Introduction to Linear Regression Analysis. Douglas C. Montgomery

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      and

      (2.10)

Thus, a convenient way to write Eq. (2.7) is

(2.11)

      The difference between the observed value yi and the corresponding fitted value is a residual. Mathematically the ith residual is

      (2.12)

      Residuals play an important role in investigating model adequacy and in detecting departures from the underlying assumptions. This topic is discussed in subsequent chapters.

Example 2.1 The Rocket Propellant Data

A rocket motor is manufactured by bonding an igniter propellant and a sustainer propellant together inside a metal housing. The shear strength of the bond between the two types of propellant is an important quality characteristic. It is suspected that shear strength is related to the age in weeks of the batch of sustainer propellant. Twenty observations on shear strength and the age of the corresponding batch of propellant have been collected and are shown in Table 2.1. The scatter diagram, shown in Figure 2.1, suggests that there is a strong statistical relationship between shear strength and propellant age, and the tentative assumption of the straight-line model y = β₀ + β₁x + ε appears to be reasonable.

       TABLE 2.1 Data for Example 2.1

Observation, i	Shear Strength, yi (psi)	Age of Propellant, xi (weeks)
1	2158.70	15.50
2	1678.15	23.75
3	2316.00	8.00
4	2061.30	17.00
5	2207.50	5.50
6	1708.30	19.00
7	1784.70	24.00
8	2575.00	2.50
9	2357.90	7.50
10	2256.70	11.00
11	2165.20	13.00
12	2399.55	3.75
13	1779.80	25.00
14	2336.75	9.75
15	1765.30	22.00
16	2053.50	18.00
17	2414.40	6.00
18	2200.50	12.50
19	2654.20	2.00
20	1753.70	21.50

Figure 2.1 Scatter diagram of shear strength versus propellant age, Example 2.1.

      To estimate the model parameters, first calculate

      and

Therefore, from Eqs. (2.11) and (2.6), we find that

      and

       TABLE 2.2 Data, Fitted Values, and Residuals for Example 2.1

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Observed Value, yi	Fitted Value,	Residual, ei
2158.70	2051.94	106.76
1678.15	1745.42	−67.27
2316.00	2330.59	−14.59
2061.30	1996.21	65.09
2207.50	2423.48	−215.98
1708.30	1921.90	−213.60
1784.70	1736.14	48.56
2575.00	2534.94	40.06