Question

The following data represent the maximum wind speed (in knots) and atmospheric pressure (in millibars) for a random sample of hurricanes that originated in the Atlantic Ocean.
Atmospheric Pressure (mb) Wind Speed (knots) Atmospheric Pressure (mb) Wind Speed (knots)
993 50 1006 40
995 60 942 120
994 60 1002 40

Required:
a. Find the y-intercept of the least-squares regression line, treating atmospheric pressure as the explanatory variable (round to four decimal places.)
b. Find the slope of the least-squares regression line, treating atmospheric pressure as the explanatory variable (round to four decimal places.)
c. Is it reasonable to interpret the y-intercept of the least-squares regression line, treating atmospheric pressure as the explanatory variable? Why or why not?

Answer 1

Answer:

Step-by-step explanation:

X                  Y                 X²                              Y²                  XY

993              50               986049                 2500              49650

995              60               990025                 3600              59700

994              60               988036                  3600              59640

1006             40               1012036                 1600              40240

942             120               887364                 14400              113040

1002             40               1004004                 1600              40080

$\sum X: 5932$   $\sum Y : 370$     $\sum X^2 : 5867514$    $\sum Y^2 = 27300$    $\sum XY : 362350$

To determine the regression:

$Mean \ (X) = \dfrac{\sum X }{n} \\ \\ = \dfrac{5932}{6} \\ \\ = 988.67$

$Mean \ (Y) = \dfrac{\sum Y}{n} \\ \\ = \dfrac{370}{6} \\ \\ = 61.67$

Intercept $b_o = \dfrac{\sum YX *\sum X^2 - \sum X \sum Y}{n(\sum X^2) - (\sum X)^2}$

$=\dfrac{370(5867514) -(5932)(370)}{6(5867514) - (5932)^2}$

= 131760.9563

Slope $b_1 = \dfrac{n(\sum XY) -(\sum X *\sum Y) }{n(\sum X^2)-(\sum X)^2}$

$b_1 = \dfrac{6(362350) -(5932*370) }{6(5867514)-(5932)^2}$

$b_1 = -1.2600$

The regression line equation $Y = b_o +b_1X$

$Y = 131760.96 -1.2600 X$

We then make a comparison of the slope of the equation to y = mx+c

slope of the equation = -1.2600

the y-intercept corresponds to when X = 0, thus:

y-intercept = 131760.9563

Yes, it is reasonable to interpret the y-intercept of the regression line, Using atmospheric pressure as an explanatory variable due to the fact that:

X is the independent variable and Y exists as the dependent variable.