Answered: The following table shows, for eight…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

The following table shows, for eight vintages of select wine, purchases per buyer (y) and the wine buyer’s rating in a year (x):
x 3.6 3.3 2.8 2.6 2.7 2.9 2.0 2.6
y 24 21 22 22 18 13 9 6
a. Estimate the regression of purchases per buyer on the buyer’s rating.
b. Interpret the slope of the estimated regression line.
c. Find and interpret the coefficient of determination.
d. Find and interpret a 90% confidence interval for the slope of the population regression line.
e. Find a 90% confidence interval for expected purchases per buyer for a vintage for which the buyer’s rating is 2.0.

Expert Solution

Step 1

“Since you have posted a question with multiple sub-parts, we will solve first three subparts for you. To get the remaining sub-parts solved, please repost the complete question and mention the sub-parts to be solved.”

Given

The data is as follows:

x	y
3.6	24
3.3	21
2.8	22
2.6	22
2.7	18
2.9	13
2	9
2.6	6

Step 2

	x	y	x*y	x^2	y^2
	3.6	24	86.4	12.96	576
	3.3	21	69.3	10.89	441
	2.8	22	61.6	7.84	484
	2.6	22	57.2	6.76	484
	2.7	18	48.6	7.29	324
	2.9	13	37.7	8.41	169
	2	9	18	4	81
	2.6	6	15.6	6.76	36
Sum	22.5	135	394.4	64.91	2595

$\begin{array}{rcl} \bar{x} & = & \frac{\sum_{i = 1}^{n} x_{i}}{n} \\ = & \frac{22.5}{8} \\ = & 2.8125 \end{array}$

$\begin{array}{rcl} \bar{y} & = & \frac{\sum_{i = 1}^{n} y_{i}}{n} \\ = & \frac{135}{8} \\ = & 16.875 \end{array}$

$\begin{array}{rcl} S S_{X X} & = & \sum_{i = 1}^{n} {x_{i}}^{2} - \frac{1}{n} \times {(\sum_{i = 1}^{n} x_{i})}^{2} \\ = & 64.91 - \frac{22 . 5^{2}}{8} \\ = & 1.62875 \end{array}$

$\begin{array}{rcl} S S_{Y Y} & = & \sum_{i = 1}^{n} {y_{i}}^{2} - \frac{1}{n} \times {(\sum_{i = 1}^{n} y_{i})}^{2} \\ = & 2595 - \frac{135^{2}}{8} \\ = & 316.875 \end{array}$

$\begin{array}{rcl} S S_{X Y} & = & \sum_{i = 1}^{n} x_{i} y_{i} - \frac{1}{n} (\sum_{i = 1}^{n} x_{i}) (\sum_{i = 1}^{n} y_{i}) \\ = & 394.4 - 22.5 \times \frac{135}{8} \\ = & 14.7125 \end{array}$

$\begin{array}{rcl} β_{1} & = & \frac{S S_{X Y}}{S S_{X X}} \\ = & \frac{14.7125}{1.62875} \\ = & 9.033 \end{array}$

$\begin{array}{rcl} β_{0} & = & \bar{y} - \bar{x} \times β_{1} \\ = & 16.875 - 2.8125 \times 9.033 \\ = & - 8.5303 \end{array}$

The regression equation is as follows:

$y = β_{0} + β_{1} x \Rightarrow y = - 8.5303 + 9.033 x$

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