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Chap 12-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 12 Simple Regression Statistics for Business and Economics 6 th Edition

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-2 Chapter Goals After completing this chapter, you should be able to: Explain the correlation coefficient and perform a hypothesis test for zero population correlation Explain the simple linear regression model Obtain and interpret the simple linear regression equation for a set of data Describe R 2 as a measure of explanatory power of the regression model Understand the assumptions behind regression analysis

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-3 Chapter Goals After completing this chapter, you should be able to: Explain measures of variation and determine whether the independent variable is significant Calculate and interpret confidence intervals for the regression coefficients Use a regression equation for prediction Form forecast intervals around an estimated Y value for a given X Use graphical analysis to recognize potential problems in regression analysis (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-4 Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Correlation was first presented in Chapter 3

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-5 Correlation Analysis The population correlation coefficient is denoted ρ (the Greek letter rho) The sample correlation coefficient is where

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-6 To test the null hypothesis of no linear association, the test statistic follows the Student’s t distribution with (n – 2 ) degrees of freedom: Hypothesis Test for Correlation

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-7 Lower-tail test: H 0 : ρ 0 H 1 : ρ < 0 Upper-tail test: H 0 : ρ ≤ 0 H 1 : ρ > 0 Two-tail test: H 0 : ρ = 0 H 1 : ρ ≠ 0 Hypothesis Test for Correlation Decision Rules /2 -t -t /2 tt t /2 Reject H 0 if t < -t n-2, Reject H 0 if t > t n-2, Reject H 0 if t < -t n-2, or t > t n-2, Where has n - 2 d.f.

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-8 Introduction to Regression Analysis 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 (also called the endogenous variable) Independent variable: the variable used to explain the dependent variable (also called the exogenous variable)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-9 Linear Regression Model The relationship between X and Y is described by a linear function Changes in Y are assumed to be caused by changes in X Linear regression population equation model Where 0 and 1 are the population model coefficients and is a random error term.

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-10 Linear component Simple Linear Regression Model The population regression model: Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-11 (continued) Random Error for this X i 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 Simple Linear Regression Model

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-12 The simple linear regression equation provides an estimate of the population regression line Simple Linear Regression Equation Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) y value for observation i Value of x for observation i The individual random error terms e i have a mean of zero

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-13 Least Squares Estimators b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared differences between y and : Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-14 The slope coefficient estimator is And the constant or y-intercept is The regression line always goes through the mean x, y Least Squares Estimators (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-15 Finding the Least Squares Equation The coefficients b 0 and b 1, and other regression results in this chapter, will be found using a computer Hand calculations are tedious Statistical routines are built into Excel Other statistical analysis software can be used

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-16 Linear Regression Model Assumptions The true relationship form is linear (Y is a linear function of X, plus random error) The error terms, ε i are independent of the x values The error terms are random variables with mean 0 and constant variance, σ 2 (the constant variance property is called homoscedasticity) The random error terms, ε i, are not correlated with one another, so that

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-17 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) b 1 is the estimated change in the average value of y as a result of a one-unit change in x Interpretation of the Slope and the Intercept

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-18 Simple Linear Regression Example 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-19 Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-20 Graphical Presentation House price model: scatter plot

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-21 Regression Using Excel Tools / Data Analysis / Regression

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-22 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 The regression equation is:

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-23 Graphical Presentation House price model: scatter plot and regression line Slope = 0.10977 Intercept = 98.248

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-24 Interpretation of the Intercept, b 0 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 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-25 Interpretation of the Slope Coefficient, b 1 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 =.10977 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-26 Measures of Variation Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error Sum of Squares where: = Average value of the dependent variable y i = Observed values of the dependent variable i = Predicted value of y for the given x i value

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-27 SST = total sum of squares Measures the variation of the y i values around their mean, y SSR = regression sum of squares Explained variation attributable to the linear relationship between x and y SSE = error sum of squares Variation attributable to factors other than the linear relationship between x and y (continued) Measures of Variation

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-28 (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 Measures of Variation

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-29 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 Coefficient of Determination, R 2 note:

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-30 r 2 = 1 Examples of Approximate r 2 Values 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-31 Examples of Approximate r 2 Values Y X Y X 0 < r 2 < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-32 Examples of Approximate r 2 Values 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

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-33 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 58.08% of the variation in house prices is explained by variation in square feet

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-34 Correlation and R 2 The coefficient of determination, R 2, for a simple regression is equal to the simple correlation squared

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-35 Estimation of Model Error Variance An estimator for the variance of the population model error is Division by n – 2 instead of n – 1 is because the simple regression model uses two estimated parameters, b 0 and b 1, instead of one is called the standard error of the estimate

