Least Squares Line of Best Fit

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Question 1 of 13

Solve for the equation of the least squares line of best fit.

¯ x = 12

$barx=12$

¯ y = 19

$bary=19$

σ_{x} = 1.8

$sigma_x=1.8$

σ_{y} = 4

$sigma_y=4$

r = 0.9

$r=0.9$

$y =$ $y=$ (2) $x$ $x$ (-5)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

m = r \cdot \frac{σ_{y}}{σ_{x}}

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

b = ¯ y - m ¯ x

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

First, use the gradient formula to find the gradient or slope of the line.

$m$ $m$	$=$ $=$	$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$	Gradient Formula

	$=$ $=$	$0.9\cdot\frac{\color{#9a00c7}{4}}{\color{#00880A}{1.8}}$	Substitute values

$m$ $m$	$=$ $=$	$2$ $2$

Next, find the

y

$y$ intercept.

$b$ $b$	$=$ $=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ $y$ intercept Formula

	$=$ $=$	$19-(\color{#007DDC}{2})12$	Substitute values

	$=$ $=$	$19 - 24$ $19-24$	Evaluate
$b$ $b$	$=$ $=$	$- 5$ $-5$

Finally, substitute the values into the gradient-intercept form

m = 2

$m=2$

b = - 5

$b=-5$

$y$ $y$	$=$ $=$	$m$ $m$ $x +$ $x+$ $b$ $b$	Gradient-Intercept Form

$y$ $y$	$=$ $=$	$2$ $2$ $x -$ $x-$ $5$ $5$	Substitute values

y = 2 x - 5

$y=2x-5$

Question 2 of 13

The scores of

5

$5$ students for their Maths and Science tests were listed in a table. Find the equation of the least squares line of best fit.

$Maths$ $\text(Maths)$	$x$ $x$	81	65	70	52	87
$Science$ $\text(Science)$	$y$ $y$	85	72	63	68	91

Round your answer to two decimal places

$S =$ $S=$ (0.69) $M$ $M$ (-26.81)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

m = r \cdot \frac{σ_{y}}{σ_{x}}

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

b = ¯ y - m ¯ x

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

First, use a calculator and find the following values.

Note that the steps may vary depending on the model of your calculator.

(1)

$(1)$ Press on Mode and select Statistics.

(2)

$(2)$ Choose the options for

A + b x

$A+bx$ among the mode options.

(3)

$(3)$ Input the

x

$x$ values followed by the

=

$=$ button on the left column.

(4)

$(4)$ Repeat the same steps for the

y

$y$ values on the right column.

(5)

$(5)$ Press AC to clear the values.

(6)

$(6)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

¯ x

$barx$ to get the mean of

x

$x$ .

(7)

$(7)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

σ_{x}

$sigma_x$ to get the standard deviation of

x

$x$ .

(8)

$(8)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

¯ y

$bary$ to get the mean of

y

$y$ .

(9)

$(9)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

σ_{y}

$sigma_y$ to get the standard deviation of

y

$y$ .

(10)

$(10)$ Press Shift $+ 1 + 5$ $+1+5$ to get the correlation coefficient

(r)

$(r)$ .

Following the steps above, the given values would be the following:

$¯ x$ $barx$	$=$ $=$	$71 (Maths)$ $71 \text((Maths))$
$¯ y$ $bary$	$=$ $=$	$75.8 (Science)$ $75.8 \text((Science))$
$σ_{x}$ $sigma_x$	$=$ $=$	$12.28$ $12.28$
$σ_{y}$ $sigma_y$	$=$ $=$	$10.53$ $10.53$
$r$ $r$	$=$ $=$	$0.8$ $0.8$

Next, use the gradient formula to find the gradient or slope of the line.

$m$ $m$	$=$ $=$	$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$	Gradient Formula

	$=$ $=$	$0.8\cdot\frac{\color{#9a00c7}{10.53}}{\color{#00880A}{12.28}}$	Substitute values

$m$ $m$	$=$ $=$	$0.69$ $0.69$

Next, find the

y

$y$ intercept.

