Econ 140 Lecture 41 More on Univariate Populations Lecture 4

Econ 140 Lecture 42 Today’s Plan Examining known distributions: Normal distribution & Standard normal curve Student’s t distribution F distribution &  2 distribution Note: should have a handout for today’s lecture with all tables and a cartoon Brief statements about: Bivariate populations and conditional probabilities Joint and marginal probabilities

Econ 140 Lecture 43 Standard Normal Curve (6) Going back to our earlier question: What is the probability that someone earns between $300 and $400 [P(300  Y  400)]? P(300  Y  400) Z1Z1 Z2Z2

Econ 140 Lecture 44 Standard Normal Curve (7) We know from using our PDF that the chance of someone earning between $300 and $400 is around 23%, so 0.24 is a good approximation Now we can ask: What is the probability that someone earns between $253 and $316? Z1Z1 P(253  Y  316) Z2Z2

Econ 140 Lecture 45 Standard Normal Curve (8) There are instructions for how you can do this using Excel: L4_1.xls. Note how to use STANDARDIZE and NORMDIST and what they represent Our spreadsheet example has 3 examples of different earnings intervals, using the same distribution that we used today Testing the Normality assumption. We know the approximate shape of the Earnings (L3_79.xls) distribution. Slightly skewed. Is normality a good assumption? Use in Excel (L4_2.xls) of NORMSINV

Econ 140 Lecture 46 Student’s T-Distribution Starting next week, we’ll be looking more closely at sample statistics In sample statistics, we have a sample that is small relative to the population size We do not know the true population mean and variance –So, we take samples and from those samples we will estimate a mean and variance S Y 2

Econ 140 Lecture 47 T-Distribution Properties Fatter tails than the Z distribution Variance is n/(n-2) where n is the number of observations When n approaches a large number (usually over 30), the t approximates the normal curve The t-distribution is also centered on a mean of zero The t lets us approximate probabilities for small samples

Econ 140 Lecture 48 F and  2 Distributions Chi-squared distribution:square of a standard normal (Z) distribution is distributed  2 with one degree of freedom (df). Chi-squared is skewed. As df increases, the  2 approximates a normal. F-distribution: deals with sample data. F stands for Fisher, R.A. who derived the distribution. F tests if variances are equal. F is skewed and positive. As sample sizes grow infinitely large the F approximates a normal. F has two parameters: degrees of freedom in the numerator and denominator.

Econ 140 Lecture 49 A recap on the story so far Probability is concerned with random events. Nearly all data is the outcome of a ‘random draw’ - a sample drawn at random. The probability of earning particular amounts –Relationship between a sample and population –Using standard normal tables Introduction to the t-distribution Introduction to the F and  2 distributions

Econ 140 Lecture 410 A quick note on bivariate probability Bivariate populations and conditional probabilities Joint and marginal probabilities

Econ 140 Lecture 411 A Simple E.C.P Example Introduce Bivariate probability with an example of empirical classical probability (ecp). Consider a fictitious computer company. We might ask the following questions: –What is the probability that consumers will actually buy a new computer? –What is the probability that consumers are planning to buy a new computer? –What is the probability that consumers are planning to buy and actually will buy a new computer? –Given that a consumer is planning to buy, what is the probability of a purchase?

Econ 140 Lecture 412 certainnull A Simple E.C.P Example(2) Think of probability as relating to the outcome of a random event (recap) All probabilities fall between 0 and 1: Probability of any event A is: Where m is the number of events A and n is the number of possible events

Econ 140 Lecture 413 A Simple E.C.P Example(3) The cumulative frequency is: The sample space (of a 1000 obs) looks like this: Before we move on we’ll look at some simple definitions

Econ 140 Lecture 414 A Simple E.C.P Example(4) If we have an event A there will be a compliment to A which we’ll call A’ or B Computing marginal probabilities –Event A consists of two outcomes, a 1 and a 2 : –The compliment B consists also of two outcomes, b 1 and b 2 : –two events are mutually exclusive if both events cannot occur –A set of events is collectively exhaustive if one of the events must occur

Econ 140 Lecture 415 A Simple E.C.P Example(5) Computing marginal probabilities Where k is some arbitrary large number If A = planned to purchase and B=actually purchased: P(planned to buy) = P(planned & did) + P(planned & did not)=

Econ 140 Lecture 416 A Simple E.C.P Example(6) If the two events, A and B, are mutually exclusive, then –General rule written as: –Example: Probability that you draw a heart or spade from a deck of cards They’re mutually exclusive events P(Heart or Spade) = P(Heart) + P(Spade) – P(Heart + Spade)=

Econ 140 Lecture 417 A Simple E.C.P Example(6) Probability that someone planned to buy or actually did buy: use the general addition rule: If A is planning to purchase, and B is actually purchasing, we can plug in the marginal probabilities to find Joint Probability: P(A and B): Planned and Actually Purchased

Econ 140 Lecture 418 Conditional Probabilities Lets leave the example for a while and consider conditional probabilities. Conditional probabilities are represented as P(Y|X) This looks similar to the conditional mean function: We’ll use this to lead into regression line inference.

Econ 140 Lecture 419 Conditional Probabilities (2) Probabilities will be defined as If we sum over j and k, we will get 1, or: We define the conditional probability as f (X|Y) –This is read “a function of X given Y” –We can define this as:

Econ 140 Lecture 420 Conditional Probabilities (3) Similarly we can define f (Y|X): Looking at our example spreadsheet, we have a sample of weekly earnings and years of education: L5_1.XLS. There are two statements on the spreadsheet that will clarify the difference between a joint and conditional probabilities

Econ 140 Lecture 421 Conditional Probabilities (4) The joint probability is a relative frequency and it asks: –How many people earn between $600 and $799 and have 10 years of education? The conditional probability asks: –How many people earn between $600 and $799 given they have 10 years of education? On the spreadsheet I’ve outlined the cells that contain the highest probability in each completed years of education –There’s a pattern you should notice

Econ 140 Lecture 422 Conditional Probabilities (5) We can use the same data to graph the conditional mean function –the graph shows the same pattern we saw in the outlined cells –The conditional probability table gives us a small distribution around each year of education

Econ 140 Lecture 423 Conditional Probabilities (6) To summarize, conditional probabilities can be written as –This is read as “The probability of X given Y” –For example: The probability that someone earns between $200 and $300, given that he/she has completed 10 years of education Joint probabilities are written as P(X&Y) –This is read as “the probability of X and Y” –For example: The probability that someone earns between $200 and $300 and has 10 years of education

Econ 140 Lecture 424 A Marketing Example Now we’ll look at joint probabilities again using the marketing example from earlier in the lecture. We will look at: –Marginal probabilities P(A) or P(B) –Joint probabilities P(A&B) –Conditional probabilities

Econ 140 Lecture 425 Marketing Example(2) Here’s the matrix Let’s look at the probability you purchased a computer given that you planned to purchase: The joint probability that you purchased and planned to purchase: 200/1000 =.2 = 20%

Econ 140 Lecture 426 What we’ve done Introduction to standardized normal (Z) distribution Introduction to the t-distribution Introduction to the F and  2 distributions Bi-variate probabilities: calculate marginal, joint, and conditional probabilities –Computer company example –Earnings and years of education