# Science

**HW5**

1. **Definitions and explanations**

a. What are the main and supporting objectives of Experimental Design?

b. Explain the fundamental difference between experimentation and what is referred to in the engineering as testing. Why do you think experimentation is more useful than testing to the product design process?

c. What is the difference between a fixed effects model and random effects model?

d. Explain the concept of interaction between factors.

e. What are the features of a 2k and 2k-2 experiments and how is it constructed?

**Single Factor ANOVA**

2. The tensile strength of portland cement is being studied. Four different mixing techniques can be used economically. The following data have been collected:

Mixing Technique | Tensile Strength (lb/in2) | |||

1 | 3129 | 3000 | 2865 | 2890 |

2 | 3200 | 3300 | 2975 | 3150 |

3 | 2800 | 2900 | 2985 | 3050 |

4 | 2600 | 2700 | 2600 | 2765 |

a. Test the hypothesis that mixing techniques affect the strength of the cement. Use *(* = 0.05.

b. Use the Tukeys method with alpha=0.05 to make comparisons between pairs of means.

c. Construct a normal probability plot of the residuals. What conclusion would you draw about the validity of the normality assumption

d. Find a 95 percent confidence interval on the mean tensile strength of the portland cement produced by each of the four mixing techniques.

3. Four different feed rates were investigated in an experiment on a CNC machine producing a component part used in an aircraft auxiliary power unit. The manufacturing engineer in charge of the experiment knows that a critical part dimension of interest may be affected by the feed rate. However, prior experience has indicated that only dispersion effects are likely to be present. That is, changing the feed rate does not affect the average dimension, but it could affect dimensional variability. The engineer makes five production runs at each feed rate and obtains the standard deviation of the critical *dimension (in 10-3 mm). The data are shown below. Assume that all runs were made in* random order.

Feed Rate | Production | Run | |||

(in/min) | 1 | 2 | 3 | 4 | 5 |

10 | 0.09 | 0.10 | 0.13 | 0.08 | 0.07 |

12 | 0.06 | 0.09 | 0.12 | 0.07 | 0.12 |

14 | 0.11 | 0.08 | 0.08 | 0.05 | 0.06 |

16 | 0.19 | 0.13 | 0.15 | 0.20 | 0.11 |

a. Does feed rate have any effect on the standard deviation of this critical dimension?

b. Use the residuals from this experiment of investigate model adequacy. Are there any problems with experimental validity? The residual plots are satisfactory.

c. Use the Tukeys method with *(* =0.05 to make comparisons between pairs of means

**Two Factors ANOVA**

4. The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data follow:

Pressure | |||

Temperature | 200 | 215 | 230 |

150 | 90.4 | 90.7 | 90.2 |

90.2 | 90.6 | 90.4 | |

160 | 90.1 | 90.5 | 89.9 |

90.3 | 90.6 | 90.1 | |

170 | 90.5 | 90.8 | 90.4 |

90.7 | 90.9 | 90.1 |

a. Analyze the data and draw conclusions. Use *(* = 0.05.

b. Prepare appropriate residual plots and comment on the model’s adequacy.

c. Under what conditions would you operate this process?

5. An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment and obtains the following data:

Depth of | Cut (in) | |||

Feed Rate (in/min) | 0.15 | 0.18 | 0.20 | 0.25 |

74 | 79 | 82 | 99 | |

0.20 | 64 | 68 | 88 | 104 |

60 | 73 | 92 | 96 | |

92 | 98 | 99 | 104 | |

0.25 | 86 | 104 | 108 | 110 |

88 | 88 | 95 | 99 | |

99 | 104 | 108 | 114 | |

0.30 | 98 | 99 | 110 | 111 |

102 | 95 | 99 | 107 |

a. Analyze the data and draw conclusions. Use *(* = 0.05.

b. Prepare appropriate residual plots and comment on the model’s adequacy.

c. Obtain point estimates of the mean surface finish at each feed rate.

