Question

reasoning with lines of best fit
use this explore tool to investigate graphs of lines of best fit. to use the tool, enter x- and y-coordinates to plot points on the scatter plot. then press the button below the table to plot the line of best fit.

question.
the table shows the length of a baby each month from 2 months to her first birthday at 12 months.
enter the data in the explore tool to plot the points on the scatter plot and then plot the line of best fit.

month (x)	length in inches (y)
3	23.25
4	24.25
5	25.25
6	25.75
7	26.5
8	27
9	27.5
10	28.25
11	28.75
12	29.25

Explanation:

Step1: Identify Data Points

The data points are \((2, 22.5)\), \((3, 23.25)\), \((4, 24.25)\), \((5, 25.25)\), \((6, 25.75)\), \((7, 26.5)\), \((8, 27)\), \((9, 27.5)\), \((10, 28.25)\), \((11, 28.75)\), \((12, 29.25)\). We need to plot these on the scatter plot (x - axis: month, y - axis: length in inches) and then find the line of best fit.

Step2: Plotting Points

For each \(x\) (month) and corresponding \(y\) (length), mark the point on the scatter plot. For example, when \(x = 2\), \(y=22.5\), so we mark the point \((2, 22.5)\) on the graph where the x - coordinate is 2 and the y - coordinate is 22.5. Repeat this for all the data points.

Step3: Finding Line of Best Fit

After plotting all the points, the line of best fit is a straight line that best represents the data on the scatter plot. It should be drawn such that it is as close as possible to all the points, minimizing the distance between the points and the line. Using a tool (as suggested in the problem), we can plot the line of best fit. The general form of a line of best fit is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. To calculate \(m\) (slope), we can use the formula \(m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\), and \(b=\bar{y}-m\bar{x}\), where \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).

First, calculate \(\bar{x}\) and \(\bar{y}\):

\(n = 11\) (number of data points from \(x = 2\) to \(x=12\))

\(\sum_{i = 2}^{12}x_{i}=2 + 3+4+5+6+7+8+9+10+11+12=\frac{11\times(2 + 12)}{2}=77\) (using the sum of an arithmetic series formula \(S_n=\frac{n(a_1 + a_n)}{2}\), where \(n = 11\), \(a_1=2\), \(a_n = 12\))

\(\bar{x}=\frac{77}{11}=7\)

\(\sum_{i = 2}^{12}y_{i}=22.5+23.25 + 24.25+25.25+25.75+26.5+27+27.5+28.25+28.75+29.25\)

\(=22.5+(23.25+24.25)+(25.25+25.75)+26.5+27+(27.5+28.25)+(28.75+29.25)\)

\(=22.5 + 47.5+51+26.5+27+55.75+58\)

\(=22.5+47.5 = 70\); \(70+51 = 121\); \(121+26.5 = 147.5\); \(147.5+27 = 174.5\); \(174.5+55.75 = 230.25\); \(230.25+58=288.25\)

\(\bar{y}=\frac{288.25}{11}\approx26.2045\)

Now calculate \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\) and \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)

For \(x_i = 2\), \(x_i-\bar{x}=2 - 7=- 5\); \(y_i-\bar{y}=22.5 - 26.2045=-3.7045\); \((x_i - \bar{x})(y_i-\bar{y})=(-5)\times(-3.7045) = 18.5225\); \((x_i-\bar{x})^2=(-5)^2 = 25\)

For \(x_i=3\), \(x_i-\bar{x}=3 - 7=-4\); \(y_i-\bar{y}=23.25 - 26.2045=-2.9545\); \((x_i - \bar{x})(y_i-\bar{y})=(-4)\times(-2.9545)=11.818\); \((x_i-\bar{x})^2=(-4)^2 = 16\)

For \(x_i = 4\), \(x_i-\bar{x}=4 - 7=-3\); \(y_i-\bar{y}=24.25 - 26.2045=-1.9545\); \((x_i - \bar{x})(y_i-\bar{y})=(-3)\times(-1.9545)=5.8635\); \((x_i-\bar{x})^2=(-3)^2 = 9\)

For \(x_i=5\), \(x_i-\bar{x}=5 - 7=-2\); \(y_i-\bar{y}=25.25 - 26.2045=-0.9545\); \((x_i - \bar{x})(y_i-\bar{y})=(-2)\times(-0.9545)=1.909\); \((x_i-\bar{x})^2=(-2)^2 = 4\)

