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

Econ 140 Lecture 43 Standard Normal Curve We need to calculate something other than our PDF, using the sample mean, the sample variance, and an assumption about the shape of the distribution function Examine the assumption later The standard normal curve (also known as the Z table) will approximate the probability distribution of almost any continuous variable as the number of observations approaches infinity

Econ 140 Lecture 44 Standard Normal Curve (2) The standard deviation (measures the distance from the mean) is the square root of the variance: 68% area under curve 95% 99.7%

Econ 140 Lecture 45 Standard Normal Curve (3) Properties of the standard normal curve –The curve is centered around –The curve reaches its highest value at and tails off symmetrically at both ends –The distribution is fully described by the expected value and the variance You can convert any distribution for which you have estimates of and to a standard normal distribution

Econ 140 Lecture 46 Standard Normal Curve (4) A distribution only needs to be approximately normal for us to convert it to the standardized normal. The mass of the distribution must fall in the center, but the shape of the tails can be different or

Econ 140 Lecture 47 Standard Normal Curve (5) If we want to know the probability that someone earns at most $C, we are asking: We can rearrange terms to get: Properties for the standard normal variate Z: –It is normally distributed with a mean of zero and a variance of 1, written in shorthand as Z~N(0,1)

Econ 140 Lecture 48 Standard Normal Curve (5) If we have some variable Y we can assume that Y will be normally distributed, written in shorthand as Y~N(µ,  2 ) We can use Z to convert Y to a normal distribution Look at the Z standardized normal distribution handout –You can calculate the area under the Z curve from the mean of zero to the value of interest –For example: read down the left hand column to 1.6 and along the top row to.4 you’ll find that the area under the curve between Z=0 and Z=1.64 is

Econ 140 Lecture 49 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 410 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 411 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 412 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 413 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 414 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 415 What we’ve done 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 In the next lectures we’ll move on to bivariate populations, which will be important for computing conditional probability examples such as P(Y|X)