Chapter 8.10, Problem 34SR

(a)

To determine

To calculate: To make a scatter plot of the data and draw a line of fit

Answer to Problem 34SR

Scatter plot of data is created

Explanation of Solution

Given information: Table showing the latitude and average temperature in July for the various U.S.cities

City	Latitude ( ${}^{\circ }N$ )	Average July High Temperature ( ${}^{\circ }F$ )
Chicago, IL	41	73
Dallas, TX	32	85
Denver, CO	39	74
New York, NY	40	77
Duluth, MN	46	66

Formula Used:

Line of Best fit is a straight line that best represents the data on scatter plot. Line of fit must pass through some of the points, none of the points or all the points

Calculation:

Data is given as

City	Latitude ( ${}^{\circ }N$ )	Average July High Temperature ( ${}^{\circ }F$ )
Chicago, IL	41	73
Dallas, TX	32	85
Denver, CO	39	74
New York, NY	40	77
Duluth, MN	46	66

Scatter plot is created by graphing the above points:

  

Conclusion:

Hence, scatter plot and line of best fit is drawn

(b)

To determine

To calculate: To write and equation for the line of fit

Answer to Problem 34SR

Theequation of line of fit is $y=-\frac{19}{12}x+\frac{407}{3}$

Explanation of Solution

Given information: Table showing the latitude and average temperature in July for the various U.S.cities

City	Latitude ( ${}^{\circ }N$ )	Average July High Temperature ( ${}^{\circ }F$ )
Chicago, IL	41	73
Dallas, TX	32	85
Denver, CO	39	74
New York, NY	40	77
Duluth, MN	46	66

Formula Used:

Equation of line passing through two points is given as

  $y-y_2=\frac{y_2-y_1}{x_2-x_1}\left(x-x_2\right)$

Slope intercept form of line is given as

  $y=mx+c$

Where,

  $m=$ Slope

  $c=$ y-intercept

Calculation:

Data is given as

City	Latitude ( ${}^{\circ }N$ )	Average July High Temperature ( ${}^{\circ }F$ )
Chicago, IL	41	73
Dallas, TX	32	85
Denver, CO	39	74
New York, NY	40	77
Duluth, MN	46	66

Consider the points on graph as follows:

  $\left(44,66\right)$ and $\left(32,85\right)$

Equation of line passing through above points is

  $\begin{array}{l}y-85=\frac{85-66}{32-44}\left(x-32\right)\\ y-85=\frac{19}{-12}\left(x-32\right)\\ -12y+1020=19x-608\\ 12y=-19x+1020+608\\ 12y=-19x+1628\\ y=-\frac{19}{12}x+\frac{1628}{12}\\ y=-\frac{19}{12}x+\frac{407}{3}\end{array}$

Conclusion:

Hence, equation of line of fit is $y=-\frac{19}{12}x+\frac{407}{3}$

(c)

To determine

To calculate: To estimate the average July high temperature for a location with latitude ${50}^{\circ }N$

Answer to Problem 34SR

Theaverage July high temperature for a location with latitude ${50}^{\circ }N$ is ${56.5}^{\circ }F$

Explanation of Solution

Given information: Table showing the latitude and average temperature in July for the various U.S.cities

City	Latitude ( ${}^{\circ }N$ )	Average July High Temperature ( ${}^{\circ }F$ )
Chicago, IL	41	73
Dallas, TX	32	85
Denver, CO	39	74
New York, NY	40	77
Duluth, MN	46	66

Calculation:

Equation of line of fit is given as follows:

  $y=-\frac{19}{12}x+\frac{407}{3}$

Average July high temperature for a location with latitude ${50}^{\circ }N$ is calculated by substituting $x=50$

  $\begin{array}{l}y=-\frac{19}{12}\left(50\right)+\frac{407}{3}\\ y=-\frac{950}{12}+\frac{407}{3}\\ y=\frac{-950+1628}{12}\\ y=\frac{678}{12}\\ y={56.5}^{\circ }F\end{array}$

Conclusion:

Hence, the average July high temperature for a location with latitude ${50}^{\circ }N$ is ${56.5}^{\circ }F$