Answered: The following table shows, for eight…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.EA: Extended Application Using Extrapolation To Predict Life Expectancy

Problem 2EA

See similar textbooks

Related questions

Concept explainers

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:

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

Step by step

Solved in 4 steps

Knowledge Booster

$Conditional Probability, Decision Trees, and Bayes' Theorem$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Similar questions