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 below. 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 β0 and β1 in the words of the problem.
c. Predict the sweetness index if amount of pectin in the orange juice is 300 ppm.
d. Find the values of SSE, s2, and s for this regression.
e. Explain why it is difficult to give a practical interpretation to s2.
f. Give a practical interpretation of the value of s.
g. Use the results of the regression to form a 95% confidence interval for the slope, b1. Interpret the result.
h. Find and interpret the coefficient of determination, r2, and the coefficient of correlation, r.
i. A 90% confidence interval for the predicted value of mean sweetness index, E(y), for run 1 is (5.649,5.838). Interpret this interval.