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Null Hypothesis Significance Testing

• A procedure that starts with stating the null hypothesis (e.g., $H_0: \mu = 0$).
• Selects a test statistic, like $\bar{X}$.
• Derives the distribution of the test statistic under $H_0$, such as $\bar{X} \sim N(0, \sigma^2/n)$.
• Computes the observed value of the test statistic, e.g., $\bar{x} = 5$.
• The p-value is calculated as the probability of observing a test statistic as extreme as, or more extreme than, the one observed, e.g., $P(|\bar{X}| \geq 5)$.
• Rejects $H_0$ if the p-value is less than the significance level $\alpha$.

Significance Level

• Defined as the probability of rejecting a true null hypothesis.
• A common choice is $\alpha = 0.05$.
• The choice of $\alpha$ depends on context; use a small $\alpha$ for high false positive costs and a large $\alpha$ for high false negative costs.

Type I and Type II Errors

• Type I Error: Rejecting $H_0$ when $H_0$ is actually true.
• Type II Error: Failing to reject $H_0$ when $H_0$ is actually false.
• $P(\text{Type I Error}) = \alpha$.
• $P(\text{Type II Error}) = \beta$.
• Power = $1 - \beta = P(\text{Reject } H_0 \mid H_0 \text{ is false})$.

Examples of t-tests

• One-sided t-test: $H_0: \mu = 0$, $H_A: \mu > 0$, test statistic $\bar{X}$, $\bar{x} = 5$, $p = P(\bar{X} > 5)$.
• Two-sided t-test: $H_0: \mu = 0$, $H_A: \mu \neq 0$, test statistic $|\bar{X}|$, $\bar{x} = 5$, $p = P(|\bar{X}| > 5)$.
• P-value for a two-sided test is twice that of a one-sided test, assuming the test statistic's sign aligns with the alternative hypothesis's direction.

t-tests in Regression with One Predictor

• Model: $Y_i = \beta_0 + \beta_1 X_i + \epsilon_i$.
• Hypotheses: $H_0: \beta_1 = 0$, $H_A: \beta_1 \neq 0$.
• Test statistic: $t = \frac{\hat{\beta_1} - 0}{SE(\hat{\beta_1})}$.
• Degrees of freedom: $n - 2$.

Systèmes d'Équations Linéaires

Introduction

• Focuses on systems of linear equations.

Equation Linéaire

• Defined as $a_1x_1 + a_2x_2 +... + a_nx_n = b$, where $a_i$ and $b$ are constants.

Système d'Équations Linéaires

• Defined as a set of $m$ linear equations with $n$ unknowns.

Solution d'un Système

• A set of values for $x_1, x_2,..., x_n$ satisfying all equations simultaneously.
• Résoudre un système: finding all solutions.

Interprétation Géométrique

• One Unknown: Point on a real line.
• Two Unknowns: Line in a plane.
• Three Unknowns: Plane in space.
• $n > 3$: Hyperplane.

Types de Solutions

• Unique solution, infinite solutions, or no solution.

Systèmes Équivalents

• Have the same solutions.

Opérations Élémentaires

1. Exchange of two equations.
2. Multiplication of an equation by a nonzero constant.
3. Addition of a multiple of one equation to another.

Méthode d'Élimination de Gauss

• Systematic method to solve linear systems by transforming them into an echelon form.

Forme Échelonnée Requirements

1. The first nonzero coefficient (pivot) is to the right of the one above it.
2. Equations without nonzero coefficients are at the bottom.

Résolution d'un Système Échelonné

• Solved by backward substitution.

Matrices

Introduction

• A matrix consists of numbers arranged in rows and columns.
• A matrix with $m$ rows and $n$ columns is of size $m \times n$.

Opérations sur les Matrices

• Addition: Element-wise for matrices of the same size.
• Multiplication by a scalar: Multiply each element by the scalar.
• Multiplication of matrices: Defined if the number of columns of $A$ equals the number of rows of $B$.

Transposée d'une Matrice

• The transpose of a matrix $A$, denoted $A^T$, is obtained by swapping rows and columns.

Matrices Spéciales

• Matrice carrée: Same number of rows and columns.
• Matrice identité: Square matrix with 1s on the main diagonal, 0s elsewhere.
• Matrice nulle: All elements are 0.

