Is there a relationship between the sweetness index and a

Question

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.