$b$ $b$	$=$ $=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ $y$ intercept Formula
	$=$ $=$	$75.68-(\color{#007DDC}{0.69})71$	Substitute values
$b$ $b$	$=$ $=$	$26.81$ $26.81$

Finally, substitute the values into the gradient-intercept form

m = 0.69

$m=0.69$

b = 26.81

$b=26.81$

$y$ $y$	$=$ $=$	$m$ $m$ $x +$ $x+$ $b$ $b$	Gradient-Intercept Form

$y$ $y$	$=$ $=$	$0.69$ $0.69$ $x +$ $x+$ $26.81$ $26.81$	Substitute values

$S$ $S$	$=$ $=$	$0.69 M + 26.81$ $0.69M+26.81$	Change $y$ $y$ and $x$ $x$ to $S$ $S$ and $M$ $M$ respectively

S = 0.69 M + 26.81

$S=0.69M+26.81$

Question 3 of 13

3. Question
The scores of $5$ $5$ students for their Maths and Science tests were listed in a table. Draw the least square line of best fit given that its equation is $S = 0.69 M + 26.81$ $S=0.69M+26.81$ .
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y =$ $y=$ $m$ $m$ $x +$ $x+$ $b$ $b$

First, create a table of values illustrating the values of Maths $(x)$ $(x)$ and Science $(y)$ $(y)$

We can use $0$ $0$ , $50$ $50$ and $100$ $100$ as test values of $x$ $x$ .

$x$ $x$ $0$ $0$ $50$ $50$ $100$ $100$

$y$ $y$

Next, recall that $S = y$ $S=y$ and $M = x$ $M=x$

Substitute the values of $x$ $x$ to $M$ $M$ to solve for the $y$ $y$ values.

$x = 0$ $x=0$

$S$ $S$ $=$ $=$ $0.69 M + 26.81$ $0.69M+26.81$

$S$ $S$ $=$ $=$ $0.69 (0) + 26.81$ $0.69(0)+26.81$ Substitute $x = 0$ $x=0$

$S$ $S$ $=$ $=$ $26.81$ $26.81$

$x = 50$ $x=50$

$S$ $S$ $=$ $=$ $0.69 M + 26.81$ $0.69M+26.81$

$S$ $S$ $=$ $=$ $0.69 (50) + 26.81$ $0.69(50)+26.81$ Substitute $x = 50$ $x=50$

$S$ $S$ $=$ $=$ $61.3$ $61.3$

$x = 100$ $x=100$

$S$ $S$ $=$ $=$ $0.69 M + 26.81$ $0.69M+26.81$

$S$ $S$ $=$ $=$ $0.69 (100) + 26.81$ $0.69(100)+26.81$ Substitute $x = 100$ $x=100$

$S$ $S$ $=$ $=$ $95.8$ $95.8$

$x$ $x$ $0$ $0$ $50$ $50$ $100$ $100$

$y$ $y$ $26.81$ $26.81$ $61.3$ $61.3$ $95.8$ $95.8$

Finally, plot these three points and draw a line connecting them.

$S = 0.69 M + 26.81$ $S=0.69M+26.81$

$x$ $x$	$0$ $0$	$50$ $50$	$100$ $100$
$y$ $y$

$x$ $x$	$0$ $0$	$50$ $50$	$100$ $100$
$y$ $y$	$26.81$ $26.81$	$61.3$ $61.3$	$95.8$ $95.8$

Question 4 of 13

The weekly exercise routine and pulse rate of

10

$10$ students were recorded on this table of values. Given that

r = - 0.99

$r=-0.99$ , find the equation of the least squares line of best fit.

$Exercise$ $\text(Exercise)$	0.5	1	2	3	4	5	6	7	8	9
$BPM$ $\text(BPM)$	80	75	70	67	62	61	55	52	48	46

Round your answer to two decimal places

$B =$ $B=$ (-3.84) $E +$ $E+$ (79.07)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

m = r \cdot \frac{σ_{y}}{σ_{x}}

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

b = ¯ y - m ¯ x

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

First, use a calculator and find the following values.

Note that the steps may vary depending on the model of your calculator.

(1)

$(1)$ Press on Mode and select Statistics.