**The 2 k Factorial Design and Hypothesis Testing**

6. An engineer is interested in the effects of cutting speed (*A*), tool geometry (*B*), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. Use *(* = 0.05. The results follow:

Treatment | Replicate | |||||

A |
B |
C |
Combination | I | II | III |

– | – | – | (1) | 22 | 31 | 25 |

+ | – | – | a |
32 | 43 | 29 |

– | + | – | b |
35 | 34 | 50 |

+ | + | – | ab |
55 | 47 | 46 |

– | – | + | c |
44 | 45 | 38 |

+ | – | + | ac |
40 | 37 | 36 |

– | + | + | bc |
60 | 50 | 54 |

+ | + | + | abc |
39 | 41 | 47 |

a. Estimate the factor effects. Which effects appear to be large?

b. Use the analysis of variance to confirm your conclusions for part (a).

c. Write down a regression model for predicting tool life (in hours) based on the results of this experiment.

d. Analyze the residuals. Are there any obvious problems?

e. Based on the analysis of main effects and interaction plots, what levels of *A*, *B*, and *C* would you recommend using?

7. The following are the burning times (in minutes) of chemical flares of two different formulations. The design engineers are interested in both the means and variance of the burning times.

Type 1 | Type 2 | ||

65 | 82 | 64 | 56 |

81 | 67 | 71 | 69 |

57 | 59 | 83 | 74 |

66 | 75 | 59 | 82 |

82 | 70 | 65 | 79 |

a. Test the hypotheses that the two variances are equal. Use *SYMBOL 97 \f “Symbol”* = 0.05.

b. Using the results of (a), test the hypotheses that the mean burning times are equal. Use *SYMBOL 97 \f “Symbol”* = 0.05. What is the* P*-value for this test?

c. Discuss the role of the normality assumption in this problem. Check the assumption of normality for both types of flares.

8. Analyze the following data. Use *(* = 0.05.

A | B | C | D | E | Response |

-1 | -1 | -1 | -1 | 1 | 38.9 |

1 | -1 | -1 | -1 | -1 | 35.3 |

-1 | 1 | -1 | -1 | -1 | 36.7 |

1 | 1 | -1 | -1 | 1 | 45.5 |

-1 | -1 | 1 | -1 | -1 | 35.3 |

1 | -1 | 1 | -1 | 1 | 37.8 |

-1 | 1 | 1 | -1 | 1 | 44.3 |

1 | 1 | 1 | -1 | -1 | 34.8 |

-1 | -1 | -1 | 1 | -1 | 34.4 |

1 | -1 | -1 | 1 | 1 | 38.4 |

-1 | 1 | -1 | 1 | 1 | 43.5 |

1 | 1 | -1 | 1 | -1 | 35.6 |

-1 | -1 | 1 | 1 | 1 | 37.1 |

1 | -1 | 1 | 1 | -1 | 33.8 |

-1 | 1 | 1 | 1 | -1 | 36 |

1 | 1 | 1 | 1 | 1 | 44.9 |

a. Determine if there are any outlier data points. Comment on the technique used to make the assessment

b. Determine what factors are statistically significant

c. Determine if there are any two factors interactions and what combinations from any interactions give the greatest result

d. Write down the model equation

e. If B high was 40 Volts and B low were 30. Determine from the model equation the expected output at 32 volts

9. The diameters of steel shafts produced by a certain manufacturing process should have a mean diameter of 0.255 inches. The diameter is known to have a standard deviation of *SYMBOL 115 \f “Symbol” *= 0.0001 inch. A random sample of 10 shafts has an average diameter of 0.2545 inches.

a. Set up the appropriate hypotheses on the mean *SYMBOL 109 \f “Symbol”*.

b. Find the *P*-value for this test.

c. Construct a 95 percent confidence interval on the mean shaft diameter.

10. Given the data below for readings by 2 appraisers on 4 parts with 3 trials,

a. Determine if the measurement system is acceptable.

b. What are the reasons for low Repeatability and Poor Reproducibility?

Part Number

1 2 3 4

Appraiser A

Trial 1 0.55 0.45 0.60 0.20

Trial 2 0.55 0.40 0.60 0.30

Trial 3 0.50 0.35 0.55 0.25

Appraiser B

Trial 1 0.55 0.35 0.60 0.15

Trial 2 0.50 0.30 0.55 0.20

Trial 3 0.45 0.25 0.50 0.15

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