Statistics I: Comprehensive Formula Sheet and Study Guide

Descriptive Statistics

Sample Mean and Variance

Descriptive statistics summarize and describe the main features of a dataset.

• Sample Mean (\( \bar{x} \)): The arithmetic average of a set of values.

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

• Sample Variance (\( s^2 \)): Measures the spread of data points around the mean.

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$

• Sample Standard Deviation (\( s \)): The square root of the variance.

$s = \sqrt{s^2}$

• Quartiles: $Q_1$, $Q_3$; Percentiles: $P_p$

Interquartile Range (IQR): $IQR = Q_3 - Q_1$

Correlation and Regression

Measuring Relationships Between Variables

• Sample Correlation Coefficient (\( r \)): Measures the strength and direction of a linear relationship between two variables.

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

• Regression Equation: Predicts the value of a dependent variable based on the independent variable.

$\hat{y} = a + bx$

• Coefficient of Determination (\( r^2 \)): Proportion of variance in the dependent variable explained by the independent variable.

$r^2$

Probability

Basic Probability Rules

• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Multiplication Rule (for independent events): $P(A \cap B) = P(A)P(B)$

• Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Random Variables and Distributions

Discrete and Continuous Random Variables

• Expected Value (Mean): $E(X) = \sum x_i P(x_i)$

• Variance: $Var(X) = \sum (x_i - \mu)^2 P(x_i)$

Binomial Distribution

• Probability of $x$ successes in $n$ trials: $P(X = x) = {n \choose x} p^x (1-p)^{n-x}$

• Mean: $\mu = np$

• Variance: $\sigma^2 = np(1-p)$

Normal Distribution

• Standardization (Z-score): $Z = \frac{X - \mu}{\sigma}$

Sampling Distributions

Sample Mean and Proportion

• Sampling Distribution of the Mean: $\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$ for large $n$ (Central Limit Theorem)

• Sampling Distribution of the Proportion: $\hat{p} \sim N(p, \sqrt{\frac{p(1-p)}{n}})$ for large $n$

Statistical Inference: Confidence Intervals

Estimating Population Parameters

• Confidence Interval for Mean (\( \sigma \) known):

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Confidence Interval for Mean (\( \sigma \) unknown):

$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• Confidence Interval for Proportion:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

Statistical Inference: Significance Tests

Hypothesis Testing

• Null Hypothesis (\( H_0 \)): The statement being tested, usually a statement of no effect or no difference.

• Alternative Hypothesis (\( H_a \)): The statement we are trying to find evidence for.

• Test Statistic (Z or t): Measures how far the sample statistic is from the null hypothesis value, in standard error units.

$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• P-value: Probability of observing a test statistic as extreme as, or more extreme than, the observed value under $H_0$.

• Decision Rule: If $p$-value $< \alpha$, reject $H_0$.

Comparing Two Groups

Two-Sample Procedures

• Difference in Means (Independent Samples):

$\bar{x}_1 - \bar{x}_2 \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Difference in Proportions:

$\hat{p}_1 - \hat{p}_2 \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Tables: Hypothesis Testing Summary

Test Statistics and Critical Values

Test	Statistic	Critical Value	Decision Rule
Z-test (mean, known variance)	$Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$	$z_{\alpha}$	Reject $H_0$ if $|Z| > z_{\alpha}$
t-test (mean, unknown variance)	$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$	$t_{\alpha, df}$	Reject $H_0$ if $|t| > t_{\alpha, df}$
Z-test (proportion)	$Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$	$z_{\alpha}$	Reject $H_0$ if $|Z| > z_{\alpha}$

Additional info:

• This formula sheet covers key concepts from introductory statistics, including descriptive statistics, probability, sampling distributions, confidence intervals, hypothesis testing, and comparing two groups.

• Students should refer to their course materials for detailed examples and interpretations of results.