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Sweetness of orange juice. The quality of the orange juice produced by a manufacturer (e.g., Minute Maid, Tropicana) is constantly monitored. There are numerous sensory and chemical components that combine to make the best tasting orange juice. For example, 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 table on p. 102. 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
5.2
220
13
5.8
306
2
5.5
227
14
259
3
6
15
5.3
284
4
5.9
210
16
383
5
224
17
5.7
271
215
18
264
7
231
19
8
5.6
268
20
263
9
239
21
232
10
212
22
11
5.4
410
23
246
12
256
24
241
(a) Find the least squares line for the data.
(b) Interpret β^0 and β^1 in the words of the problem.
(c) Predict the sweetness index if amount of pectin in the orange juice is 300 ppm.
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