# One manufacturer has developed a quantitative index of the​ "sweetness" of orange juice.

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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 accompanying table. 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 5.1 220 2 5.4 229 3 6.1 259 4 5.9 210 5 5.7 226 6 5.9 214 7 5.9 232 8 5.5 267 9 5.5 240 10 5.9 211 11 5.5 411 12 5.6 257 13 5.8 307 14 5.5 260 15 5.3 286 16 5.4 385 17 5.7 271 18 5.4 261 19 5.6 229 20 5.3 262 21 5.9 234 22 5.8 219 23 5.8 244 24 5.9 239

a) Find the least squares line for the data.

b) Interpret B0 and B1 in the words of the problem.

c)   Predict the sweetness index if the amount of pectin in the orange juice is 400 ppm. Excel Output from Analysis Toolpak is presented to answer this Question.

 Regression Statistics Multiple R 0.307 R Square 0.094 Adjusted R Square 0.053 Standard Error 0.247 Observations 24 ANOVA df SS MS F p Regression 1 0.139 0.139 2.284 0.145 Residual 22 1.339 0.061 Total 23 1.478 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 6.043 0.270 22.381 0.000 5.483 6.602 Pectin_(ppm) -0.002 0.001 -1.511 0.145 -0.004 0.001

The estimated regression equation;- y^   = 6.043 + (-0.002) * Pectin_(ppm)

b) The regression coefficient beta0 does not have a practical interpretation.

c) The regression coefficient beta1 is the estimated increase (or decrease) in sweetness index for each 1-unit increase in pectin.

In this case, sweetness decreases by -0.002 units when the pectin content is increased by one unit.

The simple regression model is used to estimate juice sweetness when the pectin in Juice is 400 ppm

y^   = 6.043 + (-0.002) * 400  =  5.243