Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending, in MPa) were determined for a sample of concrete beams of a certain type, resulting in the following data (read from a graph in the article “Effects of Aggregates and Microfillers on the Flexural Properties of Concrete,” Magazine of Concrete Research, 1997: 81–98):
MOE | 29.8 | 33.2 | 33.7 | 35.3 | 35.5 | 36.1 | 36.2 |
Strength | 5.9 | 7.2 | 7.3 | 6.3 | 8.1 | 6.8 | 7.0 |
MOE | 36.3 | 37.5 | 37.7 | 38.7 | 38.8 | 39.6 | 41.0 |
Strength | 7.6 | 6.8 | 6.5 | 7.0 | 6.3 | 7.9 | 9.0 |
MOE | 42.8 | 42.8 | 43.5 | 45.6 | 46.0 | 46.9 | 48.0 |
Strength | 8.2 | 8.7 | 7.8 | 9.7 | 7.4 | 7.7 | 9.7 |
MOE | 49.3 | 51.7 | 62.6 | 69.8 | 79.5 | 80.0 |
Strength | 7.8 | 7.7 | 11.6 | 11.3 | 11.8 | 10.7 |
- a. Construct a stem-and-leaf display of the MOE values. and comment on any interesting features.
- b. Is the value of strength completely and uniquely determined by the value of MOE? Explain.
- c. Use the accompanying Minitab output to obtain the equation of the least squares line for predicting (strength from modulus of elasticity, and then predict strength for a beam whose modulus of elasticity is 40. Would you feel comfortable using the least squares line to predict strength when modulus of elasticity is 100? Explain.
Predictor | Coef | stdev | t-ratio | P |
Constant | 3.2925 | 0.6008 | 5.48 | 0.000 |
mod elas | 0.10748 | 0.01280 | 8.40 | 0.000 |
Analysis of Variance
SOURCE | DF | SS | MS | F | P |
Regression | 1 | 52.870 | 52.870 | 70.55 | 0.000 |
Error | 25 | 18.736 | 0.749 | ||
Total | 26 | 71.605 |
d. What are the values of SSE, SST. and the coefficient of determination? Do these values suggest that the simple linear regression model effectively describes the relationship between the two variables? Explain.
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Chapter 12 Solutions
Student Solutions Manual for Devore's Probability and Statistics for Engineering and the Sciences, 9th
- A study of the properties of metal plate-connected trusses used for roof support yielded the following observations on axial stiffness index (kips/in.) for plate lengths 4, 6, 8, 10, and 12 in: 4: 329.2 409.5 311.0 326.5 316.8 349.8 309.7 6: 425.1 347.2 361.0 404.5 331.0 348.9 381.7 8: 389.4 366.2 351.0 357.1 409.9 367.3 382.0 10: 341.7 452.9 461.4 433.1 410.6 384.2 362.6 12: 414.4 441.8 419.9 410.7 473.4 441.2 465.8 USE SALT Does variation in plate length have any effect on true average axial stiffness? State the relevant hypotheses using analysis of variance. O Ho: M₁ = H₂ = 13 = H4 = 1₂ H₂: all five μ's are unequal O Ho: My H₂ H3 ‡ M4 # M5 H₂: at least two μ's are equal O Ho: My # H₂ H3 # H4 # H5 H₂: all five us are equal = = o Hỏi khi là không = 3 = Mà khô H₂: at least two μ's are unequal Test the relevant hypotheses using analysis of variance with a = 0.01. Display your results in an ANOVA table. (Round your answers to two decimal places.) Sum of Squares Source Treatments Error…arrow_forwardA study of the properties of metal plate-connected trusses used for roof support yielded the following observations on axial stiffness index (kips/in.) for plate lengths 4, 6, 8, 10, and 12 in: 4: 321.2 409.5 311.0 326.5 316.8 349.8 309.7 6: 439.1 347.2 361.0 404.5 331.0 348.9 381.7 8: 390.4 366.2 351.0 357.1 409.9 367.3 382.0 10: 362.7 452.9 461.4 433.1 410.6 384.2 362.6 12: 402.4 441.8 419.9 410.7 473.4 441.2 465.8 USE SALT Does variation in plate length have any effect on true average axial stiffness? State the relevant hypotheses using analysis of variance. ○ Ho: H₁ = H₂ = H3 = H4=H5 Ha: all five u's are unequal O Ho: H₁ H₂ H3 H4 H5 Ha: all five μ's are equal Ho H₁ = ₂ = 3 = H4 = 5 H₂: at least two μ's are unequal Ho: H₁ H₂ H3 H4 H5 Ha: at least two μ's are equal Test the relevant hypotheses using analysis of variance with a = 0.01. Display your results in an ANOVA table. (Round your answers to two decimal places.) Mean Degrees of Sum of freedom Squares Squares Source Treatments…arrow_forwardValues of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending, in MPa) were determined for a sample of concrete beams of a certain type, resulting in the following data: МОЕ 29.9 33.3 33.6 35.4 35.6 36.0 36.3 36.4 37.6 37.6 Strength 6.0 7.1 7.4 6.2 8.2 6.9 6.9 7.7 6.7 6.6 МОЕ 38.8 38.9 39.7 40.9 42.7 42.7 43.4 45.5 46.1 47.0 Strength 6.9 6.4 7.8 9.1 8.3 8.8 7.7 9.8 7.3 7.6 МОЕ 48.1 49.2 51.8 62.5 69.9 79.6 80.1 Strength 9.8 7.7 7.6 11.5 11.2 11.7 10.8 Fitting the simple linear regression model to the n = 27 observations on x = modulus of elasticity and y = flexural strength given in the data above resulted in ŷ = 7.595, s; = 0.186 when x = 40 and ŷ = 9.706, = 0.263 %3D for x = 60. The simple linear regression model also resulted in a value of s = 0.9008. (a) Explain why s; is larger when x = 60 than when x = 40. The farther x is from…arrow_forward
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