Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 13.4, Problem 51E
The article “Optimization of Surface Roughness in Drilling Using Vegetable-Based Cutting Oils Developed from Sunflower Oil” (Industrial Lubrication and Tribology, 2011: 271–276) gave the following data on x1 = spindle speed (rpm), x2 = feed rate (mm/rev), x3 = drilling depth (mm), and y 5 surface roughness (μm) when a semisynthetic cutting fluid was used:
| x1 | x2 | x3 | y | e* |
| 320 | .10 | 15 | 2.27 | −1.32 |
| 320 | .12 | 20 | 4.14 | 1.08 |
| 320 | .14 | 25 | 4.69 | 0.26 |
| 420 | .10 | 20 | 1.92 | −0.40 |
| 420 | .12 | 25 | 2.63 | −0.79 |
| 420 | .14 | 15 | 4.34 | 0.99 |
| 520 | .10 | 25 | 2.03 | 1.64 |
| 520 | .12 | 15 | 2.34 | 0.03 |
| 520 | .14 | 20 | 2.67 | −1.52 |
a. Here is partial Minitab output from fitting the model with x1, x2, and x3 as predictors (authors of the cited article used Minitab for this purpose):
| Predictor Constant | Coef | SE Coef | T | P |
| 0 . 099 | 1. 871 | 0 . 05 | 0 . 960 | |
| x1 | −0.006767 | 0.002231 | −3 . 03 | 0 . 029 |
| x2 | 45 . 67 | 11.16 | 4 . 09 | 0 . 009 |
| X3 | 0 . 01333 | 0 . 04463 | 0 .30 | 0 . 777 |
| S = 0.546589 R-Sq = 83.9% R-Sq (adj ) = 74.2 | 2% | |||
Does drilling depth provide useful information about roughness given that spindle speed and feed rate remain in the model?
b. Here is Minitab output from fitting the model with just x1 and x2 as predictors (the cited article made no mention of this model):
| Predictor Constant | Coef | SE Coef | T | P |
| 0.365 | 1.514 | 0 .24 | 0 . 817 | |
| x1 | −0 . 006767 | 0.002055 | −3 .29 | 0 . 017 |
| x2 | 45.67 | 10 .28 | 4 .44 | 0 . 004 |
| S = 0. 503400 R-Sq =83.6% R-Sq(adj) =78.1% | ||||
Carry out a test of model utility using α = .05.
c. Calculate and interpret a 95% CI for the population regression coefficient on x1.
d. The estimated standard deviation of the predicted Y when x1 = 400 and x2 = .125 is .180. Calculate a 95% CI for true average roughness under these circumstances.
e. The e* values that appear along with the data are from the regression of (b). Investigate model adequacy.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Q3) An experiment was carried out to investigate variation of solubility of chemical X in water. The
quantities in kg that dissolved in 1 liter at various temperatures are show in the table (1).
Table (1)
Temperature C Mass of X
2.1
2.6
2.9
3.3
15
20
25
30
35
4
50
5.1
70
7
Use the proper methods to answer the following questions:
a) Draw a scatter diagram to show the data.
b) Estimate the temperature based on the mass of X.
c) What quantity might be expected to dissolve at 42 C? Find the quantity that your cquation indicates
would dissolve at 10 C and comment on your answer.
The article "Influence of Freezing Temperature on Hydraulic Conductivity of Silty Clay" (J.
Konrad and M. Samson, Journal of Geotechnical and Geoenvironmental Engineering,
2000:180–187) describes a study of factors affecting hydraulic conductivity of soils. The
measurements of hydraulic conductivity in units of 108 cm/s (y), initial void ratio (x), and
thawed void ratio (x2) for 12 specimens of silty clay are presented in the following table.
y
1.01 1.12 1.04 1.30 1.01 1.04 0.955 1.15 1.23 1.28 1.23 1.30
0.84 0.88 0.85 0.95 0.88 0.86 0.85 0.89 0.90 0.94 0.88 0.90
X1
0.81 0.85 0.87 0.92 0.84 0.85 0.85 0.86 0.85 0.92 0.88 0.92
X2
Fit the model y = Bo + fix1 + e. For each coefficient, test the null hypothesis that it is
equal to 0.
Fit the model y = Bo + Bzx2 + e. For each coefficient, test the null hypothesis that it is
equal to 0.
Fit the model y = Bo + BzX1 + Bzxz + e. For each coefficient, test the null hypothesis that
it is equal to 0.
d. Which of the models in parts (a) to (c) is…
Need help with (c) and (d)
Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. An article gave the accompanying data on x = fluid-flow velocity for a 5% soluble oil (cm/sec) and y = the extent of mist droplets having diameters smaller than 10 µm (mg/m3):
x
88
177
182
354
369
442
970
y
0.39
0.60
0.50
0.66
0.61
0.69
0.92
(a) The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy?
