Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Econ 140 More on Univariate Populations Lecture 4 Lecture 4 1 Today’s Plan • • • • • Econ 140 Examining known distributions: Normal distribution & Standard normal curve Student’s t distribution F distribution & c2 distribution Note: should have a handout for today’s lecture with all tables and a cartoon Lecture 4 2 Standard Normal Curve Econ 140 • 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 Lecture 4 3 Standard Normal Curve (2) Econ 140 • The standard deviation (measures the distance from the mean) is the square root of the variance: 2 68% area under curve 95% 99.7% 3 Lecture 4 2 y 2 3 4 Standard Normal Curve (3) Econ 140 • Properties of the standard normal curve – The curve is centered around y – The curve reaches its highest value at y 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 y and 2 to a standard normal distribution Lecture 4 5 Standard Normal Curve (4) Econ 140 • 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 2 1 y Lecture 4 6 Standard Normal Curve (5) Econ 140 • If we want to know the probability that someone earns at most $C, we are asking: PY C ? (Y ) C We can (Y ) C rearrange P ( Z C*) ? terms to get: where Z (Y ) • 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) Lecture 4 7 Standard Normal Curve (5) Econ 140 • 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 0.4495 Lecture 4 8 Standard Normal Curve (6) Econ 140 • Going back to our earlier question: What is the probability that someone earns between $300 and $400 [P(300Y 400)]? 316.6 Z1 Z2 2 25608 25608 160 P(300Y 400) 300 316.6 0.104 160 300 316.6 400 316.6 Z 400 0.52 160 P (0.104 Z 0) 0.0418 P (0 Z 0.52) 0.1985 P (0.104 Z 0.52) 0.0418 0.1985 .2403 Z300 Lecture 4 400 9 Standard Normal Curve (7) Econ 140 • 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? Z1 Z2 P(253Y 316) 253 316.6 0.3975 160 316 316.6 Z2 0.0038 160 P (0.3975 Z 0) 0.1554 Z1 P (0.0038 Z 0) 0.0020 P (0.3975 Z .0038) .1554 .002 Lecture 4 253 316.6 316 .1574 15.3% 10 Standard Normal Curve (8) Econ 140 • 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 Lecture 4 11 Student’s T-Distribution Econ 140 • 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 Y and variance SY2 Lecture 4 12 T-Distribution Properties Econ 140 • 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 Lecture 4 13 F and c2 Distributions Econ 140 • Chi-squared distribution:square of a standard normal (Z) distribution is distributed c2 with one degree of freedom (df). • Chi-squared is skewed. As df increases, the c2 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. Lecture 4 14 What we’ve done Econ 140 • 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 c2 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) Lecture 4 15