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)
|
Run
|
Sweetness Index
|
Pectin (ppm)
|
|
1
|
5.2
|
220
|
13
|
5.8
|
306
|
|
2
|
5.5
|
227
|
14
|
5.5
|
259
|
|
3
|
6
|
259
|
15
|
5.3
|
284
|
|
4
|
5.9
|
210
|
16
|
5.3
|
383
|
|
5
|
5.8
|
224
|
17
|
5.7
|
271
|
|
6
|
6
|
215
|
18
|
5.5
|
264
|
|
7
|
5.8
|
231
|
19
|
5.7
|
227
|
|
8
|
5.6
|
268
|
20
|
5.3
|
263
|
|
9
|
5.6
|
239
|
21
|
5.9
|
232
|
|
10
|
5.9
|
212
|
22
|
5.8
|
220
|
|
11
|
5.4
|
410
|
23
|
5.8
|
246
|
|
12
|
5.6
|
256
|
24
|
5.9
|
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.