Matrices et Systèmes d'Équations Linéaires

• Represented as $Ax = b$, where $A$ is the coefficient matrix, $x$ is the unknown vector, and $b$ is the constants vector.

Matrice Augmentée

• Is formed by adding the constants column to the coefficients matrix.

Opérations Élémentaires sur les Lignes

• Same as elementary operations on linear system equations.

Forme Échelonnée Réduite

1. Is in echelon form.
2. The pivot of each row is equal to 1.
3. Each column with a pivot has all other elements equal to 0.

Déterminants

Introduction

• A function associating a scalar to a square matrix.

Définition

• The determinant of a matrix $A$, denoted det$(A)$ or $|A|$, results in a scalar.

Propriétés des Déterminants

• $\det(A^T) = \det(A)$.
• If $A$ has a row or column of zeros, then $\det(A) = 0$.
• If $A$ has two identical rows or columns, then $\det(A) = 0$.

Calcul des Déterminants

• $2 \times 2$ : $\det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc$.
• $3 \times 3$ : Sarrus' Rule.
• $n \times n$: Laplace expansion (cofactors).

Applications des Déterminants

• Invertibility: A matrix $A$ is invertible if and only if $\det(A) \neq 0$.
• Resolution of linear systems: Cramer's Rule.

Espaces Vectoriels

Introduction

• An algebraic structure with addition and scalar multiplication.

Définition

• An espace vectoriel is a set $V$ with addition and scalar multiplication satisfying certain axioms.

Sous-Espaces Vectoriels

• A subset $W$ of $V$ that is also a vector space with the same opérations.

Combinaisons Linéaires

• A vecteur de la forme $a_1v_1 + a_2v_2 +... + a_nv_n$, où $a_1, a_2,..., a_n$ are scalars.

Indépendance Linéaire

• The vectores $v_1, v_2,..., v_n$ are linearly independent if the only combination giving the zero vector is when all coefficients are zero.

Base et Dimension

• A base d'un espace vectoriel $V$ is a set of linearly vectors spanning $V$.
• La dimension d'un espace vectoriel $V$ is the number vectors in a base of $V$.

Transformations Linéaires

Introduction

• Function preserving addition and scalar multiplication.

Définition

• A transformation linéaire is a function $T: V \rightarrow W$ between vector spaces $V$ and $W$ that satisfies:
  • $T(u + v) = T(u) + T(v)$
  • $T(cv) = cT(v)$

Noyau et Image

• Le noyau d'une transformation linéaire $T$ is the set of vectors in $V$ sent the zero vector in $W$.
• L'image d'une transformation linéaire $T$ is the set of vectors in $W$ that are the image of one vector in $V$.

Matrice Transformation Linéaire

• Represented by a matrix.

Valeurs Propres et Vecteurs Propres

• Eigenvector $v$ satisfies $Av = \lambda v$ for some scalar $\lambda$.

Diagonalization

• A matrix $A = PDP^{-1}$, where $D$ is a diagonal matrix.

The Shape of Distributions

Dotplots

• Simple graph where each data value is shown as a dot on a number line.

Describing Shape

• Focus on major peaks, clusters, and gaps.
  • Symmetric: Mirror images on both sides.
  • Skewed to the right: Right side (larger values) longer than the left.
  • Skewed to the left: Left side longer than the right.

Stemplots

• Provide a quick shape picture with numerical values.

How to Make a Stemplot

1. Separate observations into stem and leaf.
2. List stems vertically.
3. Write leaves to the right of stems, in increasing order.
4. Provide an explanation key.

Histograms

• Convenient display for large datasets.

How to Make a Histogram

1. Divide the data range into equal classes.
2. Calculate counts or percents for each class.
3. Label axes and draw bars with heights equaling frequencies.

Describing Distributions with Numbers

Mean

• Sum of observations divided by n: $\bar{x} = \frac{\sum x_i}{n}$

Median

• Midpoint value of the distribution.

How to Find the Median

1. Arrange observations in order.
2. Odd n: $M$ is the center observation.
3. Even n: $M$ is the mean of the two center observations.

Comparing Mean and Median

• Symmetric distributions: Mean and median are close.
• Skewed distributions: Mean is farther into the tail.
• Mean is sensitive to extreme observations while median is resistant.

Quartiles

How to Find Quartiles

1. Arrange observations and find the median $M$.
2. $Q_1$ is the median to the left of $M$.
3. $Q_3$ is the median to the right of $M$.

Five-Number Summary and Boxplots

• Minimum, $Q_1$, $M$, $Q_3$, Maximum.

How to Make a Boxplot

1. Draw box from $Q_1$ to $Q_3$.
2. Mark median in the box.
3. Extend whiskers to non-outliers.
4. Mark outliers with a special symbol.

Identifying Outliers

• Suspected outlier: more than $1.5 \times IQR$ above $Q_3$ or below $Q_1$.

Standard Deviation

• Typical distance of observations from the mean: $s_x = \sqrt{\frac{1}{n-1}\sum(x_i - \bar{x})^2}$
• Variance: $s_x^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$

Choosing Measures of Center and Spread

• Median and IQR: Best for skewed distributions or with outliers.
• Mean and standard deviation: Use for reasonably symmetric distributions without outliers.

Density Curves and Normal Distributions

Density Curves

• Curve that is on or above the horizontal axis and has an area of exactly 1.
• The area under the curve represents the proportion of observations within a range.

Measuring Center and Spread for Density Curves

• Median: Equal-areas point.
• Mean: Balance point.
• Symmetric curves have the same mean and median, while skewed curves have the mean pulled towards the tail.

Normal Distributions

• Described by mean $\mu$ and standard deviation $\sigma$.
• Normal Distribution Density Curve Formula $$N(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$$
• The 68-95-99.7 rule:
  • 68% of observations are within $1\sigma$ of $\mu$.
  • 95% of observations are within $2\sigma$ of $\mu$.
  • 99.7% of observations are within $3\sigma$ of $\mu$.

Standard Normal Distribution

• Mean 0 and standard deviation 1. Also written as $N(0, 1)$.
• z-score with formula, $z = \frac{x-\mu}{\sigma}$

Finding Normal Proportions

1. State the problem.
2. Plan with a picture of the distribution.
3. Do: standardize, then use Table A to find areas.
4. Conclude. Relate in context.

Finding a Value When Given a Proportion

1. State.
2. Plan . Use a picture and shading.
3. Do: Transform from a z-score with the indicated area back to the x-units scale.
   • $x = \mu + z\sigma$
4. Conclude.Write your conclusion in the context of the problem.

Assessing Normality

• Normal probability: straight lines align as approximately Normally distributed. Systematic departure = non-normal dist.

Introduction to Commutative Algebra

Rings and Ideals

• Ring terminology: definition of a ring, commutative ring, and a ring with unity.
• Ring examples: $\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}/n\mathbb{Z}, F[x]$.
• Ideals Prime ideal and maximal ideal definitions.
• Ideal examples: In $\mathbb{Z}$, ideals are of the form $n\mathbb{Z}$ for $n \geq 0$. The prime ideals are $(0)$ and $(p)$ where $p$ is a prime number. The maximal ideals are $(p)$ where $p$ is a prime number.
• How to find or state Quotient rings: For an ideal $I$ in ring $R$, ring denoted as $R/I$.
• Ring of Homomorphisms If $f: R \to S$ ring homomorphism, $\text{ker}(f)$ ideal of $R$ and $\text{im}(f)$ subring of $S$
• Isomorphism theorems list:
  • $R/ \text{ker}(f) \cong \text{im}(f)$
  • If $I \subseteq J$ are ideals of $R$, then $(R/I)/(J/I) \cong R/J$
  • If $I, J$ are ideals of $R$, then $(I + J)/I \cong J/(I \cap J)$
• Prime and Maximal ideals. If ideal $P$ of $R$ is prime IFF $R/P$ is an integral domain. If ideal $M$ of $R$ is maximal IFF $R/M$ is a field. Every maximal idea is also prime.
• Zero divisors, Nilpotent elements and Units.
• Examples $\mathbb{Z}/6\mathbb{Z}$ :$2$ and $3$ are zero divisors, $1$ and $5$ are units.
• Localization, if $S$ is multiplicative subset of $R$, ring is denoted $S^{-1}R$.
• Examples $R \setminus P$ where $P$ is a prime ideal, ${1, f, f^2,...}$ where $f \in R$.

Modules

• Module Definition over a ring.
• Modules, Vector spaces over a field, ideals of a ring and abelian groups (modules over $\mathbb{Z}$) examples.
• Submodules, quotient modules, and homomorphisms of modules
• Isomorphism theorems for modules
• Finitely generated modules terminology.
• Free modules terminology.
• Torsion element if non-zero $r \in R$ such that $rm = 0$, $m \in M $ is a torsion element.
• Tensor product of modules: If $M$ and $N$ are $R$-modules, then $M \otimes_R N$.
• Exact Sequence (sequence modules & homomorphisms $M' \xrightarrow{f} M \xrightarrow{g} M''$) occurs if $\text{im}(f) = \text{ker}(g)$.
• Projective, injective, and flat modules characteristics.

Noetherian Rings and Modules

• Rings and Modules Definitions. Ring $R$ is Noetherian if every ideal of $R$ is finitely generated. Module $M$ is Noetherian if every submodule of $M$ is finitely generated.
• Hilbert Basis Theorem: Noetherian then $R[x]$ is Noetherian. $R$
• If ($R$ Noetherian ring), then $R[x_1,..., x_n]$ is Noetherian ring/. If ($R$ Noetherian ring) then $R/I$ is a Noetherian ring..
• Artinian rings and modules terms.
• Definitions of Artinian ring if every descending chain of ideals of $R$ terminates. Module $M$ is Artinian if every descending chain of submodules of $M$ terminates.
• Every Artinian ring is Noetherian.

Integral Extensions

• Definitions of $x \in S$ integral over $R$ defined ,if a exists monic polynomial $f(x) \in R[x]$ is present such that $f(x) = 0$.. The ring, $S$ is integral over $R$ if every element of $S$ is integral over $R$.
• Example $\mathbb{Z}[\sqrt{2}]$ is integral over $\mathbb{Z}$.
• Integral closure (integers) set of all elements of $S$ that are integral over $R$) .
• Noether's Normalization Lemma.

Valuation Rings

• Integral domain $R$ has a fraction either of the below existing for 0 element $x$
  • $x \in R$
  • $x^{-1} \in R$
• Examples $\mathbb{Z}_{(p)} = {a/b \in \mathbb{Q} \mid p \nmid b}$.

Chemical Kinetics

Reaction Rate

• The speed at which reactants are converted into products.

Factors Affecting Reaction Rate

1. Concentration of Reactants.
2. Temperature.
3. Catalysts.
4. Surface Area.
5. Pressure.
6. Light (photochemical reactions).
7. Nature of Reactants.

Rate Law

• Represents connection b/w rate reaction vs. reactant levels/contcentrations.

$$Rate = k[A]^m[B]^n$$

• variables k (rate constant), [A] and [B] , exponents (m & n) -reaction orders.

Types of Rate Laws

1. Differential Rate Law- expresses as concentration fn.
2. Integrated Rate Law- expresses level as a function of time .

Reaction Order- reaction exponents total by the rate formula.

• Zero, frist and second order rxn

Rate Constant

• Proportionality constant, temperature dependent.

Arrhenius Equation

• captures temperature dependence $$k = Ae^{\frac{-E_a}{RT}}$$
  • variable = rate constant
  • A is the pre-exponential factor
  • activation energy is $E_a$
  • is a gas constant $R$
  • T is in Kelvin

Determining Activation Energy

$$ln(k) = ln(A) - \frac{E_a}{RT}$$

• equation graphing captures $E_a$ with linear slope

Reaction Mechanisms

• Step-by-step reaction, overall chemical change occurs.

Elementary Step

• one-step element reaction rate.

Rate-Determining Step

• Step rate slowest, determines overall

Intermediates

• Made in-btw, in 1 rx and consumed by the end, do not show in balancer equation.

Molecularity

• Number of molecules involved in a rxn

Catalysis

Definition

• speeds up rxn without consumed.

Types of Catalysts

1. Homogeneous in syn with the reactants.
2. Heterogeneous different phase with the reactants.
3. Enzymes = biological catalysts.

How Catalysts Work

• new pathway, lower activation energy = speed up rxn.

Catalysis in Industry (widespread)

• Haber-Bosch, catalytic converters; Polymerization Reactions.

