1. Construction Engineering 221 Probability and Statistics

2. Problem 10 Solution to problem 10 on page 82 –First recognize that it is a binomial (counting) problem –Second recognize that the binomial calculation will be problematic because n is large (n=100) –Third- use the normal probability distribution as an approximation of the binomial distribution

3. Problem 10 Normal probability approximation: –µ=nπ, or µ = 100*.3 = 30 –sd= nπ(1-π), or 100*.3*.7 = 4.58 –P (40); z= (40-30)/4.58 = 2.18 –A(x) @ z=2.18 =.48537 –Probability of 40 hits is 1-(.5 +.48537) –P(40) =.0146, or 1.46% –I believe the book’s answer is incorrect

4. Linear Regression Sometimes we need to make predictions about the likelihood of an event (flood, traffic accident, inflation, disease, etc.) We can use statistics to sort variance into recognizable patterns to help us interpret what is “random” variance” and what is “sample” variance. Random variance is distributed throughout the population at random. Sample variance is created by membership in a sample (people who smoke and get lung cancer)

5. Linear Regression Sample variance can be correlated between -1 and +1. If a high score is correlated (occurs frequently within the sample) with a low score, then the correlation coefficient is negative. If a high score occurs frequently with a high score, the data is positively correlated

6. Linear Regression What type of correlation would you expect between: –IQ and salary? –GPA and hours studying? –GPA and hours drinking/partying? –Price of tea in China and number of wins in a season by the Chicago Cubs? –Socio-economic standing and crime rate?

7. Linear Regression Correlation coefficient r = Σ(x-ˉ)(y-ˉ)/ [Σx-ˉ] 2[ Σ(y-ˉ) 2 ] Alternate formula eq. 9-2 on page 109 Assumptions: relationship is linear both variables are random conditional variances are equal variables are bivariate normal X

8. Linear regression Example of correlation height Weight 65185 67200 69215 62140 71220 77250 75245 79235 70220

9. Linear Regression Column 1Column 2 Column 11 Column 20.9090221

10. Linear Regression Can be done with Excel spreadsheets Linear regression is a special form of correlation, attempts to find the regression line, or the line through the correlated data that best fits the data. The regression line can then be used to predict outcomes. Regression has formula y=bx +a, where –Y is the dependent variable, x is the independent variable, b is the regression coefficient, and a is a constant

11. Linear Regression When one predictor (independent) variable is used, it is called a simple regression, when more than one predictor is used, it is called multiple regression Restatement of regression formula in common terms: –Expected value of the variable to be predicted =intercept +(slope X value of predictor variable); where slope is regression coeff.

12. SUMMARY OUTPUT Regression Statistics Multiple R0.909022 R Square0.826322 Adjusted R Square0.801511 Standard Error15.15392 Observations9 Linear Regression

13. ANOVA dfSSMSFSignificance F Regression17648.0677648.06733.304410.000684 Residual71607.489229.6413 Total89255.556 CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0% Intercept -176.367.5124-2.611370.034844-335.941-16.6582-335.941-16.6582 X Variable 5.5066080.9541875.7709970.0006843.2503177.7628993.2503177.762899 Formula is: weight = -176.3 +height(5.51) So if a new person joined the team and all we knew was that he was 6’-10”, we would be able to guess his weight at w= -176.3 +(82)(5.51)= 275 pounds