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

Explanation:

Step1: Calculate the mean of latitude ($x$) and temperature ($y$)

Let \(x\) be latitude and \(y\) be temperature.
\(\bar{x}=\frac{42 + 45+39+35+32+41+40+33+30}{9}=\frac{337}{9}\approx37.44\)
\(\bar{y}=\frac{53 + 41+67+63+70+58+61+67+72}{9}=\frac{552}{9}\approx61.33\)

Step2: Calculate the numerator and denominator for slope formula \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)

\(\sum_{i = 1}^{9}(x_{i}-\bar{x})(y_{i}-\bar{y})=(42 - 37.44)(53 - 61.33)+(45 - 37.44)(41 - 61.33)+(39 - 37.44)(67 - 61.33)+(35 - 37.44)(63 - 61.33)+(32 - 37.44)(70 - 61.33)+(41 - 37.44)(58 - 61.33)+(40 - 37.44)(61 - 61.33)+(33 - 37.44)(67 - 61.33)+(30 - 37.44)(72 - 61.33)\)
\(=(4.56)(- 8.33)+(7.56)(-20.33)+(1.56)(5.67)+(-2.44)(1.67)+(-5.44)(8.67)+(3.56)(-3.33)+(2.56)(-0.33)+(-4.44)(5.67)+(-7.44)(10.67)\)
\(=-38.09 - 153.60+8.84 - 4.08 - 47.17-11.86 - 0.85 - 25.17-79.38\)
\(=-351.36\)

\(\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=(42 - 37.44)^{2}+(45 - 37.44)^{2}+(39 - 37.44)^{2}+(35 - 37.44)^{2}+(32 - 37.44)^{2}+(41 - 37.44)^{2}+(40 - 37.44)^{2}+(33 - 37.44)^{2}+(30 - 37.44)^{2}\)
\(=(4.56)^{2}+(7.56)^{2}+(1.56)^{2}+(-2.44)^{2}+(-5.44)^{2}+(3.56)^{2}+(2.56)^{2}+(-4.44)^{2}+(-7.44)^{2}\)
\(=20.79+57.15+2.43+5.95+29.59+12.67+6.55+19.71+55.35\)
\(=200.19\)

\(m=\frac{-351.36}{200.19}\approx - 1.75\approx - 1.7\)

Answer:

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