Hypothesis Testing

Introduction to Hypothesis Testing

• Hypothesis testing is a method that evaluates a claim about a population using sample data.
• A hypothesis is a statement about a population parameter.
• The null hypothesis ($H_0$) states that there is no effect or no difference and is assumed to be true.
• The alternative hypothesis ($H_1$) contradicts the null hypothesis.
  • It's also the research hypothesis.
• The goal is to determine if there's enough evidence to reject the null hypothesis in favor of the alternative.
• The null hypothesis is never "accepted".

Basic Steps of Hypothesis Testing

• State the null and alternative hypotheses.
• Choose a significance level ($\alpha$).
• Calculate the appropriate test statistic.
• Determine the p-value.
• Make a decision based on the p-value and $\alpha$.

Null and Alternative Hypothesis Details

• Null hypothesis ($H_0$) is initially assumed to be true.
• Alternative hypothesis ($H_1$) contradicts the null hypothesis.
• $H_0$ and $H_1$ must be mutually exclusive and exhaustive.

Example Null and Alternative Hypotheses

• Average height of women is greater than 5'4":
  • $H_0: \mu = 5'4"$
  • $H_1: \mu > 5'4"$
• Average IQ score of students differs from 100:
  • $H_0: \mu = 100$
  • $H_1: \mu \neq 100$
• Proportion of people supporting a candidate is less than 50%:
  • $H_0: p = 0.5$
  • $H_1: p < 0.5$

Significance Level Explained

• Significance level ($\alpha$) is the probability of rejecting the null hypothesis when it's true.
  • It represents the Type I error rate.
  • Common values are 0.05, 0.01, or 0.10.
  • $\alpha$ = P(Reject $H_0$ | $H_0$ is true)

Common Significance Level Values

• $\alpha = 0.05$: 5% chance of rejecting $H_0$ when it is actually true.
• $\alpha = 0.01$: 1% chance of rejecting $H_0$ when it is actually true.
• $\alpha = 0.10$: 10% chance of rejecting $H_0$ when it is actually true.

Significance Level Analogy

• Significance level is like the standard of evidence in a trial.
• A high significance level (e.g., $\alpha = 0.10$) increases the chance of incorrectly rejecting $H_0$.
• A low significance level (e.g., $\alpha = 0.01$) decreases the chance of incorrectly rejecting $H_0$, but increases the chance of failing to reject a false $H_0$.

Test Statistic Definition

• A test statistic is calculated from sample data.
• It determines whether to reject the null hypothesis.
• This statistic measures the differences between sample data and null hypothesis.
• It is compared to a critical value to determine whether to reject the null hypothesis.

Common Test Statistics Explained

• z-statistic: population standard deviation known or large sample size ($n \geq 30$).
• t-statistic: population standard deviation unknown and small sample size ($n < 30$).
• $\chi^2$ statistic: used for categorical data.
• F-statistic: compares variances or means of multiple groups.

P-value Explained

• P-value is the probability of getting a test statistic as extreme or more extreme than observed, assuming $H_0$ is true.
• Measures the strength of evidence against the null hypothesis.
• Smaller p-value means stronger evidence against the null hypothesis.
• p = P(Observing data | $H_0$ is true)
  • In practice, it is the likelihood of observing a test statistic as extreme or more extreme than the one observed.

P-value Interpretation

• Reject $H_0$ if p-value is less than $\alpha$.
• Fail to reject $H_0$ if p-value is greater than $\alpha$.

P-value Analogy

• The p-value is analogous to the probability of observing evidence in a trial given the defendant is innocent.
• A very small p-value suggests that observing the evidence is unlikely if the defendant were innocent, leading to rejection of $H_0$.
• A large p-value indicates the evidence is not surprising if the defendant were innocent, so we fail to reject $H_0$.

Rejecting or Failing to Reject the Null Hypothesis

• If p-value < $\alpha$, reject $H_0$.
  • There is sufficient evidence to support the alternative hypothesis ($H_1$).
  • The results are statistically significant.
• If p-value > $\alpha$, fail to reject $H_0$.
  • There is insufficient evidence to support alternative hypothesis ($H_1$).
  • The results are not statistically significant.
• Failing to reject $H_0$ is not the same as accepting $H_0$; it means there's not enough evidence to reject it.

