Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
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Textbook Question
Chapter 12.2, Problem 15E
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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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: 333.2 409.5 311.0 326.5 316.8 349.8 309.7
6: 433.1 347.2 361.0 404.5 331.0 348.9 381.7
8: 382.4 366.2 351.0 357.1 409.9 367.3 382.0
10: 350.7 452.9 461.4 433.1 410.6 384.2 362.6
12: 413.4 441.8 419.9 410.7 473.4 441.2 465.8
LUSE SALT
Does variation in plate length have any effect on true average axial stiffness? State the relevant hypotheses using analysis of variance.
O Hoi Hy #fly #Hz" Ha #Hs
H: all five μ's are equal
O Hoi H₂H₂ = H3 = HaHs
H: at least two μ's are unequal
O Hoi H₂ = H₂ = H₂ "HaHs
H: all five μ's are unequal
O Hoi H₂ #4₂ # Hz*H4 *H5
H: 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.)
Degrees of Sum of Mean
freedom Squares Squares
Error
Total…
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:
MOE
29.7
33.1
33.6
35.2
35.4
36.2
36.2
36.3
37.4
37.8
Strength
5.8
7.1
7.2
6.2
8.0
6.9
7.1
7.7
6.7
6.4
MOE
38.6
38.7
39.7
40.9
42.7
42.7
43.6
45.7
45.9
46.8
Strength
7.1
6.4
8.0
8.9
8.3
8.8
7.9
9.6
7.3
7.6
MOE
47.9
49.2
51.6
62.7
69.7
79.6
80.1
Strength
9.8
7.7
7.8
11.5
11.4
11.9
10.6
Fitting the simple linear regression model to then = 27 observations on x = modulus of elasticity and y = flexural strength given in the data above resulted in ý = 7.589, s, = 0.180 when x = 40 and ý = 9.749, s, = 0.255 for x = 60. The simple linear
regression model also resulted in a value of s = 0.8718.
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:
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…
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- 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: MOE 29.7 33.1 33.6 35.2 35.4 36.2 36.2 36.3 37.4 37.8 Strength 5.8 7.1 7.2 6.2 8.0 6.9 7.1 7.7 6.7 6.4 МОЕ 38.6 38.7 39.7 40.9 42.7 42.7 43.6 45.7 45.9 46.8 Strength 7.1 6.4 8.0 8.9 8.3 8.8 7.9 9.6 7.3 7.6 МОЕ 47.9 49.2 51.6 62.7 69.7 79.6 80.1 Strength 9.8 7.7 7.8 11.5 11.4 11.9 10.6 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 ý = 7.589, s, = 0.180 when x = 40 and ý = 9.749, s, = 0.255 for x = 60. The simple linear regression model also resulted in a value of s = 0.8718. (a) Explain why s, is larger when x = 60 than when x = 40. O The closer x is to…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: 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: 315.2 409.5 311.0 326.5 316.8 349.8 309.7 6: 405.1 347.2 361.0 404.5 331.0 348.9 381.7 8: 399.4 366.2 351.0 357.1 409.9 367.3 382.0 10: 353.7 452.9 461.4 433.1 410.6 384.2 362.6 12: 417.4 441.8 419.9 410.7 473.4 441.2 465.8 n 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: H1# H2 # Hz# H4# H5 H: at least two µ's are equal O Ho: H1 = H2 = H3= H4= H5 H: at least two u's are unequal O Ho: H1 # H2 # Hz# H4# Hs H: all five u's are equal O Ho: H1 = H2 = Hz3 = H4= Hs H: all five u,'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.) Degrees of freedom Sum of Squares Mean Source…arrow_forward
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