Question

1. the data in the table below shows the average temperature in northern latitudes:

latitude (°n)	temp (°f)
10	81
20	79
30	68
40	58
50	43
60	28
70	13
80	1

a) find the line of best fit:

b) estimate the average temperature for a city with a latitude of 48°.

Explanation:

Step1: Calculate necessary sums

Let \( x \) be Latitude (\(^\circ\)N) and \( y \) be Temp (\(^\circ\)F).
\( n = 8 \)
\( \sum x = 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 = 360 \) (Wait, no, count the rows: 0,10,20,30,40,50,60,70,80? Wait the table has 9 rows? Wait the given data: Latitude 0,10,20,30,40,50,60,70,80 (9 points). So \( n = 9 \)
\( \sum x = 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 = 360 \)
\( \sum y = 79 + 81 + 79 + 68 + 58 + 43 + 28 + 13 + 1 = 440 \)
\( \sum xy = 0*79 + 10*81 + 20*79 + 30*68 + 40*58 + 50*43 + 60*28 + 70*13 + 80*1 \)
\( = 0 + 810 + 1580 + 2040 + 2320 + 2150 + 1680 + 910 + 80 = 11570 \)
\( \sum x^2 = 0^2 + 10^2 + 20^2 + 30^2 + 40^2 + 50^2 + 60^2 + 70^2 + 80^2 \)
\( = 0 + 100 + 400 + 900 + 1600 + 2500 + 3600 + 4900 + 6400 = 20400 \)

Step2: Calculate slope (\(m\)) and intercept (\(b\))

Slope \( m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \)
\( m = \frac{9*11570 - 360*440}{9*20400 - 360^2} \)
\( 9*11570 = 104130 \), \( 360*440 = 158400 \), numerator \( = 104130 - 158400 = -54270 \)
\( 9*20400 = 183600 \), \( 360^2 = 129600 \), denominator \( = 183600 - 129600 = 54000 \)
\( m = \frac{-54270}{54000} \approx -1.005 \) (approx -1.01)

Intercept \( b = \frac{\sum y - m\sum x}{n} \)
\( b = \frac{440 - (-1.005)*360}{9} \)
\( (-1.005)*360 = -361.8 \), \( 440 + 361.8 = 801.8 \)
\( b = \frac{801.8}{9} \approx 89.09 \)

So line of best fit: \( y = -1.005x + 89.09 \) (or \( y \approx -x + 89 \) for simplicity)

Step3: Estimate temp at \( x = 48 \)

Using \( y = -x + 89 \) (approximate from line of best fit)
\( y = -48 + 89 = 41 \)
(Using more accurate \( m \approx -1.005 \), \( y = -1.005*48 + 89.09 \approx -48.24 + 89.09 \approx 40.85 \approx 41 \))

Answer:

a) Line of best fit: \( y \approx -1.01x + 89.1 \) (or \( y \approx -x + 89 \))
b) Estimated temperature: \( \approx 41^\circ\text{F} \)