Analogy for the Decision Step

• Jury finds defendant guilty: reject $H_0$.
• Jury finds defendant not guilty: fail to reject $H_0$.

Type I Error Explained

• Type I error: Rejecting $H_0$ when it is actually true (false positive).
• Probability of a Type I error = $\alpha$.
• Analogous to convicting an innocent person.

Type II Error Explained

• Type II error: Failing to reject $H_0$ when it is false (false negative).
• Probability of a Type II error = $\beta$.
• Analogous to letting a guilty person go free.

Error Types Summary

• $H_0$ True and Fail to Reject $H_0$: Correct Decision
• $H_0$ True and Reject $H_0$: Type I Error
• $H_0$ False and Fail to Reject $H_0$: Type II Error
• $H_0$ False and Reject $H_0$: Correct Decision

Controlling Errors

• Significance level ($\alpha$) controls the probability of making a Type I error.
• Probability of Type II error ($\beta$) is more difficult to control.
• Decreasing the probability of Type II error can be achieved by increasing the sample size or the alpha value.
  • This will increase the likelihood of committing a Type I error.

One-Tailed Tests

• Used with directional alternative hypotheses.
• Example: testing if women's average height is greater than 5'4".
• Critical region is in only one tail of the distribution.

Two-Tailed Tests

• Used with non-directional alternative hypotheses.
• Example: testing if students' average IQ scores differ from 100.
• Critical region is located in both tails of the distribution.

Examples of One and Two-Tailed Tests

• Average height of women is greater than 5'4":

  • $H_0: \mu = 5'4"$
  • $H_1: \mu > 5'4"$
  • One-tailed test

• Average IQ score of students is different from 100:

  • $H_0: \mu = 100$
  • $H_1: \mu \neq 100$
  • Two-tailed test

• Proportion of people supporting a candidate is less than 50%:

  • $H_0: p = 0.5$
  • $H_1: p < 0.5$
  • One-tailed test

Use Case for One vs. Two-Tailed Tests

• Use One-tailed tests when you have a specific directional hypothesis
• Use Two-tailed tests when you do not have a specific directional hypothesis
• Two-tailed tests are more conservative than one-tailed tests.
• Generally recommended to use a two-tailed test unless there is a good reason to use a one-tailed test

Example Hypothesis Test

• Problem: Researcher thinks average student IQ is above 100. 25 students sampled, mean IQ 105, standard deviation 15. Test at $\alpha = 0.05$.

Steps To Solve the Practice Problem

1. State the null and alternative hypotheses.

   • $H_0: \mu = 100$
   • $H_1: \mu > 100$
   • One-tailed test

2. Choose a significance level ($\alpha$).

   • $\alpha = 0.05$

3. Calculate the test statistic.

   • Using a t-test due to unknown population standard deviation and small sample size
   • Formula: $t = \frac{\bar{x} - \mu}{s/\sqrt{n}} = \frac{105-100}{15/\sqrt{25}} = \frac{5}{3} = 1.67$

4. Determine the p-value.

   • Degrees of freedom: $df = n - 1 = 25 - 1 = 24$.
   • P-value between 0.054 and 0.1 is for a one-tailed test with t = 1.67 and df = 24.
   • $p \approx 0.054$

5. Make a decision.

   • Fail to reject as p-value is greater than $\alpha$ = 0.05.
   • There isn't enough evidence to support that the average student IQ score is greater than 100.

Einstein Coefficients

• Considers two energy levels of an atom, lower level 1 and upper level 2, with corresponding energies $E_1$ and $E_2$ ($E_1 < E_2$).

