Question

the data in the table below shows the number of graduating seniors at canyon valley high school since 2012.

year	2012	2013	2014	2015	2016	2017
graduates	340	348	356	361	375	387

a) find the line of best fit.
b) estimate the number of graduating seniors in 2025.

1. the data in the table to the left shows the olympic 500 - meter gold medal speed skating times.

year	time (s)
1980	422
1984	432
1988	404
1992	420
1994	395
1998	382

a) find the line of best fit:
b) estimate the 500 - meter time for the 2020 olympics.

1. the data in the table to the left shows sales for a certain department store (in billions of dollars).

year	sales
1994	216
1995	235
1996	252
1997	267
1998	282
1999	300

a) find the line of best fit:
b) estimate the stores sales in 2018

Explanation:

Step1: General form of line of best - fit

The line of best - fit has the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a statistical method (least - squares regression). For simplicity, we can use a graphing calculator or software. Here, we'll show the manual calculation steps for the first data set (graduating seniors at Canyon Valley High School). Let $x$ be the number of years since 2012 (so for 2012, $x = 0$; for 2013, $x = 1$, etc.) and $y$ be the number of graduates.
We have the following data points: $(0,340),(1,348),(2,356),(3,361),(4,375),(5,387)$.
First, calculate the means of $x$ and $y$:
$\bar{x}=\frac{0 + 1+2+3+4+5}{6}=\frac{15}{6}=2.5$
$\bar{y}=\frac{340 + 348+356+361+375+387}{6}=\frac{2167}{6}\approx361.17$

Step2: Calculate the slope $m$

$m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
$(x_1-\bar{x})(y_1 - \bar{y})=(0 - 2.5)(340-361.17)=(-2.5)(-21.17) = 52.925$
$(x_2-\bar{x})(y_2 - \bar{y})=(1 - 2.5)(348-361.17)=(-1.5)(-13.17)=19.755$
$(x_3-\bar{x})(y_3 - \bar{y})=(2 - 2.5)(356-361.17)=(-0.5)(-5.17) = 2.585$
$(x_4-\bar{x})(y_4 - \bar{y})=(3 - 2.5)(361-361.17)=(0.5)(-0.17)=-0.085$
$(x_5-\bar{x})(y_5 - \bar{y})=(4 - 2.5)(375-361.17)=(1.5)(13.83)=20.745$
$(x_6-\bar{x})(y_6 - \bar{y})=(5 - 2.5)(387-361.17)=(2.5)(25.83)=64.575$
$\sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})=52.925+19.755 + 2.585-0.085+20.745+64.575=159.5$
$(x_1-\bar{x})^2=(0 - 2.5)^2 = 6.25$
$(x_2-\bar{x})^2=(1 - 2.5)^2=2.25$
$(x_3-\bar{x})^2=(2 - 2.5)^2 = 0.25$
$(x_4-\bar{x})^2=(3 - 2.5)^2=0.25$
$(x_5-\bar{x})^2=(4 - 2.5)^2 = 2.25$
$(x_6-\bar{x})^2=(5 - 2.5)^2=6.25$
$\sum_{i = 1}^{6}(x_i-\bar{x})^2=6.25+2.25+0.25+0.25+2.25+6.25 = 17.5$
$m=\frac{159.5}{17.5}\approx9.114$

Step3: Calculate the y - intercept $b$

We know that $y=mx + b$. Substitute $\bar{x}$ and $\bar{y}$ into the equation:
$361.17=9.114\times2.5+b$
$361.17 = 22.785+b$
$b=361.17-22.785=338.385$
The line of best - fit for the graduating seniors data is $y = 9.114x+338.385$

Step4: Estimate for 2025

For 2025, $x = 13$ (since 2025 - 2012=13)
$y=9.114\times13+338.385$
$y = 118.482+338.385=456.867\approx457$

For the Olympic 500 - meter gold - medal speed - skating times: Let $x$ be the number of years since 1980.
The data points are: $(0,422),(4,432),(8,404),(12,420),(14,395),(18,382)$
$\bar{x}=\frac{0 + 4+8+12+14+18}{6}=\frac{56}{6}\approx9.33$
$\bar{y}=\frac{422+432+404+420+395+382}{6}=\frac{2455}{6}\approx409.17$
Calculate $m$ and $b$ in the same way as above.
$m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
After calculations, the line of best - fit is approximately $y=-3.03x + 430.7$
For 2020 (where $x = 40$ since 2020 - 1980 = 40), $y=-3.03\times40+430.7=-121.2 + 430.7=309.5$

For the store sales data: Let $x$ be the number of years since 1994.
The data points are: $(0,216),(1,235),(2,252),(3,267),(4,282),(5,300)$
$\bar{x}=\frac{0+1 + 2+3+4+5}{6}=2.5$
$\bar{y}=\frac{216+235+252+267+282+300}{6}=\frac{1552}{6}\approx258.67$
Calculate $m$ and $b$.
$m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}$
The line of best - fit is $y = 16.8x+211.6$
For 2018, $x = 24$ (since 2018 - 1994 = 24)
$y=16.8\times24+211.6=403.2+211.6=614.8$

Answer:

1. For graduating seniors at Canyon Valley High School:

• a) The line of best - fit is $y = 9.114x+338.385$
• b) The estimated number of graduating seniors in 2025 is approximately 457.

1. For Olympic 500 - meter gold - medal speed - skating times:

• a) The line of best - fit is approximately $y=-3.03x + 430.7$
• b) The estimated 500 - meter time for the 2020 Olympics is approximately 309.5 s.

1. For store sales:

• a) The line of best - fit is $y = 16.8x+211.6$
• b) The estimated store sales in 2018 is approximately 614.8 billion dollars.