Econ 440: Lecture 2 - Experimental Design and Probability

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Questions and Answers

In the function u(x) = −3(x − 2)², what does the value of x that maximizes u(x) represent?

The lowest point on the graph

The average value of x

The point of maximum utility (correct)

The point of minimum utility

What would be the appropriate regression specification to test whether average GPA differs between men and women?

GPA = β0 + ε

GPA = β0 + β1*GPA + ε

GPA = β0 + β1*Gender + ε (correct)

GPA = β0 + β1*Gender + β2*GPA + ε

What does the notation ∂u/∂x represent in the context of the given function?

The average change in utility per unit of x

The derivative of utility with respect to x (correct)

The marginal cost of increasing x

The total utility of the function

Which of the following options best explains why experiments are necessary in economic studies?

They provide conclusive evidence of causation. Signup and view all the answers

What is the significance of using systems of equations in economic analysis?

They help in finding equilibrium in markets. Signup and view all the answers

What does the concept of consumer surplus reflect in economic theory?

The difference between what consumers are willing to pay and what they actually pay Signup and view all the answers

What is the primary benefit of increasing sample size in statistical analysis?

It increases the sensitivity of detecting true effects. Signup and view all the answers

Why might doubling the number of observations by making a second laceration on each mouse be considered flawed?

The two lacerations on each mouse are not independent. Signup and view all the answers

What is a potential pitfall of experimental design mentioned in the content?

Changing multiple treatment variables simultaneously. Signup and view all the answers

What role does probability play in the context of social sciences?

It provides a framework for evaluating the rationality of decisions. Signup and view all the answers

In the example given about rolling a die, what does the set Ω represent?

The collection of all potential outcomes. Signup and view all the answers

What does a high statistical power indicate in hypothesis testing?

A low chance of committing a Type II error. Signup and view all the answers

What is meant by 'observations not being fully independent' in experimental design?

The effect observed in one measurement affects another. Signup and view all the answers

What is a consequence of using a poor choice of context in an experiment?

Misleading results and interpretations. Signup and view all the answers

What does probability judgment entail in social sciences?

Assigning a numerical value reflecting belief in an event's occurrence. Signup and view all the answers

What is a significant disadvantage of a within-subjects design?

It can lead to order effects impacting treatment outcomes. Signup and view all the answers

What type of design allows each subject to complete multiple treatments?

Within-subjects design Signup and view all the answers

Which of the following best describes why certain stakes may be inappropriate for an experiment?

Low stakes do not encourage participants to engage seriously. Signup and view all the answers

What is a common issue that arises from the experimenter demand effect?

Subjects may change their behavior based on perceived expectations. Signup and view all the answers

Why is it important to calibrate the size of the stakes in an experiment?

To avoid making the decisions too trivial or serious. Signup and view all the answers

Which strategy best minimizes confounding variables in a lab experiment?

Maintaining a controlled environment to limit distractions. Signup and view all the answers

Which statement about hypothetical stakes in experiments is accurate?

Hypothetical stakes considered might lack real-world relevance. Signup and view all the answers

What is the primary interest behind testing the law of demand in a food choice experiment?

To measure the correlation between food price and quantity demanded. Signup and view all the answers

In experimental design, what is one of the key advantages of a within-subjects design over a between-subjects design?

It often requires a smaller sample size. Signup and view all the answers

What is a critical aspect to avoid when setting the context for an experiment?

Including unnecessary complexity that distracts participants. Signup and view all the answers

What does it mean for two events A and B to be independent?

P(A|B) = P(A) Signup and view all the answers

In the context of Bayes' Rule, how can you calculate P(A1 | B)?

P(B|A1)P(A1) / P(B) Signup and view all the answers

What is the correct interpretation of the probability P(A and B)?

The joint probability of both A and B occurring Signup and view all the answers

If a woman has a positive mammogram result, what is the probability that she has breast cancer?

27% Signup and view all the answers

According to Bayes' Rule, the probability of B can be expressed using which of the following?

P(B|A1)P(A1) + P(B|A2)P(A2) Signup and view all the answers

What does the axiom P(Ω) = 1 signify in probability theory?