Einstein Coefficients: Absorption

• Atom in level 1 absorbs a photon with energy $E = E_2 - E_1$, transitioning to level 2.
• Absorption rate is proportional to atom density in level 1 ($n_1$) and radiation field energy density at the transition frequency ($\rho_{\nu}$).
• $R_{12} = B_{12} n_1 \rho_{\nu}$ where $B_{12}$ is the Einstein coefficient for absorption

Einstein Coefficients: Spontaneous Emission

• Atom in level 2 spontaneously emits a photon with energy $E = E_2 - E_1$ and transitions to level 1.
• Spontaneous emission rate is proportional to atom density in level 2 ($n_2$).
• $R_{21} = A_{21} n_2$ where $A_{21}$ is the Einstein coefficient for spontaneous emission.

Einstein Coefficients: Stimulated Emission

• Atom in level 2 emits a photon with energy $E = E_2 - E_1$, stimulated by another photon of the same energy.
• Stimulated emission rate is proportional to atom density in level 2 ($n_2$) and radiation field energy density at the transition frequency ($\rho_{\nu}$).
• $R_{21} = B_{21} n_2 \rho_{\nu}$ where $B_{21}$ is the Einstein coefficient for stimulated emission.

Relations between Einstein Coefficients

• In thermal equilibrium, upward and downward transition rates balance: $$n_1 B_{12} \rho_{\nu} = n_2 A_{21} + n_2 B_{21} \rho_{\nu}$$
• The equation is rearranged to solve for $\rho_{\nu}$: $$\rho_{\nu} = \frac{A_{21}}{B_{12} \frac{n_1}{n_2} - B_{21}}$$
• In LTE, the Boltzman distribution gives the energy levels: $$\frac{n_1}{n_2} = \frac{g_1}{g_2} e^{\frac{E_2 - E_1}{kT}} = \frac{g_1}{g_2} e^{\frac{h\nu}{kT}}$$
  • Here $g_1$ and $g_2$ are the weights for levels 1 and 2, $h$ is Planck's constant, $k$ is Boltzman's constant and $T$ is temperature.
• Substituted into the equation for $\rho_{\nu}$: $$\rho_{\nu} = \frac{A_{21}}{B_{12} \frac{g_1}{g_2} e^{\frac{h\nu}{kT}} - B_{21}}$$
• Comparing with the Planck Function gives the following: $$B_{12} = \frac{g_2}{g_1} B_{21}$$ $$A_{21} = \frac{2h\nu^3}{c^2} B_{21}$$
• These equations are fundamental in understanding the interaction between matter and radiation

Economics Defined

• Economics examines how societies allocate scarce resources to produce valuable goods/services and distribute them among individuals.

Core Economic Ideas

• Resources/Goods are scarce.
• Society must use resources efficiently.
• How people make decisions faced with scarcity

Micro vs Macro Economics

• Micro: studies the behavior of individual units
  • Markets
  • Businesses
  • Households
• Macro: analyzes overall economic performance
  • Inflation
  • Unemployment
  • Economic Growth

Positive vs Normative Economics

• Positive economics is based on analysis and empirical evidence and how the world actually works
  • "Increasing minimum wage reduce employment"
• Normative involves value judgements of how the world ought to be
  • "The government should increase the mimimum wage."

Three Economic Problems

• What goods/services are produced, and in what quantities?
• How are these goods/services produced?
• For whom are these goods/services produced?

Economic Systems

• Market Economy: Decisions made by individuals and firms.
• Centrally planned Economy: Government decisions are made by the government
• Mixed Economy: Combination of both systems

Tools of Economics

• Marginal analysis studies decisions at the "margin"
  • Marginal cost
  • Marginal benefit
  • A rational decision implies that marginal benefit >/ marginal costs
• Economic models are a simplification of relaity which allow analysis of relationships
  • Simplified assumptions makes the analysis easier
  • Variables: Factors that can affect results
• Data economics is numerical information that is utilized to analyze economies
  • Prices, Quantities and Interest rates are some examples

Production Possibilities Frontier

• Maximum combos of goods/services an economy can produce while efficiently using all resources

