temperatures: following are average temperatures, in degrees fahrenheit, during the months of january and…

temperatures: following are average temperatures, in degrees fahrenheit, during the months of january and july for 6 u.s. cities. use the ti - 84 plus calculator as needed.\n\n| city | january temperature | july temperature |\n| ---- | ---- | ---- |\n| new york | 32.1 | 76.5 |\n| st. louis | 29.6 | 80.2 |\n| denver | 29.2 | 73.4 |\n| chicago | 22.0 | 73.3 |\n| houston | 51.8 | 83.6 |\n| washington, d.c. | 34.9 | 79.2 |\n\nsend data to excel\n\npart: 0 / 4\n\npart 1 of 4\n\n(a) compute the least - squares regression line for predicting july temperature from january temperature.\nthe equation for the least - squares regression line is $hat{y}=square$. round the slope and y - intercept to four decimal places as needed.

temperatures: following are average temperatures, in degrees fahrenheit, during the months of january and july for 6 u.s. cities. use the ti - 84 plus calculator as needed.\n\n| city | january temperature | july temperature |\n| ---- | ---- | ---- |\n| new york | 32.1 | 76.5 |\n| st. louis | 29.6 | 80.2 |\n| denver | 29.2 | 73.4 |\n| chicago | 22.0 | 73.3 |\n| houston | 51.8 | 83.6 |\n| washington, d.c. | 34.9 | 79.2 |\n\nsend data to excel\n\npart: 0 / 4\n\npart 1 of 4\n\n(a) compute the least - squares regression line for predicting july temperature from january temperature.\nthe equation for the least - squares regression line is $hat{y}=square$. round the slope and y - intercept to four decimal places as needed.

Answer

Explanation:

Step1: Define variables

Let $x$ be the January temperature and $y$ be the July temperature. We have $n = 6$ data - points: $(x_1,y_1)=(32.1,76.5),(x_2,y_2)=(29.6,80.2),(x_3,y_3)=(29.2,73.4),(x_4,y_4)=(22.0,73.3),(x_5,y_5)=(51.8,83.6),(x_6,y_6)=(34.9,79.2)$.

Step2: Calculate sums

$\sum_{i = 1}^{n}x_i=32.1 + 29.6+29.2+22.0+51.8+34.9 = 199.6$ $\sum_{i = 1}^{n}y_i=76.5 + 80.2+73.4+73.3+83.6+79.2 = 466.2$ $\sum_{i = 1}^{n}x_i^2=32.1^2+29.6^2+29.2^2+22.0^2+51.8^2+34.9^2$ $=1030.41+876.16+852.64+484+2683.24+1218.01 = 7144.46$ $\sum_{i = 1}^{n}x_iy_i=32.1\times76.5+29.6\times80.2+29.2\times73.4+22.0\times73.3+51.8\times83.6+34.9\times79.2$ $=2455.65+2373.92+2143.28+1602.6+4330.48+2764.08 = 15670.01$

Step3: Calculate slope $b_1$

$b_1=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$ $=\frac{6\times15670.01 - 199.6\times466.2}{6\times7144.46-199.6^2}$ $=\frac{94020.06-93053.52}{42866.76 - 39840.16}$ $=\frac{966.54}{3026.6}=0.3194$

Step4: Calculate y - intercept $b_0$

$\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{199.6}{6}\approx33.2667$ $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{466.2}{6}=77.7$ $b_0=\bar{y}-b_1\bar{x}=77.7-0.3194\times33.2667$ $=77.7 - 10.6254=67.0746$

Answer:

$\hat{y}=67.0746 + 0.3194x$