Sweetness of orange juice. Refer to the simple linear regression of sweetness index y and amount of pectin, x , for n = 24 orange juice samples, presented in Exercise.
Sweetness of orange juice The quality of the orange juice produced by a manufacturer 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 is 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 during 24 production runs at a juice-manufacturing plant are shown in the table and saved in the OJUICE file. Suppose a manufacturer wants to use simple linear regression to predict the sweetness ( y ) from the amount of pectin ( x ).
a. Find the least squares line for the data.
b. Interpret in the words of the problem.
c. Predict the sweetness index if the amount of pectin in the orange juice is 300 ppm. [ Note: A measure of reliability of such a prediction is discussed in Section 9.6 .]
Run | Sweetness Index | Pectin (ppm) |
1 | 5.2 | 220 |
2 | 5.5 | 227 |
3 | 5.9 | 241 |
4 | 6.0 | 259 |
5 | 5.9 | 210 |
6 | 5.8 | 224 |
7 | 6.0 | 215 |
8 | 5.8 | 231 |
9 | 5.6 | 268 |
10 | 5.6 | 239 |
11 | 5.9 | 212 |
12 | 5.4 | 410 |
13 | 5.6 | 256 |
14 | 5.8 | 306 |
15 | 5.5 | 259 |
16 | 5.3 | 284 |
17 | 5.3 | 383 |
18 | 5.7 | 271 |
19 | 5.5 | 264 |
20 | 5.7 | 227 |
21 | 5.3 | 263 |
22 | 5.9 | 232 |
23 | 5.8 | 220 |
24 | 5.8 | 246 |
Note: The data in the table are authentic. For reasons of confidentiality, the name of the manufacturer cannot be disclosed.
The SPSS printout of the analysis is shown at the top of page 520. A 90% confidence interval for the mean sweetness index E ( y ) for each value of x is shown on the SPSS spreadsheet on the next page. Select an observation and interpret this interval.
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