Statistics - Regression Projects

The processing of raw coal involves "washing," in which coal ash (non-organic, incombustible material)

Problem 1: The processing of raw coal involves "washing," in which coal ash (non-organic, incombustible material) is removed. The article "Quantifying Sampling Precision for Coal Ash Using Gy's Discrete Model of the Fundamental Error': (Journal of Coal Quality, 1989, 33-39) provides data relating the percentage of ash to the density of a coal particle. The average percentage ash for five densities of coal particles was measured. The data are presented in the following table:

Density (g/cm ³ )	Percent ash
1.25	1.93
1.325	4.63
1.375	8.95
1.45	15.05
1.55	23.31

Construct a scatterplot of percent ash (y) versus density (x). Verify that a linear model is appropriate.
Compute the least-squares line for predicting percent ash from density.
If two coal particles differed in density by 0.1 g/cm3, by how much would you predict their percent ash to differ?
Predict the percent ash for particles with density 1.40 g/cm3.
Compute the fitted values.
Compute the residuals. Which point has the residual with the largest magnitude?
Compute the correlation between density and percent ash.

Problem 2: The data are shown in the following table:

	Y	X
1	40	825
2	42	830
3	49	890
4	46	895
5	44	890
6	48	910
7	46	915
8	43	960
9	53	990
10	52	1010
11	54	1012
12	57	1030
13	58	1050

(a) Fit a simple linear regression model with y-green liquor Na2S concentration and x-production. Draw a scatter diagram of the data and the resulting least squares fitted model.

(b) Find the fitted value of y corresponding to x = 910 and the associated residual.

(d) Find a 99% confidence interval on mean Na2S concentration when production x = 910 tons/day.

Problem 3:

The following are measurements of the air velocity and evaporation coefficient of burning fuel droplets in an impulse engine:

Air velocity (cm / sec) x	Evaporation coefficient (mm ² / sec) y
20	0.18
60	0.37
100	0.35
140	0.78
180	0.56
220	0.75
260	1.18
300	1.36
340	1.17
380	1.65

(a) Find the best fit line

(b) Test ${{H}_{0}}:\beta =0$ against ${{H}_{1}}:\beta \ne 0$, for $\alpha =0.05$.

Problem 4:

The following are data on the number of twists required to break a certain kind of forged alloy bar and the percentages of two alloying elements present in the metal:

Number of twists — y	Percent of element A X1	Percent of element B x2
41	1	5
49	2	5
69	3	5
65	4	5
40	1	10
50	2	10
58	3	10
57	4	10
31	1	15
36	2	15
44	3	15
57	4	15
19	1	20
31	2	20
33	3	20
43	4	20

Find the least square regression plane and use the equation to estimate the number of twists required to break one of the bars when x ₁ = 2.5 and x ₂ = 12.

Problem 5: The pull strength of a wire bond is an important characteristic. The table below gives information on pull strength (y), die height (x1), post height (x2), loop height (x3), wire length (x4), bond width on the die (x5), and bond width on the post (x6).

y	x1	x2	x3	x4	x5	x6
8.0	5.2	19.6	29.6	94.9	2.1	2.3
8.3	5.2	19.8	32.4	89.7	2.1	1.8
8.5	5.8	19.6	31.0	96.2	2.0	2.0
8.8	6.4	19.4	32.4	95.6	2.2	2.1
9.0	5.8	18.6	28.6	86.5	2.0	1.8
9.3	5.2	18.8	30.6	84.5	2.1	2.1
9.3	5.6	20.4	32.4	88.8	2.2	1.9
9.5	6.0	19.0	32.6	85.7	2.1	1.9
9.8	5.2	20.8	32.2	93.6	2.3	2.1
10.0	5.8	19.9	31.8	86.0	2.1	1.8
10.3	6.4	18.0	32.6	87.1	2.0	1.6
10.5	6.0	20.6	33.4	93.1	2.1	2.1
10.8	6.2	20.2	31.8	83.4	2.2	2.1
11.0	6.2	20.2	32.4	94.5	2.1	1.9
11.3	6.2	19.2	31.4	83.4	1.9	1.8
11.5	5.6	17.0	33.2	85.2	2.1	2.1
11.8	6.0	19.8	35.4	84.1	2.0	1.8
12.3	5.8	18.8	34.0	86.9	2.1	1.8
12.5	5.6	18.6	34.2	83.0	1.9	2.0

Fit a multiple linear regression model using x2, x3, x4, and x5 as the regressors.
Use the model from part (a) to predict pull strength when x2 = 20, x3 = 30, x4= 90, and x5 = 2.0.

Problem 6:

We take a pair of dice, one red and one green, roll them a few times and get the following results:

Red die Green die

3 5

2 2

5 6

3 1

4 3

Test the null hypothesis for $\rho =0$, at $\alpha =0.05$

Problem 7: (a) Fit a multiple regression model using Y (amount of heat) as the response variable and the X’s as the response variables

(b) Remove the variables that are not enhancing the model. Use Prob. to enter = 0.1, alpha = 0.10

X ₁	X ₂	X ₃	X ₄	Y
7	29	6	6C	78
1	29	15	52	74
11	56	8	20	104
11	31	8	47	87
7	52	6	33	95
11	55	9	22	109
3	71	17	6	102
1	31	22	44	72
2	54	18	22	93
21	47	4	26	115
1	40	23	34	83
11	66	9	12	113
10	68	8	12	109

Price: $23.24

Solution: The downloadable solution consists of 14 pages, 924 words and 7 charts.
Deliverable: Word Document

The processing of raw coal involves "washing," in which coal ash (non-organic, incombustible material)

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