(2)

$(2)$ Choose the options for

A + b x

$A+bx$ among the mode options.

(3)

$(3)$ Input the

x

$x$ values followed by the

=

$=$ button on the left column.

(4)

$(4)$ Repeat the same steps for the

y

$y$ values on the right column.

(5)

$(5)$ Press AC to clear the values.

(6)

$(6)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

¯ x

$barx$ to get the mean of

x

$x$ .

(7)

$(7)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

σ_{x}

$sigma_x$ to get the standard deviation of

x

$x$ .

(8)

$(8)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

¯ y

$bary$ to get the mean of

y

$y$ .

(9)

$(9)$ Press Shift $+ 1 + 4$ $+1+4$ and select the option for

σ_{y}

$sigma_y$ to get the standard deviation of

y

$y$ .

Following the steps above, the given values would be the following:

$¯ x$ $barx$	$=$ $=$	$4.55 (Exercise)$ $4.55 \text((Exercise))$
$¯ y$ $bary$	$=$ $=$	$61.6 (BPM)$ $61.6 \text((BPM))$
$σ_{x}$ $sigma_x$	$=$ $=$	$2.80$ $2.80$
$σ_{y}$ $sigma_y$	$=$ $=$	$10.87$ $10.87$

Next, use the gradient formula to find the gradient or slope of the line.

r

$r$

=

$=$

- 0.99

$-0.99$

$m$ $m$	$=$ $=$	$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$	Gradient Formula

	$=$ $=$	$-0.99\cdot\frac{\color{#9a00c7}{10.87}}{\color{#00880A}{2.80}}$	Substitute values

$m$ $m$	$=$ $=$	$- 3.84$ $-3.84$

Next, find the

y

$y$ intercept.

$b$ $b$	$=$ $=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ $y$ intercept Formula
	$=$ $=$	$61.6-(\color{#007DDC}{-3.84})4.55$	Substitute values
$b$ $b$	$=$ $=$	$79.07$ $79.07$

Finally, substitute the values into the gradient-intercept form

m = - 3.84

$m=-3.84$

b = 79.07

$b=79.07$

$y$ $y$	$=$ $=$	$m$ $m$ $x +$ $x+$ $b$ $b$	Gradient-Intercept Form

$y$ $y$	$=$ $=$	$- 3.84$ $-3.84$ $x +$ $x+$ $79.07$ $79.07$	Substitute values

$B$ $B$	$=$ $=$	$- 3.84 E + 79.07$ $-3.84E+79.07$	Change $y$ $y$ and $x$ $x$ to $B$ $B$ and $E$ $E$ respectively

B = - 3.84 E + 79.07

$B=-3.84E+79.07$

Question 5 of 13

5. Question
The weekly exercise routine and pulse rate of $10$ $10$ students were recorded to this table of values. Draw the least square line of best fit given that its equation is $B = - 3.84 E + 79.07$ $B=-3.84E+79.07$ .
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y =$ $y=$ $m$ $m$ $x +$ $x+$ $b$ $b$

First, create a table of values illustrating the values of Exercise $(x)$ $(x)$ and BPM $(y)$ $(y)$

We can use $0$ $0$ , $5$ $5$ and $10$ $10$ as test values of $x$ $x$ .

$x$ $x$ $0$ $0$ $5$ $5$ $10$ $10$

$y$ $y$

Next, recall that $B = y$ $B=y$ and $E = x$ $E=x$

Substitute the values of $x$ $x$ to $E$ $E$ to solve for the $y$ $y$ values.

