As always, show your work and follow the HW format. You may use Excel, but must show sample calculations.

Transcription

1 As always, show your work and follow the HW format. You may use Excel, but must show sample calculations. 1. Single Mean. A new roof truss is designed to hold more than 5000 pounds of snow load. You test 20 trusses and obtain a mean of 5025 pounds and standard deviation of 80 pounds. Use the 8-step method to determine if the mean is greater than 5000 pounds, at a significance of μ 2. H o : μ = 5000 lb 3. H 1 : μ > 5000 lb 4. α = TS = T (because unknown and n < 30) 6. Reject H o if TS>CV, where CV = t α,ν = t 0.05,19 = 1.73 [Table A.2 or T.INV(0.95,19)] 7. Calculate necessary values: = a. T o = X μ o = 1.40 s n < Fail to Reject H o at α = 0.05; the capacity of the roof truss appears to NOT be greater than 5000 lb.

2 2. Two Means. Two different types of tubing should have different maximum pressures. Use the sample data given below to assess this using the 8-step method, using a significance of Sample Parameter Tube 1 Tube 2 Size Mean, psi Standard Deviation, psi μ 1 μ 2 2. H o : μ 1 μ 2 = 0 3. H 1 : μ 1 μ α = TS = T* (because s unknown and unequal, and both n < 30) 6. Reject H o if TS >CV, where CV = t α/2,ν = t 0.025,36 = 2.03 [Table A.2 or T.INV(0.95,36] a. ν = (s1 2 n1 +s 2 2 n2 )2 ( s 1 2 n1 )2 ( s2 2 n1+1 + n2 )2 n2+1 2 = ( Calculate necessary values: a. T o = ( ) 0 = )0.5 ( )2 ( ) ( 25 ) = < Fail to Reject H o at α = 0.05; The pressure capacities are not significantly different.

3 3. Distribution Checking. Can parking spot pavement defects be predicted by the Poisson Distribution? You evaluate almost 1800 parking spots and observe the following. Defects in parking spot, d i Parking spots with d i defects, n i Use the 8-step method with a significance level of 0.1. r will be the average number of defects per parking spot. Determine r as the (total number of defects observed) / (total number of parking spots evaluated). Solution: Distribution Checking Table Count n i p i e i c i Sums Sample Calculations: r = ( )/1795 = Pr(X = x) = rx x! e r, e.g., for X = 0: e = and Pr(X 4) = 1 sum of 0 to 3 counts. 0! e i = p i 1795, e.g., for the first row, e 1 = = c i = (n i - e i ) 2 / e i, e.g., for the first row, c 1 = ( ) 2 / = C o equals the sum of the c i column = 19.8.

4 1. C (Stand in for all n i = e i ) 2. H o : C = 0 (All n i = e i ) 3. H 1 : C > 0 (At least one n i e i ) 4. α = TS = C 6. Reject H o if TS > CV, where ν =5-1-1=3, and CV = χ 2 0.1,3 = 6.25 [Table A.3 or CHISQ.INV.RT(0.1,3)] 7. Calculate necessary values: see Table, c o = Reject or Fail to Reject based on rejection equation: 19.8 > 6.25; Reject H o at significance level 0.1, the Poisson distribution does not appears to model the parking spot pavement defects.

5 4. Simple Linear Regression. Given the data below, use the 8-step method to determine if the overall linear relationship is significant at the 0.01 significance level. Use a TS > CV rejection region. You will need to calculate f o and f,u,v. X Y s xy s xx y i ss R ss T ss E X Y S xy S xx NA SS R SS T SS E NA F (Stand in for slope coefficients, β j ) 2. H o : F = 0 (All β j = 0) 3. H 1 : F > 0 (At least one β j 0) 4. α = TS = F 6. Reject H o if TS > CV, u = p - 1 = 2-1 and v = n - p = 5-2, CV = f 0.01,1,4 = 21.2 [Table A.5 or F.INV.RT(0.01,1,4) or F.INV(0.9,1,3)] 7. Calculate necessary values: TS = f o = MS R /MS E =(SS R /(p-1))/(ss E /(n-p))=(202.38/1)/(19.62/4) = Reject or Fail to Reject based on rejection equation: 41.3 > 21.2; Reject H o at significance level 0.01, a significant linear relationship exists between the dependent variable and the independent variable.

6 5. Multiple Linear Regression. Use A and B to predict Y. Use the Excel Data Analysis Add-in. A B Y Include the Excel Data Analysis Add-in output and use it to answer the following questions. (a) Use the 8-step method to determine if the overall linear relationship is significant at the 0.01 significance level. Use a p-value < rejection region. (b) Use the 8-step method to determine if 1 is not equal to zero (at the 0.01 significance level). Use a p-value < rejection region. (c) Use the 8-step method to determine if 2 is not equal to zero (at the 0.01 significance level). Use a p-value < rejection region. ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Intercept X Variable X Variable (a) 1. F (Stand in for slope coefficients, β j ) 2. H o : F = 0 (All β j = 0) 3. H 1 : F > 0 (At least one β j 0) 4. α = TS = F 6. Reject H o if P-Value < α, from Table P-Value = Calculate necessary values: NA 8. Reject or Fail to Reject based on rejection equation: < 0.01; Reject H o at significance level 0.01, a significant linear relationship exists between the dependent variable and the independent variables.

7 (b) 1. β 1 2. H o : β 1 = 0 3. H o : β α = T 6. Reject H o if p-value < α, where p-value is determined from the Excel Data Analysis Addin Table given above. p-value = (From Table above) 7. Calculate necessary values: NA 8. Reject or Fail to Reject H o : > 0.01; Fail to Reject H o at α = 0.05, β 1 is not significantly different from zero. (c) 1. β 2 2. H o : β 2 = 0 3. H o : β α = T 6. Reject H o if p-value < α where p-value is determined from the Excel Data Analysis Add-in Table given above. p-value = (From Table above) 7. Calculate necessary values: NA 8. Reject or Fail to Reject H o : < 0.05; Reject H o at α = 0.01, β 2 is significantly different from zero.