At least one event in the sample space is guaranteed to occur. Signup and view all the answers

Which of the following statements correctly describes a probability function?

It assigns probabilities between 0 and 1 inclusive to all events. Signup and view all the answers

If P({1}) = $\frac{1}{6}$, what would be the probability of rolling a 2?

$\frac{1}{6}$ Signup and view all the answers

What is the correct way to express the addition axiom for mutually exclusive events A and B?

P(A or B) = P(A) + P(B) Signup and view all the answers

In a frequentist perspective, what does a probability represent?

The average outcome over many trials. Signup and view all the answers

How can the probability of flipping two heads in a row be calculated?

Using the counting perspective by identifying all possible outcomes. Signup and view all the answers

What is the formula for conditional probability?

P(A|B) = \frac{P(A and B)}{P(B)} Signup and view all the answers

Given P({3, 4, 5, 6}) = $\frac{2}{3}$, what can be inferred about P({1, 2})?

P({1, 2}) must be $\frac{1}{3}$. Signup and view all the answers

According to the axioms of probability, what can be said about the probabilities assigned to overlapping events?

They require adjustment to avoid double counting. Signup and view all the answers

Which of the following is NOT a feature of the counting perspective in defining probabilities?

Assigning biased probabilities based on previous outcomes. Signup and view all the answers

Study Notes

Quiz 0 Overview

Consumer utility function: ( U(c_1, c_2) = 12 \ln(c_1) + 12 \ln(c_2) )
Marginal Rate of Substitution (MRS) needs calculation at given consumption levels
Utility function ( u(x) = -3(x - 2)^2 ) requires finding the first derivative and maximizing value of ( x )
T-test for average GPA requires calculating the t-statistic against zero
Regression analysis needed to compare average GPAs across genders

Experimental Design

Experiments clarify causal relationships
Between-subjects design: each subject experiences one treatment; often requires more subjects
Within-subjects design: each subject experiences multiple treatments; controls individual differences but may suffer from order effects

Incentives in Experiments

Material incentives can motivate choices: cash, goods, or social considerations
Importance of stake calibration to task difficulty
Hypothetical stakes can be used effectively, e.g., job offers with uncertain gains

Context in Experiments

Control context to minimize distractions during experiments
Avoiding public observations can reduce confounding variables unless public choice is the treatment variable
Experimenter demand effect may bias subject behavior; clear value-free language is essential

Statistical Independence

Increasing statistical power often leads to better detection of effects
Doubling sample size by overlapping measurements can invalidate independence, yielding unreliable data

Common Pitfalls in Experiment Design

Simultaneously changing multiple variables can undermine clarity
Poor contexts or visual placements can bias results
Overlooking order effects and proper independence in observations can skew findings

Importance of Probability

Social scientists need probability to predict behavior and evaluate rationality
Probability judgment involves estimating event likelihoods based on belief strength

Probability Basics

Sample space ( Ω ) consists of all outcomes (e.g., rolling a die)
Events are subsets of outcomes; probability assigns values between 0 and 1

Axioms of Probability

Axiom 1: ( P(Ω) = 1 ) signifies certainty of outcomes
Axiom 2 (Addition Axiom): ( P(A \text{ or } B) = P(A) + P(B) ) for mutually exclusive events

Probability Sources

Frequentist perspective: long-run averages determine probabilities
Counting events yields probabilities based on equally likely outcomes, supporting the addition rule

Conditional Probability

Defined as ( P(A|B) = \frac{P(A \text{ and } B)}{P(B)} )
Events A and B are independent if ( P(A|B) = P(A) )
The multiplication rule links joint probabilities to conditional probabilities

Bayes’ Rule

Example utilizes breast cancer statistics and mammogram results to calculate real probabilities
Bayes’ rule formalizes the relationship between prior knowledge and new evidence:
- ( P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} )
Incorporates the law of total probability for broader applicability in more complex scenarios.

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Description

This quiz covers key concepts from Lecture 2 of Econ 440, focusing on utility functions and marginal rates of substitution in consumer theory. Students will explore questions related to consumption today and tomorrow, as well as economic interpretations of the results. Prepare to analyze utility functions and their implications in economic decision-making.