Summary

Chemical Reaction Engineering - Resistance to Transfer

Chemical Reactions Involving Solids

$A(fluid) + B(solid) \rightleftharpoons C(fluid) + D(solid)$

• Overall rate factors:*

1. Fluid resistance.
2. Solid resistance.
3. Surface reaction rate.
4. Equilibrium.

The Rate Equation

• Simplest case: $A(fluid) \rightleftharpoons R(fluid)$*
• Reacting on a flat surface.*

Different Situations of Reactions

• (very slow): surface reaction much slower than diffusion. Then concentration is : $C_{As} = C_{Ab}$.
• (very fast reaction): surface reaction much faster than diffusion: $C_{As} \approx 0$.

Flux of A to surface

$N_A = k_g (C_{Ab} - C_{As})$

The rate of reaction on the surface is:

$N_A = k'' C_{As}$

Combining these equations: $N_A = k_{overall} C_{Ab}$

where

$\frac{1}{k_{overall}} = \frac{1}{k_g} + \frac{1}{k''}$

Extending the reaction A ightleftharpoons R

$N_A = k_g (C_{Ab} - C_{As}) = k'' (C_{As} - C_{Rs}) = k_g (C_{Rb} - C_{Rs})$

$-N_A = k_{overall} (C_{Ab} - C_{Ab}^*)$

• where $C_{Ab}^*$ is the concentration of $A$ in the bulk fluid which would be in equilibrium with the solid surface, and: $\frac{1}{k_{overall}} = \frac{1}{k_g} + \frac{1}{k''} + \frac{1}{k_g K}$ where $K = \frac{C_{Rs}}{C_{As}}$ is the equilibrium constant.

Rate controlling steps

• If fluid resistance dominates. ($\frac{1}{k_g} >> \frac{1}{k''}$)
• If surface reaction dominates.($\frac{1}{k''} >> \frac{1}{k_g} $)

Overall resistance

• Relative, depends on rxn conditions.

Example : hydrogenation of ethylene, w/ Ni catalyst particles and excess of Hydrogen

$C_2H_4 + H_2 \rightarrow C_2H_6$

• At low temps, result shows rate as:$-r' = k_1 p_{C_2H_4}$*
• At high temps the result shows rate as : $-r' = k_2 p_{C_2H_4}^{0}$*At low temperature the surface reaction is the rate controlling step; At high temperature mass transfer is the rate controlling step.

Film Thickness

• Thickness = mass transfer
  • can speed it up via velocity increase

Sherwood number formula

$Sh = 2 + 1.1Re^{0.6}Sc^{1/3}$ where: $Sh = \frac{k_g d_p}{\mathcal{D}}$ is the Sherwood number $Re = \frac{d_p v \rho}{\mu}$ is the Reynolds number $Sc = \frac{\mu}{\rho \mathcal{D}}$ is the Schmidt number

Chemical Bond

Intramolecular Forces

• attractive strength to hold atoms together in a molecule.
• Kinds: ionic, covalent, and metallic bonds.

Intermolecular Forces

attract b/w atoms, affects molec properties ( boiling pont, etc...)

• Type of interaction (Dipole-dipole,Hydrogen bonding, London, Vander walls)

Chemical Bonds

forces that hold atoms in bond with molecular or ionic compound.

Lewis Dot

• valence shown.

Ionic Bond

electron transfer, results in positive cation and negative anion.

Born Haber Cycle

• connects steps in ions forming with lattice Energy
• requires seperation to occur (gaseous) directly proportional to their number by product.

Lattice Energy formula

$$E=k \frac{Q_{1} Q_{2}}{r}$$

• Where = $E$ (potential energy), $k$ = constant, both $q$ (charges on the ions), and $r$ (distance between the ions)

Properties of Ionic Compounds

• solid, hard, high melting point or solution

Covalent Bond

sharing to two atom. held together

Lewis Structure

valence shown.

Octect Rule

Single, Double & Triple bonds

• Single = one sharing.
• Double = 2 pairs sharing occur.
• Multiple pairs for triples.

Bond Length

Bond Energy

Electronegativity

ability to pull atoms.

• increase and decrease periods group from thermochemical data.

Electronegativity Difference and Bond Type

can make bonds.

Dipole moment formula

$$\mu=Q \times r$$

• $\mu$ is the dipole moment, Q = charge, r = distance; dbyes

Percent Ionic Character

$$% \text { Ionic Character }=\left(\frac{\text { measured dipole moment }}{\text { calculated dipole moment assuming } 100 % \text { electron transfer }}\right) \times 100 %$$