Question

When survey data indicated that a company needed to improve its​ package-sealing process, an experiment was conducted to determine the factors in the​ bag-sealing equipment that might be affecting the ease of opening the bags without tearing the inner liner of the bag. Data were collected on 19 bags and the plate gap on the​ bag-sealing equipment was used to predict the tear rating of a bag. The results are displayed in the accompanying table and the regression equation is the following. Complete parts​ (a) through​ (c).

ModifyingAbove Upper Y with caret Subscript iYi=0.75380.7538+0.5302Xi​,

with Summation from i equals 1 to n∑i=1nYi=14.6414.64​,

Summation from i equals 1 to n∑i=1nUpper Y Subscript i Superscript 2Y2i=38.6848​,

and Summation from i equals 1 to n∑i=1nXiYi=19.73.

Bag   Plate gap (X)  Tear rating (Y)
1   -0.3   0.03
2   -0.30   0.06
3   1.50   0.41
4   1.50   0.82
5   -0.30   0.36
6   0.00   0.37
7   0.30   0.75
8   0.00   1.98
9   0.00   0.24
10   -1.80   0.17
11   -1.80   0.13
12   2.40   3.72
13   -1.80   0.03
14   0.00   0.52
15   -2.70   0.01
16   -1.80   0.13
17   1.80   0.44
18   2.10   4.06
19   0.30   0.07

a. Determine the coefficient of​ determination, r2​, and interpret its meaning. (Fill in the Blank)

r2 = ___? ​(Round to four decimal places as​ needed.)

What is the meaning of the coefficient of​ determination? (Choose Below)

A. It measures the variability in the actual plate gap from the predicted plate gap.

B. It measures the variability in the actual tear rating from the predicted tear rating.

C. It is the proportion of the variation in the plate gap that is explained by the variability in the tear rating.

D. It is the proportion of the variation in the tear rating that is explained by the variability in the plate gap.

b. Determine the standard error of the estimate.

SYX = ___? ​(Round to four decimal places as​ needed.)

c. How useful do you think this regression model is for predicting the tear rating based on the plate gap in the​ bag-sealing equipment? (Choose Below)

Since the value of r2 is (fairly close to 0, fairly close to 1, fairly close to 0.5, equal to 1, equal to 0) and the value of SYX is (relatively large, relatively small) the regression model is (not very useful, fairly useful) for predicting the tear rating.

Answer 1

Solution

Let

x = plate gap and

y = tear rating of the bag.

Back-up Theory

The linear regression model: Y = β₀ + β₁X + ε, ………………………………..........................................................………..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_cap = β_0cap + β_1capX, …………….................................................……….(2)

where β_1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and β_0cap = Ybar – β_1cap.Xbar..………………............……….…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i …………………………….........................................................………………….….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i …………………………….........................................................………………….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² ………………………............................................................…………………..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² ……………………….............................................................………………..………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ……………..........................................................……………………………….(8)

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) ……........................................................………………………..……….. (9)

Coefficient of determination = r² …………………...........................................................………………………………….(10)

Estimate of σ² is given by s² = (Syy – β_1cap²Sxx)/(n - 2)…………………………………...........................................…..(11)

Interpretation of r²:

r² represents the proportion of the variation in the response variable that is explained by the variation

in the predictor variable and is called coefficient of determination............................................................... (12)

Now, to work out the solution,

Final answers are given below. Details of Calculations follow at the end.

Part (a)

Coefficient of​ determination, r2 = ​0.4038 [vide (9) and (10)] Answer 1

Vide (12), meaning of the coefficient of​ determination is:

It is the proportion of the variation in the tear rating that is explained by the variability in the plate gap.

Option D Answer 2

Part (b)

Standard error of the estimate.

SYX = 0.9501 [vide (11)] Answer 3

Part (c)

How useful is this regression model is for predicting the tear rating based on the plate gap in the​ bag-sealing equipment?

Since the value of r² is (fairly close to 0, fairly close to 1, fairly close to 0.5, equal to 1, equal to 0) and the value of S_YX is (relatively large, relatively small), the regression model is (not very useful, fairly useful) for predicting the tear rating. Answer 4

Details of Calculations

n	19	Vide
Xbar	-0.0474	4
ybar	0.7526	5
Sxx	38.56736842	6
Syy	25.74396842	7
Sxy	20.02436842	8
s^2	0.902777489	11
s	0.950146035
r	0.635493447	9
r^2	0.403851922	10

DONE