Question

latitude and longitude describe locations on the earth with respect to the equator and prime meridian. the tabl 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: Create a scatter plot and find the line of best fit

We can use a graphing calculator or software to input the data points (latitude as \(x\) - values and temperature as \(y\) - values). For example, using a TI - 84 Plus:

1. Press STAT, then EDIT. Enter the latitude values in \(L_1\) and the temperature values in \(L_2\).
2. Press STAT, then CALC, and select LinReg(ax + b) (linear regression).

The linear regression equation is of the form \(y=ax + b\), where \(a\) is the slope.

Step2: Calculate the slope using the formula \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\)

Let \(n = 9\).
\(\sum_{i=1}^{9}x_i=42 + 45+39+35+32+41+40+33+30=\sum_{i = 1}^{9}x_i = 337\)
\(\sum_{i=1}^{9}y_i=53 + 41+67+63+70+58+61+67+72=\sum_{i = 1}^{9}y_i=552\)
\(\sum_{i = 1}^{9}x_i^{2}=42^{2}+45^{2}+39^{2}+35^{2}+32^{2}+41^{2}+40^{2}+33^{2}+30^{2}\)
\(=1764 + 2025+1521+1225+1024+1681+1600+1089+900=\sum_{i = 1}^{9}x_i^{2}=12839\)
\(\sum_{i=1}^{9}x_iy_i=(42\times53)+(45\times41)+(39\times67)+(35\times63)+(32\times70)+(41\times58)+(40\times61)+(33\times67)+(30\times72)\)
\(=2226+1845+2613+2205+2240+2378+2440+2211+2160=\sum_{i = 1}^{9}x_iy_i = 20318\)

Substitute into the formula:

$$
LATEXBLOCK0
$$

Answer:

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