Chapter 12.4, Problem 46E

Astringency is the quality in a wine that makes the wine drinker’s mouth feel slightly rough, dry, and puckery. The paper “Analysis of Tannins in Red Wine Using Multiple Methods: Correlation with Perceived Astringency” (Amer. J. of Enol. and Vitic., 2006: 481–485) reported on an investigation to assess the relationship between perceived astringency and tannin concentration using various analytic methods. Here is data provided by the authors on x = tannin concentration by protein precipitation and y = perceived astringency as determined by a panel of tasters.

x	.718	.808	.924 1.000	.667	.529	.514	.559
y	.428	.480	.493 .978	.318	.298	−.224	.198

x	.766	.470	.726	.762	.666	.562	.378	.779
y	.326	−.336	.765	.190	.066	−.221	−.898	.836

x	.674	.858	.406	.927	.311	.319	.518	.687
y	.126	.305	−.577	.779	−.707	−.610	−.648	−.145

x	.907	.638	.234	.781	.326	.433	.319	.238
y	1.007	−.090	−1.132	.538	−1.098	−.581	−.862	-.551

Relevant summary quantities are as follows:

$\begin{array}{l}\sum x_i=19.404,\sum y_i=-.549,\sum x_i^2=13.248032,\\ \sum y_i^2=11.835795,\sum x_i{y}_i=3.497811\\ S_{xx}=13.248032-{(19.404)}^2/32=1.48193150,\\ S_{yy}=11.82637622\\ S_{xy}=3.497811-(19.404)(-.549)/32\\ \text{ = 3}\text{.83071088}\end{array}$

1. a. Fit the simple linear regression model to this data. Then determine the proportion of observed variation in astringency that can be attributed to the model relationship between astringency and tannin concentration.
2. b. Calculate and interpret a confidence interval for the slope of the true regression line.
3. c. Estimate true average astringency when tannin concentration is .6, and do so in a way that conveys information about reliability and precision.
4. d. Predict astringency for a single wine sample whose tannin concentration is .6, and do so in a way that conveys information about reliability and precision.
5. e. Does it appear that true average astringency for a tannin concentration of .7 is something other than 0? State and test the appropriate hypotheses.