$x = 0$ $x=0$

$B$ $B$ $=$ $=$ $- 3.84 E + 79.07$ $-3.84E+79.07$

$B$ $B$ $=$ $=$ $- 3.84 (0) + 79.07$ $-3.84(0)+79.07$ Substitute $x = 0$ $x=0$

$B$ $B$ $=$ $=$ $79.07$ $79.07$

$x = 5$ $x=5$

$B$ $B$ $=$ $=$ $- 3.84 E + 79.07$ $-3.84E+79.07$

$B$ $B$ $=$ $=$ $- 3.84 (5) + 79.07$ $-3.84(5)+79.07$ Substitute $x = 5$ $x=5$

$B$ $B$ $=$ $=$ $59.87$ $59.87$

$x = 10$ $x=10$

$B$ $B$ $=$ $=$ $- 3.84 E + 79.07$ $-3.84E+79.07$

$B$ $B$ $=$ $=$ $- 3.84 (10) + 79.07$ $-3.84(10)+79.07$ Substitute $x = 10$ $x=10$

$B$ $B$ $=$ $=$ $40.67$ $40.67$

$x$ $x$ $0$ $0$ $5$ $5$ $10$ $10$

$y$ $y$ $79.07$ $79.07$ $59.87$ $59.87$ $40.67$ $40.67$

Finally, plot these three points and draw a line connecting them.

$B = - 3.84 E + 79.07$ $B=-3.84E+79.07$

$x$ $x$	$0$ $0$	$5$ $5$	$10$ $10$
$y$ $y$

$x$ $x$	$0$ $0$	$5$ $5$	$10$ $10$
$y$ $y$	$79.07$ $79.07$	$59.87$ $59.87$	$40.67$ $40.67$

Question 6 of 13

The height and weight of

10

$10$ girls were listed in a table of values. Find the equation of the least squares line of best fit.

$Height$ $\text(Height)$	162	167	179	162	149	154	144	157	172	160
$Weight$ $\text(Weight)$	74	65	87	54	53	61	57	71	79	58

Round your answer to two decimal places

$W =$ $W=$ (0.86) $H$ $H$ (-72.22)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

m = r \cdot \frac{σ_{y}}{σ_{x}}

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ $y$ intercept Formula

b = ¯ y - m ¯ x

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

First, use a calculator and find the following values.

Note that the steps may vary depending on the model of your calculator.

(1)

$(1)$ Press on Mode and select Statistics.

(2)

$(2)$ Choose the options for

A + b x

$A+bx$ among the mode options.

(3)

$(3)$ Input the

x

$x$ values followed by the

=

$=$ button on the left column.

(4)

$(4)$ Repeat the same steps for the

$y$ values on the right column.

$(5)$ Press AC to clear the values.

$(6)$ Press Shift $+1+4$ and select the option for

$barx$ to get the mean of

$x$ .

$(7)$ Press Shift $+1+4$ and select the option for

$sigma_x$ to get the standard deviation of

$x$ .

$(8)$ Press Shift $+1+4$ and select the option for

$bary$ to get the mean of

$y$ .

$(9)$ Press Shift $+1+4$ and select the option for

$sigma_y$ to get the standard deviation of

$y$ .

$(10)$ Press Shift $+1+5$ to get the correlation coefficient

$(r)$ .

Following the steps above, the given values would be the following:

$barx$	$=$	$160.6 \text((Height))$
$bary$	$=$	$65.9 \text((Weight))$
$sigma_x$	$=$	$9.90$
$sigma_y$	$=$	$10.88$
$r$	$=$	$0.78$

Next, use the gradient formula to find the gradient or slope of the line.

$m$	$=$	$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$	Gradient Formula

	$=$	$0.78\cdot\frac{\color{#9a00c7}{10.88}}{\color{#00880A}{9.90}}$	Substitute values

$m$	$=$	$0.86$

Next, find the

$y$ intercept.

$b$	$=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ intercept Formula
	$=$	$65.9-(\color{#007DDC}{0.86})160.6$	Substitute values
$b$	$=$	$-72.22$

Finally, substitute the values into the gradient-intercept form

$m=0.86$

$b=-72.22$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$0.86$ $x$ $-72.22$	Substitute values

$W$	$=$	$0.86H-72.22$	Change $y$ and $x$ to $W$ and $H$ respectively

$W=0.86H-72.22$

Question 7 of 13

7. Question
The height and weight of $10$ girls were listed in a table of values. Given that the equation of the least square line of best fit is $W=0.86H-72.22$ , find the weight of a girl with a height of $175 \text(cm)$ .

Round your answer to two decimal places
- $W=$ (77.73) $\text(kg)$
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Since we are given the value $H=175$ , we can substitute it to the formula and solve for the weight $(W)$ .

