Anyone who has been outdoors on a summer evening has probably heard crickets. Did you know that it is possible to use the cricket as a thermometer? Crickets tend to chirp more frequently as temperatures increase. This phenomenon was studied in detail by George W. Pierce, a physics professor at Harvard. In the following data, x is a random variable representing chirps per second and y is a random variable representing temperature (°F). These data are also available for download at the online study center. X 20.0 16.0 19.8 18.4 17.1 15.5 14.7 17.1
Y 88.6 71.6 93.3 84.3 80.6 75.2 69.7 82.0

X 15.4 16.2 15.0 17.2 16.0 17.0 14.4
Y 69.4 83.3 79.6 82.6 80.6 83.5 76.3

Required:
a. Draw a scatter diagram displaying the data.
b. Find X- bar, Y- bar, a and b. Then find the equation of the least squares line y=a+bx.
c. Graph the least squares line on your scatter diagram. Be sure to use the point ( X-bar, Y- bar) as one of the points on the line.

a) attached below

b) X - bar = 16.65

Y - bar =  80.04

y = 25.26 + 3.29 ( X )

c) attached in diagram below

Step-by-step explanation:

n = 15

∑ x = 20.0 + 16.0 + 19.8 + 18.4 + 17.1 + 15.5 + 14.7 + 17.1 + 15.4 + 16.2 + 15.0 + 17.2  + 16.0 + 17.0 + 14.4  = 249.8

∑ y =  88.6 + 71.6 + 93.3 + 84.3 + 80.6 + 75.2 + 69.7 + 82.0 + 69.4 + 83.3 + 79.6 + 82.6 + 80.6 + 83.5 + 76.3  = 1200.6

∑ xy = ( 20 * 88.6 ) + ( 16 *71.6 ) + ( 19.8 * 93.3 ) + ( 18.4 * 84.3 ) + ( 17.1 * 80.6 ) + (15.5 * 75.2 ) + ( 14.7 * 69.7 ) + ( 17.1 * 82.0 ) + ( 15.4 * 69.4 ) + ( 16.2 * 83.3 ) + ( 15 *79.6 ) + ( 17.2 * 82.6 ) + ( 16 * 80.6 ) + ( 17 * 83.5 ) + ( 14.4 * 76.3 )

 = 20127.47

∑x^2 =  20.0^2 + 16^2 + 19.8^2 + 18.4^2 + 17.1^2 + 15.5^2 + 14.7^2 + 17.1^2 + 15.4^2 + 16.2^2 + 15.0^2 + 17.2^2  + 16.0^2 + 17.0^2 + 14.4^2

   = 4200.56

∑ y^2 = 88.6^2 + 71.6^2 + 93.3^2 + 84.3^2 + 80.6^2 + 75.2^2 + 69.7^2 + 82.0 ^2 + 69.4^2 + 83.3^2 + 79.6^2 + 82.6^2 + 80.6^2 + 83.5^2  + 76.3^2

   = 96725.86

a) Scatter diagram showing given data

Attached below

B)

X - bar = ∑x / n  = 249.8 / 15 = 16.65

Y - bar = ∑y / n  = 1200.6 / 15 = 80.04

where: Y = Temperature and X = Chirps per seconds

b = [ n ∑ xy  - ( ∑x ∑y ) ] / [ n∑x^2 - ( ∑x )^2 ]

   = [ 15 * 20127.47 - ( 249.8  * 1200.6 )] / [15* 4200.56 - ( 249.8 )^2 ]

   = 2002.17 / 608.36  = 3.29

a = Y-bar - b( X-bar )

  = 80.04 - 3.29( 16.65) = 25.26

equation for least squares line

y = 25.26 + 3.29 ( X )

C) Graph the least squares line on scattered diagram

attached below              

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4