Question

Number of Components	Inspection Time
33	85
14	50
7	31
18	59
16	52
12	41
24	72
43	100
6	21
12	42
18	64
8	25
31	79
13	49
12	30
20	62
18	52
20	59
24	73
43	101
17	59
13	45
22	67
13	45
24	69

a-1. Estimate the linear, quadratic, and cubic regression models. Report the Adjusted R² for each model. (Round answers to 4 decimal places.)

a-2. Which model has the best fit?

• Linear model

• Quadratic model

• Cubic model

b. Use the best model to predict the time required to inspect a device with 37 components. (Round coefficient estimates to at least 4 decimal places and final answer to 2 decimal places.)

Answer 1

We use the following data to run the regression.

Inspection Time (t)	Number of Components (x)	X^2	X^3
85	33	1089	35937
50	14	196	2744
31	7	49	343
59	18	324	5832
52	16	256	4096
41	12	144	1728
72	24	576	13824
100	43	1849	79507
21	6	36	216
42	12	144	1728
64	18	324	5832
25	8	64	512
79	31	961	29791
49	13	169	2197
30	12	144	1728
62	20	400	8000
52	18	324	5832
59	20	400	8000
73	24	576	13824
101	43	1849	79507
59	17	289	4913
45	13	169	2197
67	22	484	10648
45	13	169	2197
69	24	576	13824

a-1) We obtain the following results -

For linear regression:

Regression Statistics
Multiple R	0.97
R Square	0.94
Adjusted R Square	0.93
Standard Error	5.40
Observations	25.00
ANOVA
	df	SS	MS	F	Significance F
Regression	1.00	9849.22	9849.22	338.20	0.00
Residual	23.00	669.82	29.12
Total	24.00	10519.04
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	17.52	2.42	7.25	0.00	12.52	22.52	12.52	22.52
Number of Components (x)	2.07	0.11	18.39	0.00	1.83	2.30	1.83	2.30

For quadratic regression -

Regression Statistics
Multiple R	0.90
R Square	0.81
Adjusted R Square	0.80
Standard Error	9.31
Observations	25.00
ANOVA
	df	SS	MS	F	Significance F
Regression	1.00	8524.06	8524.06	98.27	0.00
Residual	23.00	1994.98	86.74
Total	24.00	10519.04
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	39.56	2.58	15.33	0.00	34.22	44.90	34.22	44.90
X^2	0.04	0.00	9.91	0.00	0.03	0.05	0.03	0.05

For cubic regression -

Regression Statistics
Multiple R	0.83
R Square	0.68
Adjusted R Square	0.67
Standard Error	12.01
Observations	25.00
ANOVA
	df	SS	MS	F	Significance F
Regression	1.00	7204.19	7204.19	49.99	0.00
Residual	23.00	3314.85	144.12
Total	24.00	10519.04
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	46.59	2.84	16.42	0.00	40.72	52.46	40.72	52.46
X^3	0.00	0.00	7.07	0.00	0.00	0.00	0.00	0.00

a-2) Based on the R sqaure and adjusted r square, we see the linear model has the best fit.

a-3) To inspect a device with 37 components using the linear regression we use the following equation-

t = 17.52 + 2.07 * X = 17.52 + 2.07 * 37 = 94