Production Possibilities Illustration

A. 0 Alimentos and 15 Vestido

B. 1 Alimentos and 14 Vestido

C. 2 Alimentos and 12 Vestido

D. 3 Alimentos and 9 Vestido

E. 4 Alimentos and 5 Vestido

F. 5 Alimentos and 0 Vestido

Opportunity Cost

• Ammount of good that you most forfeit to gain another unit of another good
• the table above increases in the production amount for Alimentos will reduce Vestido amount for the productions

Economic Growth

• The PPF is displayed outwardly to show that the economy can produce larger number of goods/services

• Causes: more resources or improved technology

The Circular Flow

Definition

It is a visual model of the economy, shows how dollars flow through markets among households and firms

Actors

• Households own the factors of production that are sold to firms in order to generate revenue
• Firms buy factors to production to generate factors of productioon
• Factors such as land, labor and capital are key.
• Money is constantly flowing

What is a Graph

• A graph G = (V, E) is composed of V which is the set of verices and nodes and E which is the set of edges which are links.
• Each edge is a pair of vertices
• Directed Graphs are when the edges are ordered pairs
• Undirected graphs are when edges are unordered pairs
• Weighted graphs are when edges have costs Bipartite graphs can be divided into disjoint sets, every edge connects a vertex in one set to another vertice in the other

Representations of Graphs

• Adjacency matrix: a $V \times V$ matrix where $A_{ij} = 1$ if there is an edge from vertex $i$ to vertex $j$, and 0 otherwise.
• Adjacency list: a list of vertices adjacent to each vertex.

Connectivity

• Connectivity checks if there is a path between to vertices
• Shortest path calculates shortest path between two vertices
• connectivity is yes if the graph is connected
• Strong connectivity test measures if you can get from any node to any other node in the graph

Search Algorithms Explained

• Find all vertices reachable from a given vertex
• Find all vertices that can reach a given vertex

Min Spanning Tree Explained

• The min spanning tree includes the set of edges that connects all the vertices with the lowest total weight

Max Flow Explained

• Max flow finds the maximum rate at which a fluid can be sent from a source vertex to a sink vertex

Travelling Salesman Problem Explained

• The travelling salesman problem calculates a short tour to visit each vertices once.

Adjacency matrix in computers

• A[i,j] = 1 if there is an edge between node i and j.
  • Easy to check for edges and implement
  • Fast at O(1)
  • Disadvantage in memory and iteration speed O(V^2)

Adjacency List in Computers

• Each node stores a list of its neighbors and nodes
  • requires O(E) space
  • Iterate is O(E)
  • Disadvantage in memory and complixity

Breadth First Search (BFS)

• Explores the graph
  • Implementation:
    • Starts at source vertex
    • marks it visited after searching through each
    • Time compexity is O(v+e)

Depth First Searh (DFS)

• Examines Graph branch by branch Implementation:
  • Start at a source vertex.
  • Mark the vertex as visited.
  • For each unvisited neighbor of the vertex:
    • Recursively call DFS on the neighbor. Time complexity: O(V + E).

Comparisons

• BFS:
  • Finds the shortest path in unweighted graphs.
  • Can be used to find connected components. DFS:
  • Can be used to detect cycles.
  • Can be used to find topological ordering.

Topological Sort

• A directed acyclic graph (DAG) that is a linear ordering of its vertices
• algorithm
• The algorithm finds the incoming egde and adds all vertices and edges.
• The time Complexity is O(E + V)

Dijstra's Algorithm

• A weighted graph with non-negative edge weights will find the shortest part with this algorithm
• O(E log V) with heap
• O(E + V log v) with Fibonacci heap
• Algorithm:
  • The intial vertex is set to 0, and the distance to vertices sets to infiinity

Bell man Ford algorithm

• Used to find the shortest paths
• Time complexity of O(v E)
  • Iterate v-1 times
  • if u + weight to U-> V is less than V, update the values

Floyd, Warshal Algorithm.

• Used tofind the ehortest pasts between all Pairs of verices between a weighted graphs with Possible negative edge weights
• Tiem complexity of O(V3) For each vertex k from 1 to V: - For each vertex i from - 1 to V: -For each vertex j from 1 to V: