HW2 - Solutions

HW2 - Solutions - Stat 102 Spring 2012 Homework#2 Solutions...

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1 Stat 102 – Spring 2012 Homework #2 Solutions 3.4 a) Based on the original information, we have: 1 . 831 5 . 192 153 6 . 138 120 98 69 60 7 . 100 5 . 17 3 . 15 4 . 14 5 . 11 12 476 121 100 81 64 49 36 25 56 10 9 8 7 6 5 2 = + + + + + + = = + + + + + + = = + + + + + + = = + + + + + + = i i i i i y x y x x Hence, using Equations (3.3) and (3.4), the estimates 0 b and 1 b are 1 . 7 7 28 5 . 7 7 . 100 9107 . 0 5 . ) ( 7 1 476 ) 7 . 100 )( ( 7 1 1 . 831 ) ( 7 1 7 1 1 0 2 2 2 1 = Ο Π Ξ Μ Ν Λ Ο Π Ξ Μ Ν Λ = = = = = x b y b x x y x y x b i i i i i i This gives us a least-squares regression line of x y 9107 . 0 1 . 7 ˆ + = , and thus, we can get y y ˆ 2 ) ˆ ( y y 12 6535 . ) 5 )( 9107 . 0 ( 1 . 7 = + 0.1201 11.5 5642 . ) 6 9107 . 0 ( 1 . 7 = + 1.1325 14 4749 . 13 ) 7 9107 . 0 ( 1 . 7 = + 0.2757 15 3856 . ) 8 )( 9107 . 0 ( 1 . 7 = + 0.3775 15.4 2963 . ) 9 )( 9107 . 0 ( 1 . 7 = + 0.0108 15.3 207 . 16 ) )( 9107 . 0 ( 1 . 7 = + 0.8226 17.5 1177 . ) )( 9107 . 0 ( 1 . 7 = + 0.1462 Using the last column of this table, we can calculate 7597 . 0 ) 1462 . 0 8226 . 0 0108 . 0 3775 . 0 2757 . 0 1325 . 1 1201 . 0 ( 2 7 1 + + + + + + = e s . Also, since 2 2 2 ) 1 ( x n x s n i x = , this gives us 1436 . 0 7597 . 0 ) 8 )( 7 ( 476 1 ) 7597 . 0 ( ) 1 ( 1 2 2 1 = = x e b s n s s . With this out of the way, we can now proceed to testing the hypotheses 0 : 0 : 1 1 0 = β a H H
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2 At a 5% level of significance, our decision rule is Reject 0 H if 571 . 2 5 , 025 . 0 = > t t Do not reject 0 H if 571 . 2 t Our test statistic is 3419 . 6 1436 . 0 9107 . 0 1 1 = b s b t . Since the value of our test statistic (6.3419) is greater than the critical value (2.571), we will reject 0 H (interpretation in (b)). Alternately, we can fit the linear regression in JMP. The relevant part of the output is below. Parameter Estimates Term Estimate Std Error t Ratio Prob>|t| Intercept 7.1 1.183832 6.00 0.0018 Production 0.9107143 0.143561 6.34 0.0014 b) Since we rejected the null hypothesis in (a), we have enough evidence to conclude that a linear relationship with a non-zero slope does exist between production and overhead costs. c) We now test the hypotheses 1 : 1 : 1 1 0 = β a H H Since we are still using a 5% level of significance, our decision rule is the same decision rule we used in (a). Our test statistic, however, is now 6219 . 0 1436 . 0 1 9107 . 0 1 1 1 = b s b t . Since the absolute value (0.6219) of our test statistic is less than the critical value (2.571), we do not reject 0 H (interpretation in (d)). d) From (c), we do not have enough evidence to say that the slope of the line is not actually 1; i.e., the slope of the line is not significantly different from 1.
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