When survey data indicated that a company needed to improve its package-sealing

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 ni=1nYi=14.6414.64,

Summation from i equals 1 to ni=1nUpper Y Subscript i Superscript 2Y2i=38.6848,

and Summation from i equals 1 to ni=1nXiYi=19.73.

Bag         Plate gap (X)      Tear rating (Y)

1              -0.3                        0.01

2              0.00                        0.08

3              1.50                        0.46

4              1.80                        0.82

5              0.30                        0.32

6              0.00                        0.35

7              0.00                        0.75

8              -0.30                      1.96

9              -0.30                      0.26

10           -1.50                      0.14

11           -1.50                      0.16

12           2.40                        3.94

13           -1.80                      0.02

14           -0.30                      0.59

15           -2.40                      0.03

16           -1.50                      0.11

17           2.10                        0.43

18           2.10                        4.12

19           0.30                        0.09

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.

Explanation / Answer

The statistical software output for this problem is:

Simple linear regression results:
Dependent Variable: Tear Rating (y)
Independent Variable: Plate gap (x)
Tear Rating (y) = 0.75378259 + 0.53021811 Plate gap (x)
Sample size: 19
R (correlation coefficient) = 0.61058199
R-sq = 0.37281037
Estimate of error standard deviation: 1.0055051

Parameter estimates:

Analysis of variance table for regression model:

Hence,

a) r2 = 0.3728

Meaning: Option D is correct.

b) Syx = 1.0055

c) Fairly close to 0.5; relatively small; fairly useful

Parameter Estimate Std. Err. Alternative DF T-Stat P-value Intercept 0.75378259 0.23073882 0 17 3.2668217 0.0045 Slope 0.53021811 0.16679596 0 17 3.1788426 0.0055

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