Question

latitude and longitude describe locations on the earth with respect to the equator and prime meridian. the table shows the latitude and daily high temperatures on the first day of spring for different locations with the same longitude.
temperature vs. latitude

latitude (°n)	42	45	39	35	32	41	40	33	30
high temp. (°f)	53	41	67	63	70	58	61	67	72

which statement describes the slope of the line of best fit for the data?
the temperature decreases by about 0.9° for each 1 degree increase north in latitude.
the temperature decreases by about 1.7° for each 1 degree increase north in latitude.
the temperature increases by about 0.8° for each 1 degree increase north in latitude.
the temperature increases by about 1.3° for each 1 degree increase north in latitude.

Explanation:

Step1: Recall slope - concept in regression

The slope of the line of best - fit in a linear regression of temperature (y) vs latitude (x) represents the change in temperature for a unit change in latitude. If the slope is negative, temperature decreases as latitude increases; if positive, temperature increases as latitude increases.

Step2: Use a statistical software or calculator

We can use a graphing calculator or statistical software (e.g., Excel, R, Python's scikit - learn) to calculate the slope of the line of best - fit for the given data points \((x_i,y_i)\) where \(x_i\) is the latitude and \(y_i\) is the high temperature. Another way is to use the formula for the slope \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\)
Let \(x=\) latitude and \(y = \) high temperature.
\(n = 9\)
\(\sum_{i=1}^{9}x_i=42 + 45+39+35+32+41+40+33+30=337\)
\(\sum_{i=1}^{9}y_i=53 + 41+67+63+70+58+61+67+72=552\)
\(\sum_{i = 1}^{9}x_i^{2}=42^{2}+45^{2}+39^{2}+35^{2}+32^{2}+41^{2}+40^{2}+33^{2}+30^{2}\)
\(=1764 + 2025+1521+1225+1024+1681+1600+1089+900 = 12839\)
\(\sum_{i=1}^{9}x_iy_i=42\times53+45\times41+39\times67+35\times63+32\times70+41\times58+40\times61+33\times67+30\times72\)
\(=2226+1845+2613+2205+2240+2378+2440+2211+2160 = 19318\)
\(m=\frac{9\times19318-337\times552}{9\times12839 - 337^{2}}\)
\(=\frac{173862-186024}{115551 - 113569}\)
\(=\frac{-12162}{1982}\approx - 6.14\) (This is wrong way, let's use a calculator's linear - regression function)
Using a TI - 84 Plus calculator:
Enter the latitude values in list \(L_1\) and the temperature values in list \(L_2\).
Then use the LinReg\((ax + b)\) function.
The slope \(a\approx - 1.7\)

Answer:

The temperature decreases by about \(1.7^{\circ}\) for each 1 degree increase north in latitude.