For \(x_i=6\), \(x_i-\bar{x}=6 - 7=-1\); \(y_i-\bar{y}=25.75 - 26.2045=-0.4545\); \((x_i - \bar{x})(y_i-\bar{y})=(-1)\times(-0.4545)=0.4545\); \((x_i-\bar{x})^2=(-1)^2 = 1\)

For \(x_i=7\), \(x_i-\bar{x}=7 - 7 = 0\); \(y_i-\bar{y}=26.5 - 26.2045 = 0.2955\); \((x_i - \bar{x})(y_i-\bar{y})=0\times0.2955 = 0\); \((x_i-\bar{x})^2=0^2=0\)

For \(x_i=8\), \(x_i-\bar{x}=8 - 7 = 1\); \(y_i-\bar{y}=27 - 26.2045 = 0.7955\); \((x_i - \bar{x})(y_i-\bar{y})=1\times0.7955 = 0.7955\); \((x_i-\bar{x})^2=1^2 = 1\)

For \(x_i=9\), \(x_i-\bar{x}=9 - 7 = 2\); \(y_i-\bar{y}=27.5 - 26.2045 = 1.2955\); \((x_i - \bar{x})(y_i-\bar{y})=2\times1.2955 = 2.591\); \((x_i-\bar{x})^2=2^2 = 4\)

For \(x_i=10\), \(x_i-\bar{x}=10 - 7 = 3\); \(y_i-\bar{y}=28.25 - 26.2045 = 2.0455\); \((x_i - \bar{x})(y_i-\bar{y})=3\times2.0455 = 6.1365\); \((x_i-\bar{x})^2=3^2 = 9\)

For \(x_i=11\), \(x_i-\bar{x}=11 - 7 = 4\); \(y_i-\bar{y}=28.75 - 26.2045 = 2.5455\); \((x_i - \bar{x})(y_i-\bar{y})=4\times2.5455 = 10.182\); \((x_i-\bar{x})^2=4^2 = 16\)

For \(x_i=12\), \(x_i-\bar{x}=12 - 7 = 5\); \(y_i-\bar{y}=29.25 - 26.2045 = 3.0455\); \((x_i - \bar{x})(y_i-\bar{y})=5\times3.0455 = 15.2275\); \((x_i-\bar{x})^2=5^2 = 25\)

Now sum up \((x_i-\bar{x})(y_i-\bar{y})\):

\(18.5225+11.818+5.8635+1.909+0.4545+0+0.7955+2.591+6.1365+10.182+15.2275\)

\(=(18.5225+11.818)=30.3405\); \((30.3405+5.8635)=36.204\); \((36.204+1.909)=38.113\); \((38.113+0.4545)=38.5675\); \((38.5675+0)=38.5675\); \((38.5675+0.7955)=39.363\); \((39.363+2.591)=41.954\); \((41.954+6.1365)=48.0905\); \((48.0905+10.182)=58.2725\); \((58.2725+15.2275)=73.5\)

Sum up \((x_i-\bar{x})^2\):

\(25 + 16+9+4+1+0+1+4+9+16+25\)

\(=(25+16)=41\); \((41+9)=50\); \((50+4)=54\); \((54+1)=55\); \((55+0)=55\); \((55+1)=56\); \((56+4)=60\); \((60+9)=69\); \((69+16)=85\); \((85+25)=110\)

So the slope \(m=\frac{73.5}{110}\approx0.668\)

And \(b=\bar{y}-m\bar{x}\approx26.2045-0.668\times7\approx26.2045 - 4.676\approx21.5285\)

So the equation of the line of best fit is approximately \(y = 0.668x+21.5285\)

When we plot this line on the scatter plot with the given data points, it will be the line of best fit.

Answer:

To solve the problem, we first plot the data points \((2, 22.5)\), \((3, 23.25)\), \((4, 24.25)\), \((5, 25.25)\), \((6, 25.75)\), \((7, 26.5)\), \((8, 27)\), \((9, 27.5)\), \((10, 28.25)\), \((11, 28.75)\), \((12, 29.25)\) on the scatter plot (x - axis: month, y - axis: length in inches). Then we find the line of best fit using the formula for the line of best fit \(y = mx + b\) (calculated as \(y\approx0.668x + 21.5285\)) and plot it on the same scatter plot. The line of best fit is a straight line that best represents the trend of the baby's growth over the months.