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One manufacturer has developed a quantitative index of the "sweetness" of orange juice. (The higher the index, the sweeter the juice). Is there a relationship between the sweetness index and a chemical measure such as the amount of water-soluble pectin (parts per million) in the orange juice? Data collected on these two variables for 24 production runs at a juice manufacturing plant are shown in the accompanying table. Suppose a manufacturer wants to use simple linear regression to predict the sweetness (y) from the amount of pectin (x).

Run Sweetness_Index Pectin_(ppm)
1 5.1 220
2 5.4 229
3 6.1 259
4 5.9 210
5 5.7 226
6 5.9 214
7 5.9 232
8 5.5 267
9 5.5 240
10 5.9 211
11 5.5 411
12 5.6 257
13 5.8 307
14 5.5 260
15 5.3 286
16 5.4 385
17 5.7 271
18 5.4 261
19 5.6 229
20 5.3 262
21 5.9 234
22 5.8 219
23 5.8 244
24 5.9 239

a) Find the least squares line for the data.

b) Interpret B0 and B1 in the words of the problem.

c)   Predict the sweetness index if the amount of pectin in the orange juice is 400 ppm.

 

 

Excel Output from Analysis Toolpak is presented to answer this Question. 

 

Regression Statistics          
Multiple R 0.307          
R Square 0.094          
Adjusted R Square 0.053          
Standard Error 0.247          
Observations 24          
             
ANOVA            
  df SS MS F p  
Regression 1 0.139 0.139 2.284 0.145  
Residual 22 1.339 0.061      
Total 23 1.478        
             
  Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 6.043 0.270 22.381 0.000 5.483 6.602
Pectin_(ppm) -0.002 0.001 -1.511 0.145 -0.004 0.001

The estimated regression equation;- y^   = 6.043 + (-0.002) * Pectin_(ppm)

b) The regression coefficient beta0 does not have a practical interpretation.

c) The regression coefficient beta1 is the estimated increase (or decrease) in sweetness index for each 1-unit increase in pectin.

In this case, sweetness decreases by -0.002 units when the pectin content is increased by one unit. 

The simple regression model is used to estimate juice sweetness when the pectin in Juice is 400 ppm

y^   = 6.043 + (-0.002) * 400  =  5.243

 

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