \sum_{i=1}^n(y_i-\bar{y_i})^2\]\\
\[SSR = \sum_{i=1}^n(\hat{y_i}-\bar{y})^2\]\\
\[SSE = \sum_{i=1}^n(y_i-\hat{y_i})^2\]\\
SSTotal = SSR + SSE
\[r^2 = \frac{SS^2_{xy}}{SS_{xx}\cdot SS_{yy}} = \frac{SSR}{SSTotal}\]
\(r = \sqrt{r^2}\) when \(b_1 > 0\) or \(r = -\sqrt{r^2}\) when \(b_1 \leq 0\)\\
\textbf{\uline{C7S4: Checking Assumptions and Transforming Data}}\\
Residual Plots:\\
Assumption 1: At any given value of x, the mean of potential errors is 0. Even number above and below zero line and don't follow a trend.\\
Assumption 2: The vaiance of potential errors is always the same, no matter what the value of x is. Spread neither increasing nor decreasing.\\
Assumption 3: At any given value of x, the distribution of potential errors is normal. Dont see deviance from line on a Normal Q-Q plot.\\
\(p(i) = P(Z\leq z_{(i)} = \frac{3i-1}{3n-1}\)\\
Assumption 4: The error terms are independent of each other. Shouldn't see a trend in residuals with time.\\
If assumptions are invalid we can transform data (square root, ln, power, etc.)
\end{multicols}
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