(b) What proportion of observed variation in mist can be attributed to the simple linear regression relationship between velocity and mist? (Round your answer to three decimal places.) (c) The investigators were particularly interested in the impact on mist of increasing velocity from 100 to 1000 (a…
Chapter 13 Solutions
Probability and Statistics for Engineering and the Sciences
Ch. 13.1 - Suppose the variables x = commuting distance and y...Ch. 13.1 - Prob. 2ECh. 13.1 - Prob. 3ECh. 13.1 - Prob. 4ECh. 13.1 - As the air temperature drops, river water becomes...Ch. 13.1 - The accompanying scatterplot is based on data...Ch. 13.1 - Prob. 7ECh. 13.1 - Prob. 8ECh. 13.1 - Consider the following four (x, y) data sets; the...Ch. 13.1 - a. Show that i=1nei=0 when the eis are the...
Ch. 13.1 - Prob. 11ECh. 13.1 - Prob. 12ECh. 13.1 - Prob. 13ECh. 13.1 - If there is at least one x value at which more...Ch. 13.2 - No tortilla chip aficionado likes soggy chips, so...Ch. 13.2 - Polyester fiber ropes are increasingly being used...Ch. 13.2 - The following data on mass rate of burning x and...Ch. 13.2 - Failures in aircraft gas turbine engines due to...Ch. 13.2 - Prob. 19ECh. 13.2 - Prob. 20ECh. 13.2 - Mineral mining is one of the most important...Ch. 13.2 - Prob. 22ECh. 13.2 - Prob. 23ECh. 13.2 - Kyphosis refers to severe forward flexion of the...Ch. 13.2 - Prob. 25ECh. 13.3 - The following data on y 5 glucose concentration...Ch. 13.3 - The viscosity (y) of an oil was measured by a cone...Ch. 13.3 - Prob. 29ECh. 13.3 - The accompanying data was extracted from the...Ch. 13.3 - The accompanying data on y 5 energy output (W) and...Ch. 13.3 - Prob. 32ECh. 13.3 - Prob. 33ECh. 13.3 - The following data resulted from an experiment to...Ch. 13.3 - The article The Respiration in Air and in Water of...Ch. 13.4 - Cardiorespiratory fitness is widely recognized as...Ch. 13.4 - A trucking company considered a multiple...Ch. 13.4 - Let y = wear life of a bearing, x1 = oil...Ch. 13.4 - Let y = sales at a fast-food outlet (1000s of ),...Ch. 13.4 - The article cited in Exercise 49 of Chapter 7 gave...Ch. 13.4 - The article A Study of Factors Affecting the Human...Ch. 13.4 - An investigation of a die-casting process resulted...Ch. 13.4 - Prob. 43ECh. 13.4 - The accompanying Minitab regression output is...Ch. 13.4 - The article Analysis of the Modeling Methodologies...Ch. 13.4 - A regression analysis carried out to relate y =...Ch. 13.4 - Efficient design of certain types of municipal...Ch. 13.4 - An experiment to investigate the effects of a new...Ch. 13.4 - Prob. 49ECh. 13.4 - Prob. 50ECh. 13.4 - The article Optimization of Surface Roughness in...Ch. 13.4 - Utilization of sucrose as a carbon source for the...Ch. 13.4 - Prob. 53ECh. 13.4 - Prob. 54ECh. 13.5 - The article The Influence of Honing Process...Ch. 13.5 - Prob. 56ECh. 13.5 - In the accompanying table, we give the smallest...Ch. 13.5 - Prob. 58ECh. 13.5 - Prob. 59ECh. 13.5 - Pillar stability is a most important factor to...Ch. 13.5 - Prob. 61ECh. 13.5 - Prob. 62ECh. 13.5 - Prob. 63ECh. 13.5 - Prob. 64ECh. 13 - Curing concrete is known to be vulnerable to shock...Ch. 13 - Prob. 66SECh. 13 - The article Validation of the Rockport Fitness...Ch. 13 - Feature recognition from surface models of...Ch. 13 - Air pressure (psi) and temperature (F) were...Ch. 13 - An aeronautical engineering student carried out an...Ch. 13 - An ammonia bath is the one most widely used for...Ch. 13 - The article An Experimental Study of Resistance...Ch. 13 - The accompanying data on x = frequency (MHz) and y...Ch. 13 - Prob. 74SECh. 13 - Prob. 75SECh. 13 - The article Chemithermomechanical Pulp from Mixed...Ch. 13 - Prob. 77SECh. 13 - Prob. 78SECh. 13 - Prob. 79SECh. 13 - Prob. 80SECh. 13 - Prob. 81SECh. 13 - Prob. 82SECh. 13 - Prob. 83SE
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Similar questions
- Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as determined by ester carboxyl band absorbance, improves the wrinkle resistance of the fabric (at the expense of reducing mechanical strength). The accompanying data on x = absorbance and y = wrinkle resistance angle was read from a graph in the paper "Predicting the Performance of Durable Press Finished Cotton Fabric with Infrared Spectroscopy".t 半 0.115 0.126 0.183 0.246 0.282 0.344 0.355 0.452 0.491 0.554 0.651 334 342 355 363 365 372 381 392 400 412 420 Here is regression output from Minitab: Predictor Coef SE Coef P Constant 321.878 2.483 129.64 0.000 absorb 156.711 6.464 24.24 0.000 S = 3.60498 R-Sq = 98.5% R-Są (adj) - 98.3% SOURCE DF MS F P Regression 1 7639.0 7639.0 587.81 0.000 Residual Error 9 117.0 13.0 Total 10 7756.0 (a) Does the simple linear regression model appear to be…arrow_forwardI need help with part (c) only, I need t statistic and p value. Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. An article gave the accompanying data on x = fluid-flow velocity for a 5% soluble oil (cm/sec) and y = the extent of mist droplets having diameters smaller than 10 µm (mg/m3): x 88 177 182 354 369 442 970 y 0.39 0.60 0.50 0.66 0.61 0.69 0.92 (a) The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy? Yes, a scatter plot shows a reasonable linear relationship.No, a scatter plot does not show a reasonable linear relationship. (b) What proportion of observed variation in mist can be attributed to the simple linear regression relationship between velocity and…arrow_forwardWrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as determined by ester carboxyl band absorbance, improves the wrinkle resistance of the fabric (at the expense of reducing mechanical strength). The accompanying data on x = absorbance and y = wrinkle resistance angle was read from a graph in the paper "Predicting the Performance of Durable Press Finished Cotton Fabric with Infrared Spectroscopy".t x 0.115 0.126 0.183 0.246 0.282 0.344 0.355 0.452 0.491 0.554 0.651 y 334 342 355 363 365 372 381 400 392 412 420 Here is regression output from Minitab: Predictor Constant absorb S = 3.60498 Coef 321.878 156.711 SOURCE Regression Residual Error Total R-Sq= 98.5% DF SE Coef 2.483 6.464 1 9 10 SS 7639.0 117.0 7756..0 T 129.64 24.24 P 0.000 0.000. R-Sq (adj) 98.3% MS 7639.0 13.0 F 587.81 (a) Does the simple linear regression model appear to be appropriate?…arrow_forward
- Accumulation of mercury (Hg) in fish is hypothesized to correlate with the fish size. Barbonymus schwanenfeldii is a species commonly found in a dam in Sarawak. Table 1 shows data of Hg concentration (mg/kg) present in three sizes of fish caught in the dam in triplicates. (i, ii, iii was already answered.)arrow_forwardThe article "Modeling Resilient Modulus and Temperature Correction for Saudi Roads" (H. Wahhab, I. Asi, and R. Ramadhan, Journal of Materials in Civil Engineering, 2001:298– 305) describes a study designed to predict the resilient modulus of pavement from physical properties. The following table presents data for the resilient modulus at 40°Cin10® kPa (y), the surface area of the aggregate in m²/kg (x1), and the softening point of the asphalt in °C (х). y X1 X2 1.48 5.77 60.5 1.70 7.45 74.2 2.03 8.14 67.6 2.86 8.73 70.0 2.43 7.12 64.6 3.06 6.89 65.3 2.44 8.64 66.2 1.29 6.58 64.1 3.53 9.10 68.6 1.04 8.06 58.8 1.88 5.93 63.2 1.90 8.17 62.1 1.76 9.84 68.9 2.82 7.17 72.2 1.00 7.78 54.1 The full quadratic model is y = + P,x, + PzX, + Pz*jXz + Pxx¡ + Bzx; + €. Which submodel of this full model do you believe is most appropriate? Justify your answer by fitting two or more models and comparing the results.arrow_forwardThe article “Wastewater Treatment Sludge as a Raw Material for the Production of Bacillus thuringiensis Based Biopesticides” (M. Tirado Montiel, R. Tyagi, and J. Valero, Water Research, 2001:3807–3816) presents measurements of total solids, in g/L, for seven sludge specimens. The results (rounded to the nearest gram) are 20, 5, 25, 43, 24, 21, and 32. Assume the distribution of total solids is approximately symmetric. a) Can you conclude that the mean concentration of total solids is greater than 14 g/L? Compute the appropriate test statistic and find the P-value. b) Can you conclude that the mean concentration of total solids is less than 30 g/L? Compute the appropriate test statistic and find the P-value. c) An environmental engineer claims that the mean concentration of total solids is equal to 18 g/L. Can you conclude that the claim is false?arrow_forward
