MGB 203B Homework # LSD = 1 1

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1 MGB 0B Homework # 4.4 a α =.05: t = =.05 LSD = α /,n k t.05, 7 t α /,n k MSE + = = 4.8 n i n j 0 0 i =, j = i =, j = i =, j = Conclusion: µ differs from µ and µ. b C = ()/ =, α =.05, α = α E / C =.067: t α /,n k = t.008, 7 =.55 (from Excel) LSD = E t α /,n k MSE + = = 0.0 n i n j 0 0 i =, j = i =, j = i =, j = Conclusion: µ and µ differ. c q (k, ν = α ) q. 05 (,7).5 MSE ϖ = q α (k, ν) =.5 n g 700 = i =, j = i =, j = i =, j = Conclusion: µ and µ differ. 4.5 a = µ = µ H 0 : µ H : At least two means differ. 0(5.7) + 0(49.7) + 0(44.) x = =

2 SST = n j (x j x) = 0( ) +0( ) + 0( ) =,78 j s j SSE = ( n ) = (0 )(94.6) + (0 )(5.6) + (0 )(9.9) =,86 SST,78 Treatments k = SST =,78 = = k SSE,86 Error n k = 87 SSE =,86 = = n k 87 MST MSE = = F =.70, p-value =.086. There is enough evidence to infer that speed of promotion varies between the three sizes of engineering firms b q α (k, ν) = q. 05 (,87).40 ϖ =.40 = i =, j = i =, j = i =, j = The means of small and large firms differ. Answer (v) is correct. 4.0 a H 0 : µ = µ = µ = µ 4 = µ H : At least two means differ. A B C D E F G ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total F = 4.46, p-value =.007. There is enough evidence to infer that there are differences in the effect of the new assessment system between the five boroughs.

3 b Multiple Comparisons LSD Omega Treatment Treatment Difference Alpha = 0.05 Alpha = 0.05 Borough A Borough B Borough C Borough D Borough E Borough B Borough C Borough D Borough E Borough C Borough D Borough E Borough D Borough E The mean assessments in borough A differs from the means in boroughs B and C. c The assessments for each borough are required to be normally distributed with equal variances. d The histograms are approximately bell-shaped and the sample variances are similar. 4.7 Treatments Blocks Error Total 9,0 > 7 a Rejection region: F Fα,k,n k b+ = F.0,, = 4.60 Conclusion: F = 7.99, p-value = There is enough evidence to conclude that the treatment means differ. b Rejection region: F > Fα,b,n k b+ = F.0,9, 7 =.5 Conclusion: F = 6.05, p-value =.000. There is enough evidence to conclude that the block means differ. 4.7 a. k =, b = 5, Grand mean = 0.4 SS(Total) = k b i= ij ( x x) = (7 0.4) + (0 0.4) + ( 0.4) + (9 0.4) + ( 0.4)

4 k + ( 0.4) + (8 0.4) + (6 0.4) + ( 0.4) + (0 0.4) + ( 8 0.4) + (9 0.4) + ( 0.4) + (6 0.4) + ( 0.4) = 99.6 j = SST = b(x[t] x) = 5[(0 0.4) + (.8 0.4) + ( ) ] 5. 6 b i = SSB = k(x[b] x) = [(9 0.4) + (9 0.4) + (.7 0.4) + (9. 0.4) + ( 0.4) ] 48. i= SSE = SS(Total) SST SSB = = 5.7 b SS(Total) = k b i= ij ( x x) = (7 0.4) + (0 0.4) + ( 0.4) + (9 0.4) + ( 0.4) k + ( 0.4) + (8 0.4) + (6 0.4) + ( 0.4) + (0 0.4) + ( 8 0.4) + (9 0.4) + ( 0.4) + (6 0.4) + ( 0.4) = 99.6 j j = SST = n (x x) = 5(0 0.4) + 5(.8 0.4) + 5( ) 5. 6 SSE = SS(Total) SST = = 84.0 c The variation between all the data is the same for both designs. d The variation between treatments is the same for both designs. e Because the randomized block design divides the sum of squares for error in the one-way analysis of variance into two parts Treatments Blocks 9, Error a = µ = µ H 0 : µ H : At least two means differ. Rejection region: F > Fα,k,n k b+ = F.05,, 8. F =.86, p-value =.4. There is not enough evidence to conclude that there are differences in sales ability between the holders of the three degrees. b H 0 : µ = µ = = µ 0 H : At least two means differ. 4

5 F = 6.64, p-value = 0. There is sufficient evidence to indicate that there are differences between the blocks of students. The independent samples design would not be recommended. c The commissions for each type of degree are required to be normally distributed with the same variance. d The histograms are bell shaped and the sample variances are similar Factor A Factor B Interaction Error Total a Rejection region: F > Fα,(a )(b ),n ab F.05,6, 84 =.5 F =.9. There is not enough evidence to conclude that factors A and B interact. b Rejection region: F > Fα,a,n ab F.05,, 84 =.76 F =.7. There is not enough evidence to conclude that differences exist between the levels of factor A. c Rejection region: F > Fα,b,n ab F.05,, 84 =.5 F = There is enough evidence to conclude that differences exist between the levels of factor B a Detergents and temperatures b The response variable is the whiteness score. c There are a = 5 detergents and b = temperatures A B C D E F G ANOVA Source of Variation SS df MS F P-value F crit Sample Columns Interaction Within Total d Test for interaction: F =.78, p-value =.007. There is sufficient evidence to conclude that detergents and temperatures interact. The F-tests in Parts e and f are irrelevant. 5

6 4. H 0 : µ = µ = µ = µ 4 H : At least two means differ A B C D E F G ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total F = 9.7, p-value = 0. There is sufficient evidence to infer that there are differences in changes to the TSE depending on the loss the previous day. 5.5 H 0 : p =.05, p =.07, p =.04, p =.84 4 Cell i H : At least one pi is not equal to its specified value. f i e f e ) i ( i i ( f i ei ) / ei 9 50(.05) = (.07) = (.04) = (.84) = Total χ = 7.9 Rejection region: χ > χα,k = χ.05, = 7.8 χ = 7.9, p-value = There is enough evidence to infer that the reported side effects of the placebo differ from that of the cold remedy. 5.9 H 0 : The two variables economic option and political affiliation) are independent Cell i H : The two variables are dependent f i e f e ) i ( i i ( f i ei ) / ei 0 444()/000 = (557)/000 = (4)/000 = ()/000 =

7 5 67 0(557)/000 = (4)/000 = ()/000 = (557)/000 = (4)/000 = ()/000 = (557)/000 = (4)/000 = Total χ = Rejection region: α (r )(c.0, 6 χ > χ, = χ = 6.8 χ = , p-value = 0. There is sufficient evidence to infer that political affiliation affects support for economic options H 0 : The two variables (cold and group) are independent H : The two variables are dependent χ = 4.9, p-value =.468. There is not enough evidence to infer there are differences between the four groups. A B C D E Contingency Table Group Cold TOTAL TOTAL chi-squared Stat 4.9 df p-value chi-squared Critical