MAT 254 – Probability and Statistics Sections 1,2 & Spring

1) Importance and basic concepts of Probability and Statistics. Introduction to Statistics and data analysis  2) Data collection and presentation 3) Measures of central tendency; mean, median, mode 4) Probability 5) Conditional probability 6) Discrete probability distributions 7) Continuous probability distributions Midterm Exam (April 1, 17:30) 8) Hypothesis testing (2 weeks) 9) Student t-test(2 weeks) 10) Chi-square 11) Correlation and regression analysis 12) REVIEW Final Exam (May 25- June 7) web.adu.edu.tr/user/oboyaci MAT254 - Probability & Statistics2

3 CORRELATION The correlations term is used when: 1) Both variables are random variables, 2) The end goal is simply to find a number that expresses the relation between the variables REGRESSION The regression term is used when 1) One of the variables is a fixed variable, 2) The end goal is use the measure of relation to predict values of the random variable based on values of the fixed variable

Copyright © 2010 Pearson Addison-Wesley. All rights reserved

Copyright © 2010 Pearson Addison-Wesley. All rights reserved

Copyright © 2010 Pearson Addison-Wesley. All rights reserved

Copyright © 2010 Pearson Addison-Wesley. All rights reserved

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 14-8  A scatter plot (or scatter diagram) is used to show the relationship between two variables  Correlation analysis is used to measure strength of the association (linear relationship) between two variables ◦ Only concerned with strength of the relationship ◦ No causal effect is implied

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 14-9 y x y x y y x x Linear relationshipsCurvilinear relationships

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap y x y x y y x x Strong relationshipsWeak relationships (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap y x y x No relationship (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Correlation measures the strength of the linear association between two variables  The sample correlation coefficient r is a measure of the strength of the linear relationship between two variables, based on sample observations (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Unit free  Range between -1 and 1  The closer to -1, the stronger the negative linear relationship  The closer to 1, the stronger the positive linear relationship  The closer to 0, the weaker the linear relationship

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap r = +.3r = +1 y x y x y x y x y x r = -1 r = -.6r = 0

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap where: r = Sample correlation coefficient n = Sample size x = Value of the independent variable y = Value of the dependent variable Sample correlation coefficient: or the algebraic equivalent:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Tree Height Trunk Diameter yxxyy2y2 x2x  =321  =73  =3142  =14111  =713

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Trunk Diameter, x Tree Height, y (continued) r = → relatively strong positive linear association between x and y

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Regression analysis is used to: ◦ Predict the value of a dependent variable based on the value of at least one independent variable ◦ Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain Independent variable: the variable used to explain the dependent variable

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Only one independent variable, x  Relationship between x and y is described by a linear function  Changes in y are assumed to be caused by changes in x

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Error values (ε) are statistically independent  Error values are normally distributed for any given value of x  The probability distribution of the errors is normal  The distributions of possible ε values have equal variances for all values of x  The underlying relationship between the x variable and the y variable is linear

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Positive Linear Relationship Negative Linear Relationship Relationship NOT Linear No Relationship

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Linear component The population regression model: Population y intercept Population Slope Coefficient Random Error term, or residual Dependent Variable Independent Variable Random Error component

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap (continued) Random Error for this x value y x Observed Value of y for x i Predicted Value of y for x i xixi Slope = β 1 Intercept = β 0 εiεi

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap The sample regression line provides an estimate of the population regression line Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) y value Independent variable The individual random error terms e i have a mean of zero

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared residuals

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  The formulas for b 1 and b 0 are: algebraic equivalent for b 1 : and

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  b 0 is the estimated average value of y when the value of x is zero  b 1 is the estimated change in the average value of y as a result of a one-unit change in x

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)  A random sample of 10 houses is selected ◦ Dependent variable (y) = house price in $1000s ◦ Independent variable (x) = square feet

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap House Price in $1000s (y) Square Feet (x)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  House price model: scatter plot and regression line Slope = Intercept =

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  b 0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values) ◦ Here, no houses had 0 square feet, so b 0 = just indicates that, for houses within the range of sizes observed, $98, is the portion of the house price not explained by square feet

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  b 1 measures the estimated change in the average value of Y as a result of a one-unit change in X ◦ Here, b 1 = tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  The sum of the residuals from the least squares regression line is 0 ( )  The sum of the squared residuals is a minimum (minimized )  The simple regression line always passes through the mean of the y variable and the mean of the x variable  The least squares coefficients are unbiased estimates of β 0 and β 1

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  Total variation is made up of two parts: Total sum of Squares Sum of Squares Regression Sum of Squares Error where: = Average value of the dependent variable y = Observed values of the dependent variable = Estimated value of y for the given x value

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  SST = total sum of squares ◦ Measures the variation of the y i values around their mean y  SSE = error sum of squares ◦ Variation attributable to factors other than the relationship between x and y  SSR = regression sum of squares ◦ Explained variation attributable to the relationship between x and y (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap (continued) XiXi y x yiyi SST =  (y i - y) 2 SSE =  (y i - y i ) 2  SSR =  (y i - y) 2  _ _ _ y  y y _ y 

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap  The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable  The coefficient of determination is also called R-squared and is denoted as R 2 where

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap R 2 = +1 y x y x R 2 = 1 Perfect linear relationship between x and y: 100% of the variation in y is explained by variation in x

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap y x y x 0 < R 2 < 1 Weaker linear relationship between x and y: Some but not all of the variation in y is explained by variation in x (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap R 2 = 0 No linear relationship between x and y: The value of Y does not depend on x. (None of the variation in y is explained by variation in x) y x R 2 = 0 (continued)

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Coefficient of determination (continued) Note: In the single independent variable case, the coefficient of determination is where: R 2 = Coefficient of determination r = Simple correlation coefficient

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap House Price in $1000s (y) Square Feet (x) Estimated Regression Equation: Predict the price for a house with 2000 square feet

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is ($1,000s) = $317,850 (continued)

MAT254 - Probability & Statistics END OF THE LECTURE… 44