- The efficiency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). The article “Statistical Process Control of a Phosphate Coating Line” (Wire J. Intl., May 1997: 78–81) gave the accompanying data on tank temperature (x) and efficiency ratio (y).Temp. 170 172 173 174 174 175 176Ratio .84 1.31 1.42 1.03 1.07 1.08 1.04Temp. 177 180 180 180 180 180 181Ratio 1.80 1.45 1.60 1.61 2.13 2.15 .84Temp. 181 182 182 182 182 184 184Ratio 1.43 .90 1.81 1.94 2.68 1.49 2.52Temp. 185 186 188Ratio 3.00 1.87 3.08a. Construct stem-and-leaf displays of both temperature and efficiency ratio, and comment on interesting features.b. Is the value of efficiency ratio completely and uniquely determined by tank temperature? Explain your reasoning.c. Construct a scatterplot of the data. Does it appear that efficiency ratio could be very well predicted by the value of temperature? Explain your reasoning.arrow_forwardComputer chips often contain surface imperfections.For a certain type of computer chip, theprobability mass function of the number of defects X is presented in the following table.arrow_forwardAn experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsións have been added during mixing) to that of unmodified mortar resulted in x = 18.11 kgf/cm2 for the modified mortar (m = 42) and y = 16.88 kgf/cm2 for the unmodified mortar (n = 31). Let ₁ and ₂ be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. (a) Assuming that o₁ = 1.6 and ₂ = 1.3, test Ho: ₁ - ₂ = 0 versus H₂: H₁ - H₂> 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) Z = P-value = State the conclusion in the problem context. O Fail to reject Ho. The data suggests that the difference in average tension bond strengths exceeds 0. Fail to reject Ho. The data does not suggest that the difference in average tension bond strengths…arrow_forward
- An experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in x = 18.11 kgf/cm² for the modified mortar (m = 42) and y = 16.82 kgf/cm² for the unmodified mortar (n = 30). Let μ₁ and μ₂ be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. (a) Assuming that ₁ = 1.6 and ₂ = 1.3, test Ho: ₁ - ₂ = 0 versus H₂ : ₁ - ₂ > 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. O Reject Ho. The data does not suggest that the difference in average tension bond strengths exceeds 0. O Fail to reject Ho. The data does not suggest that the difference in average tension bond strengths…arrow_forwardThe article "Two Different Approaches for RDC Modelling When Simulating a Solvent Deasphalting Plant" (J. Aparicio, M. Heronimo, et al., Computers and Chemical Engineering, 2002:1369–1377) reports flow rate (in dmh) and specific gravity measurements for a sample of paraffinic hydrocarbons. The natural logs of the flow rates (y) and the specific gravity measurements (x) are presented in the following table. х -1.204 0.8139 -0.580 0.8171 0.049 0.8202 0 673 0.8233 1.311 0.8264 1.959 0.8294 2.614 0.8323 3.270 0.8352 Fit the linear model y = Bo + B,x + ɛ. For each coefficient, test the hypothesis that the coefficient is equal to 0. Fit the quadratic model y = Bo + B,x + B2x? + E. For each coefficient, test the a. b. hypothesis that the coefficient is equal to 0. Fit the cubic model y = Bo + Bix + B2x² + Bax + E. For each coefficient, test the C. hypothesis that the coefficient is equal to 0. Which of the models in parts (a) through (c) is the most appropriate? Explain. Using the most…arrow_forwardAn experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in x = 18.18 kgf/cm2 for the modified mortar (m = 42) and y = 16.86 kgf/cm for the unmodified mortar (n = 30). Let µ1 and Hz be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. (a) Assuming that o1 = 1.6 and o2 = 1.3, test Ho: µ1 - 42 = 0 versus H3: µ1 – 42 > 0 at level 0.01. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = State the conclusion in the problem context. Fail to reject Ho: The data does not suggest that the difference in average tension bond strengths exceeds from 0. o Reject Ho: The data does not suggest that the difference in average tension bond…arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Correlation Vs Regression: Difference Between them with definition & Comparison Chart; Author: Key Differences;https://www.youtube.com/watch?v=Ou2QGSJVd0U;License: Standard YouTube License, CC-BY
Correlation and Regression: Concepts with Illustrative examples; Author: LEARN & APPLY : Lean and Six Sigma;https://www.youtube.com/watch?v=xTpHD5WLuoA;License: Standard YouTube License, CC-BY