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-36 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-37 Comparing Standard Errors YY X X s e is a measure of the variation of observed y values from the regression line The magnitude of s e should always be judged relative to the size of the y values in the sample data i.e., s e = $41.33K is moderately small relative to house prices in the $200 - $300K range

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-38 Inferences About the Regression Model The variance of the regression slope coefficient (b 1 ) is estimated by where: = Estimate of the standard error of the least squares slope = Standard error of the estimate

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-39 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-40 Comparing Standard Errors of the Slope Y X Y X is a measure of the variation in the slope of regression lines from different possible samples

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-41 Inference about the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H 0 : β 1 = 0(no linear relationship) H 1 : β 1 0(linear relationship does exist) Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope s b1 = standard error of the slope

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-42 House Price in $1000s (y) Square Feet (x) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700 Estimated Regression Equation: The slope of this model is 0.1098 Does square footage of the house affect its sales price? Inference about the Slope: t Test (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-43 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 From Excel output: CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 t b1b1

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-44 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 Test Statistic: t = 3.329 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 tb1b1 Decision: Conclusion: Reject H 0 /2=.025 -t n-2,α/2 Do not reject H 0 0 /2=.025 -2.30602.3060 3.329 d.f. = 10-2 = 8 t 8,.025 = 2.3060 (continued) t n-2,α/2

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-45 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 P-value = 0.01039 There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept98.2483358.033481.692960.12892 Square Feet0.109770.032973.329380.01039 P-value Decision: P-value < α so Conclusion: (continued) This is a two-tail test, so the p-value is P(t > 3.329)+P(t < -3.329) = 0.01039 (for 8 d.f.)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-46 Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858) CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 d.f. = n - 2

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-47 Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance Confidence Interval Estimate for the Slope (continued)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-48 F-Test for Significance F Test statistic: where where F follows an F distribution with k numerator and (n – k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model)

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-49 Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 With 1 and 8 degrees of freedom P-value for the F-Test

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-50 H 0 : β 1 = 0 H 1 : β 1 ≠ 0 =.05 df 1 = 1 df 2 = 8 Test Statistic: Decision: Conclusion: Reject H 0 at = 0.05 There is sufficient evidence that house size affects selling price 0 =.05 F.05 = 5.32 Reject H 0 Do not reject H 0 Critical Value: F = 5.32 F-Test for Significance (continued) F

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-51 Prediction The regression equation can be used to predict a value for y, given a particular x For a specified value, x n+1, the predicted value is

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-52 Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850 Predictions Using Regression Analysis

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-53 Relevant Data Range When using a regression model for prediction, only predict within the relevant range of data Relevant data range Risky to try to extrapolate far beyond the range of observed X’s

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-54 Estimating Mean Values and Predicting Individual Values Y X x i y = b 0 +b 1 x i Confidence Interval for the expected value of y, given x i Prediction Interval for an single observed y, given x i Goal: Form intervals around y to express uncertainty about the value of y for a given x i y

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-55 Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular x i Notice that the formula involves the term so the size of interval varies according to the distance x n+1 is from the mean, x

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-56 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular x i This extra term adds to the interval width to reflect the added uncertainty for an individual case

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-57 Estimation of Mean Values: Example Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price y i = 317.85 ($1,000s) Confidence Interval Estimate for E(Y n+1 |X n+1 ) The confidence interval endpoints are 280.66 and 354.90, or from $280,660 to $354,900

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-58 Estimation of Individual Values: Example Find the 95% confidence interval for an individual house with 2,000 square feet Predicted Price y i = 317.85 ($1,000s) Confidence Interval Estimate for y n+1 The confidence interval endpoints are 215.50 and 420.07, or from $215,500 to $420,070

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-59 Finding Confidence and Prediction Intervals in Excel In Excel, use PHStat | regression | simple linear regression … Check the “confidence and prediction interval for x=” box and enter the x-value and confidence level desired

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-60 Input values Finding Confidence and Prediction Intervals in Excel (continued) Confidence Interval Estimate for E(Y n+1 |X n+1 ) Confidence Interval Estimate for individual y n+1 y

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-61 Graphical Analysis The linear regression model is based on minimizing the sum of squared errors If outliers exist, their potentially large squared errors may have a strong influence on the fitted regression line Be sure to examine your data graphically for outliers and extreme points Decide, based on your model and logic, whether the extreme points should remain or be removed

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Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 12-62 Chapter Summary Introduced the linear regression model Reviewed correlation and the assumptions of linear regression Discussed estimating the simple linear regression coefficients Described measures of variation Described inference about the slope Addressed estimation of mean values and prediction of individual values

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