Answered: Listed below are altitudes (in…

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

Listed below are altitudes (in thousands of feet) and outside temperatures (in degrees Fahrenheit) recorded by the author of our textbook while flying from New Orleans to Atlanta.
Altitude
3
10
14
22
28
31
33
Temp
57
37
24
-5
-30
-41
-54
Does a strong linear correlation exist between altitude and temperature? How do you know?
If a strong linear correlation exists, answer the following:
a.) Find a regression model (ax+b) for temperature as a function of altitude.
b.) If the altitude is 18,500 feet, what would be the expected temperature?
c.) If the temperature is at the freezing mark, what would be the expected altitude?

Expert Solution

Step 1

Use the given data to form excel table:

	X	Y	X*Y	X*X	Y*Y
	3	57	171	9	3249
	10	37	370	100	1369
	14	24	336	196	576
	22	-5	-110	484	25
	28	-30	-840	784	900
	31	-41	-1271	961	1681
	33	-54	-1782	1089	2916
Sum	141	-12	-3126	3623	10716

$\begin{array}{rcl} r & = & \frac{n (\sum x y) - \sum x \cdot \sum y}{\sqrt{[n \sum x^{2} - (\sum x)^{2}] [n \sum y^{2} - (\sum y)^{2}]}} \\ = & \frac{7 (- 3126) - (141) (- 12)}{\sqrt{[7 (3623) - (141)^{2}] [(7) (10716) - (- 12)^{2}]}} \\ \approx & - 0.9968 \end{array}$

There exists a strong linear negative correlation between altitude and temperature, because the value of correlation coefficient is very close to -1.

Step 2

	X	Y	X*Y	X*X
	3	57	171	9
	10	37	370	100
	14	24	336	196
	22	-5	-110	484
	28	-30	-840	784
	31	-41	-1271	961
	33	-54	-1782	1089
Sum	141	-12	-3126	3623

The regression equation is given by the formula Y = a + b*X.

Determine the value of a and b.

$\begin{array}{rcl} a & = & \frac{\sum Y \cdot \sum X^{2} - \sum X \cdot \sum X Y}{n \cdot \sum X^{2} - {(\sum X)}^{2}} \\ = & \frac{(- 12) \cdot (3623) - (141) \cdot (- 3126)}{7 \cdot (3623) - {(141)}^{2}} \\ \approx & 72.5 \end{array}$

$\begin{array}{rcl} b & = & \frac{n \cdot \sum X Y - \sum X \cdot \sum Y}{n \cdot \sum X^{2} - {(\sum X)}^{2}} \\ = & \frac{(7) \cdot (- 3126) - (141) \cdot (- 12)}{7 \cdot (3623) - {(141)}^{2}} \\ \approx & - 3.684 \end{array}$

The regression equation is Y = 72.5 - 3.684*X.

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