This document outlines the objectives for Exam 2, covering the essential concepts from chapters 7 through 11.
Understanding Parameters and Statistics: Accurately define what constitutes a parameter in the context of a population and a statistic in the context of a sample.
Identifying and Appreciating Sampling Distributions:
Application of the Central Limit Theorem (CLT):
Understanding Observational Studies vs. Experiments:
Identifying Components of Experiments: Accurately identify the experimental units, explanatory variables, treatments or factors, levels, and response variables in various research scenarios.
Understanding Experimental Design Graphs:
Evaluating Experimental Designs: Recognize the critical factors in designing an experiment, including control, the principle of control, randomization, and replication, and assess whether an experimental design can be considered good” and the utility of single blind, and double blind experiments and when blinding is not possible.
Recognizing Sampling Methods: Classify a study’s sampling method and why one method may be preffered given context.
Identifying Sampling Issues: Identify common issues in sampling such as bias, convenience sampling, self-selection, undercoverage, and nonresponse, and understand their impacts on study results.
The table below summarizes the formulas for one-sample z, one-sample t, two-sample t (independent), and two-sample t (matched pair) tests.
\[ \begin{array}{|c|c|c|c|} \hline \textbf{One-Sample Z} & \textbf{One-Sample t} & \textbf{2-Sample t (Independent)} & \textbf{2-Sample t (Matched Pair)} \\ \hline \bar{x} \pm z_{\frac{\alpha}{2}} \frac{\sigma}{\sqrt{n}} & \bar{x} \pm t_{\frac{\alpha}{2},n-1} \frac{s}{\sqrt{n}} & \bar{x}_1-\bar{x}_2 \pm t_{\frac{\alpha}{2},\nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} & \bar{d} \pm t_{\frac{\alpha}{2},n-1} \frac{s_d}{\sqrt{n}} \\ \hline \end{array} \]
\[ \begin{array}{|c|c|} \hline \textbf{One-Sample Z} & \textbf{One-Sample t} \\ \hline n = \left( \frac{\sigma z_{\alpha/2} }{ME} \right)^2 & n = \left( \frac{s' t_{\alpha/2, n'-1}}{ME} \right)^2\\ \hline \end{array} \]
\[ \begin{array}{|c|c|c|} \hline \textbf{Test Type} & \textbf{Lower Bound} & \textbf{Upper Bound} \\ \hline \textbf{One-Sample Z} & \mu > \bar{x} - z_{\alpha} \frac{\sigma}{\sqrt{n}} & \mu < \bar{x} + z_{\alpha} \frac{\sigma}{\sqrt{n}} \\ \textbf{One-Sample t} & \mu > \bar{x} - t_{\alpha, n-1} \frac{s}{\sqrt{n}} & \mu < \bar{x} + t_{\alpha, n-1} \frac{s}{\sqrt{n}} \\ \textbf{2-Sample t (Independent)} & \mu > \bar{x}_1 - \bar{x}_2 - t_{\alpha, \nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} & \mu < \bar{x}_1 - \bar{x}_2 + t_{\alpha, \nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \\ \textbf{2-Sample t (Matched Pair)} & \mu > \bar{d} - t_{\alpha, n-1} \frac{s_d}{\sqrt{n}} & \mu < \bar{d} + t_{\alpha, n-1} \frac{s_d}{\sqrt{n}} \\ \hline \end{array} \]
\[ S_p^2=\left[\frac{n_{\text{A}}-1}{n_{\text{A}}+n_{\text{B}}-2}\right] S_{\text{A}}^2 + \left[\frac{n_{\text{B}}-1}{n_{\text{A}}+n_{\text{B}}-2}\right] S_{\text{B}}^2 \]
\[ \textbf{df}=n_{\text{A}}+n_{\text{B}}-2 \]
\[ \textbf{df} = \dfrac{\left(\dfrac{\sigma_{\text{A}}^2}{n_{\text{A}}} + \dfrac{\sigma_{\text{B}}^2}{n_{\text{B}}}\right)^2}{\dfrac{1}{n_{\text{A}} - 1}\left(\dfrac{\sigma_{\text{A}}^2}{n_{\text{A}}}\right)^2 + \dfrac{1}{n_{\text{B}} - 1}\left(\dfrac{\sigma_{\text{B}}^2}{n_{\text{B}}}\right)^2} \]
\[ \nu = \dfrac{\left(\dfrac{s_{\text{A}}^2}{n_{\text{A}}} + \dfrac{s_{\text{B}}^2}{n_{\text{B}}}\right)^2}{\dfrac{1}{n_{\text{A}} - 1}\left(\dfrac{s_{\text{A}}^2}{n_{\text{A}}}\right)^2 + \dfrac{1}{n_{\text{B}} - 1}\left(\dfrac{s_{\text{B}}^2}{n_{\text{B}}}\right)^2} \]
Type I and Type II Errors, and Power:
Sample Size, Type II Error, and Power: Understand the direct relationship between sample size (\(n\)) and its impact on Type II error and power, including how to calculate \(n\) for a specified level of power (\(1−\beta\)) or Type II error (\(\beta\)) for a given alternative mean \(\mu_a\), when \(\alpha\) and \(\beta\) or \(1-\beta\) are provided.
Test Statistic and P-value:
\[ \begin{array}{|c|c|c|} \hline \textbf{Test Type} & \textbf{Test Statistic Formula} & \textbf{Degrees of Freedom} \\ \hline \textbf{One-Sample Z} & z_{ts} = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} & N/A \\ \textbf{One-Sample t} & t_{ts} = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} & n - 1 \\ \textbf{2-Sample t (Independent)} & t_{ts}' = \frac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} & \nu \\ \textbf{2-Sample t (Matched Pair)} & \frac{\bar{d} - \Delta_0}{s_D / \sqrt{n}} & n - 1 \\ \hline \end{array} \]
\[ \begin{array}{|c|c|c|c|c|c|} \hline \textbf{Hypothesis Test} & \textbf{Null Hypothesis} & \textbf{Alternative Hypothesis} &\textbf{Confidence Interval or Bound}& p\textbf{-value } (\textbf{z-code}) & p\textbf{-value } (\textbf{t-code}) \\ \hline \textbf{Upper Tailed} & H_0: \mu \leq \mu_0 & H_a: \mu > \mu_0 & \text{Lower Bound} & P(Z>z_{\text{ts}}): \textbf{pnorm}(z_{\text{ts}},\text{lower.tail} = \textbf{FALSE}) & P(T>t_{\text{ts}}):\textbf{pt}(t_{\text{ts}}, \textbf{df} = n-1, \text{lower.tail} = \textbf{FALSE})\\ \textbf{Lower Tailed} & H_0: \mu \geq \mu_0 & H_a: \mu < \mu_0 & \text{Upper Bound} & P(Z<z_{\text{ts}}): \textbf{pnorm}(z_{\text{ts}},\text{lower.tail} = \textbf{TRUE}) & P(T<t_{\text{ts}}):\textbf{pt}(t_{\text{ts}}, \textbf{df} = n-1, \text{lower.tail} = \textbf{TRUE})\\ \hline \textbf{Two-Tailed} & H_0: \mu = \mu_0 & H_a: \mu \neq \mu_0 & \text{Interval} & 2P(Z>|z_{\text{ts}}|): 2\textbf{pnorm}(\text{abs}(z_{\text{ts}}),\text{lower.tail} = \textbf{FALSE}) & 2P(T>|t_{\text{ts}}|): 2\textbf{pt}(\text{abs}(t_{\text{ts}}), \textbf{df} = n-1,\text{lower.tail} = \textbf{FALSE})\\ \hline \end{array} \]