The Hong Kong University of Science & Technology ISOM551 Introductory Statistics for Business Assignment 4 Suggested Solution

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1 8 TUNG, Yik-Man The Hong Kong University of cience & Technology IOM55 Introductory tatistics for Business Assignment 4 uggested olution Note All values of statistics are obtained by Excel Qa Theoretically, the Least quare (L) Estimation of, can be obtained by minimizing the E function of, That is, by minimizing E f (, ) ( y ) i i xi with respect to, By considering the derivative of E f (, ) as a zero row vector, E E or and, we can mathematically obtain the L estimator of, to be Y cov( X, Y) X and xy respectively xx var( X ) ince X 7 and Y 7, var( X ) 4 9, cov( X, Y) ( 7)( 7) i X i Yi, xy cov( X, Y) 8567 so 7483 and Y X xx var( X ) 49 Hence the estimated L regression line isy X The slope coefficient in our case represents the sensitivity of return from stock price of Dalton s company with respect to the overall market return in a sense that if there is % increase (decrease) in the overall market return, there will be on average 7483% increase (decrease) in the return from stock price of Dalton s company Qb As we want to test H (at most means insensitive to the market) V H (Larger than means sensitive to the market) (One sided test!) with 5 Under the regression assumptions and H we have ~ N(, ) However is usually unknown 936 in practice, so we estimate it by s E 94 and the test 8 8 statistic follows tudent s T distribution with 8 degrees of freedom, so the value of our s xx 7483 test statistic is t t5, and we reject H at 5% level of 88 significance That means the observed data supports that and it is sensitive to the market xx

2 8 TUNG, Yik-Man Qc If the overall market s return next year is 5% or X 5, then our estimated regression model forecasts the return of Dalton s stock next year Y X 5 would be ubstitute these into the prediction interval formula gives Y X 5 94 t 5 (5 7) [66,49] Hence a 9% prediction interval ofy X 5is [ 66,49] (5 7) 49 9 Qd Denote the amount of total variation of Dalton company s stock return explained by our estimated regression model as R, then we know that total variation in Dalton s stock return can be decomposed into E R That means the percentage of total variation in Dalton s stock return explained by the model R is R E xy xy cov( X, Y) cov( X, Y) R var( ) var( ) xx X Y X Y or 86% Actually, it is just the square of sample correlation coefficient of X and Y Alternatively, using Excel, we may obtain the following result The R quare above represents the proportion of variation of Y explained by the estimated regression model Qe If the annual market return is 4% or X 4, then our estimated regression model forecasts the expected return of Dalton s stock next year E ( Y X 4) would be ubstitute these into the confidence interval formula gives Y 94 t (4 7) [3,973] Hence a 98% confidence interval of E( Y X 4) is [ 3,973] (4 7) 49 9

3 8 TUNG, Yik-Man Qa The predicted sales in July by the estimated regression model is given by E Y X 68, X 4, X 8, X ) ( o the predicted sales in July is around 3579 thousands of dollars Qb We can reject the null hypothesis that the coefficient of price is zero at any common significant level since the test of H V H has very small p-value which indicates the null should be rejected usually at most significant level and implies that is strongly statistically significant E Qc The coefficient of determination is R 959 ; 7775 The adjusted coefficient of determination is k n 4 R adj ( R )( ) (959 )( ) 9388 n n ( k ) 3 5 Qd From the given condition, the required test is H 8 V H 8at 5(One 8 sided test!) The test statistic ist and is follows tudents-t distribution with degrees of freedom, so the value of value of test statistic ist t5 949 o we reject H 8 at 5% level of significant level Qe Q3a Let the advertising expenditure of March in thousands of dollars be X4, then if everything else remains the same except and X4, X3 and X are both increased by, then the expected increase of the total sales is E( Y X 4 X 3 X ) is ( ) 8 68 or $8,68 The sample correlation coefficients between dependent variable Y with independent variables X, X, and X3 respectively are computed by Excel as follows Correlation Y and X 965 Y and X Y and X3-993 Based on the above results, it seems that X is expected to be the most useful variable in predicting the house price using simple linear regression model since its correlation coefficient with Y is 965 and is the largest among the three If two independent variables are concerned, then it seems that X and X are more appropriate since the correlation coefficients of Y with X and X are the largest 3

4 8 TUNG, Yik-Man Q3b Let cov( Y, X ), to see whether the data supports that Y is positively and linearly related to X, the reasonable way we can do is perform the test H V H (one sided test) at for example 5 The test statistic ist 6 and the value of test statistic (6) is t T5 o we reject H at 5% level of 777 significance And the observed data strongly support that Y and X are strongly and positively linearly correlated Q3ci We first consider the 99% prediction interval of Y X which is given by [( Y X ) t (6) 5 ( X ) n xx ( 56786) [ ] [7745,4355] ince the maximum (minimum) of X has value 34 (975), so it seems quite reliable to predict Y given X= However, the explanatory power of this prediction interval is another matter we need to consider Q3cii The parameter 54 represents the expected selling price of a house in thousands dollars when the size in square feet is zero or E ( Y X ) However this is impossible in practice as the size of a house cannot be of zero square feet The parameter 4 represents the expected increase of the expected selling price of a house in thousands dollars when the size in square feet is increase by That means when size in square feet is increased by, the expected increase of the expected selling price of a house is increased by 4 thousands dollars Q3ciii The required test is H E( Y X 5) 35 V H E( Y X 5) 35 at 5 Y 35 The test statistic ist, (5 5679) 8 xx where s E and xx var( X) The test 6 6 statistic follows tudent-t distribution with 6 degrees of freedom Now the observed E ( Y X 5) , so the value of test (6) statistic ist t5 (5 5679) ] 4

5 8 TUNG, Yik-Man Hence H E( Y X 5) 35 is rejected at 5% level of significance and the observed data supports what the researcher s claim Q3d Using Excel, the regression result is as follows o the estimated multiple regression model isy 4447 X46X 59X3 represents the change in expected selling price of a house in thousands dollars if there is one more room in the house, keeping everything remains unchanged Q3e Q3f Assume that everything else remains the same except the X is changed from 6 to 9 Then the change in X is 3, so the change in mean selling price of a house in thousands of dollars is E ( Y X 3) That means the change in mean selling price of a house is ,367 8 dollars tep (tarts from the original model which includes X, X, X3 and Intercept) ince X has largest p-value, ie 5686 > (based on LTAY), so we remove X in the next step tep (tarts from the model which removed X and includes X, X3 and Intercept) ince no more variable has p-value larger than (based on LTAY), so we stop o based on backward elimination method, we conclude that the best regression is Y X4358X3 5