$W$ $=$ $0.86H-72.22$

$W$ $=$ $0.86(175)-72.22$ Substitute $H=175$

$W$ $=$ $77.73$

$W=77.73 \text(kg)$
Question 8 of 13

8. Question
The height and arm span of $8$ female swimmers were listed in a table of values. Find the means, standard deviation and the correlation coefficient of the two set of values.

$\text(Height)$ 189 190 197 186 195 185 192 196

$\text(Arm Span)$ 190 191 199 187 198 186 194 200

Round your answer to three decimal places
- $(i) barx=$ (191.25)
  
  $(ii) bary=$ (190.125)
  
  $(iii) sigma_x=$ (4.235)
  
  $(iv) sigma_y=$ (5.110)
  
  $(v) r=$ (0.99)
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Let $x$ represent $text(Height) = H$

Let $y$ represent $text(Arm Span) = S$

First, use a calculator and find the following values.

Note that the steps may vary depending on the model of your calculator.

$(1)$ Press on Mode and select Statistics.

$(2)$ Choose the options for $A+bx$ among the mode options.

$(3)$ Input the $x$ values followed by the $=$ button on the left column.

$(4)$ Repeat the same steps for the $y$ values on the right column.

$(5)$ Press AC to clear the values.

$(6)$ Press Shift $+1+4$ and select the option for $barx$ to get the mean of $x$ .

$(7)$ Press Shift $+1+4$ and select the option for $sigma_x$ to get the standard deviation of $x$ .

$(8)$ Press Shift $+1+4$ and select the option for $bary$ to get the mean of $y$ .

$(9)$ Press Shift $+1+4$ and select the option for $sigma_y$ to get the standard deviation of $y$ .

$(10)$ Press Shift $+1+5$ to get the correlation coefficient $(r)$ .

Following the steps above, the given values would be the following:

$barx$ $=$ $191.25 \text((Height))$

$bary$ $=$ $190.125 \text((Arm Span))$

$sigma_x$ $=$ $4.235$

$sigma_y$ $=$ $5.110$

$r$ $=$ $0.99$

$barx=191.25$

$bary=190.125$

$sigma_x=4.235$

$sigma_y=5.110$

$r=0.99$

Question 9 of 13

The height and arm span of

$8$ female swimmers were listed in a table of values. Find the equation of the least squares line of best fit.

$\text(Height)$	189	190	197	186	195	185	192	196
$\text(Arm Span)$	190	191	199	187	198	186	194	200

Round your answer to three decimal places

$S=$ (1.195) $H$ (-35.419)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

From the previous question, the given values would be the following:

$barx$	$=$	$191.25 \text((Height))$
$bary$	$=$	$190.125 \text((Arm Span))$
$sigma_x$	$=$	$4.235$
$sigma_y$	$=$	$5.110$
$r$	$=$	$0.99$

Use the gradient formula to find the gradient or slope of the line.

$m$	$=$	$r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$	Gradient Formula

	$=$	$0.99\cdot\frac{\color{#9a00c7}{5.11}}{\color{#00880A}{4.235}}$	Substitute values

$m$	$=$	$1.195$

Next, find the

$y$ intercept.

$b$	$=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ intercept Formula
	$=$	$190.125-(\color{#007DDC}{1.195})191.25$	Substitute values
$b$	$=$	$-35.419$

Finally, substitute the values into the gradient-intercept form

$m=1.195$

$b=-35.419$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$1.195$ $x$ $-35.419$	Substitute values

$S$	$=$	$1.195H-35.419$	Change $y$ and $x$ to $S$ and $H$ respectively

$S=1.195H-35.419$

Question 10 of 13

10. Question
The height and arm span of $8$ female swimmers were listed in a table of values. Given that the equation of the least square line of best fit is $S=1.195H-35.419$ , find the arm span of a girl with a height of $201 \text(cm)$ .

Round your answer to three decimal places
- $S=$ (204.776) $\text(cm)$
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Since we are given the value $H=201$ , we can substitute it to the formula and solve for the arm span $(S)$ .

$S$ $=$ $1.195H-35.419$

$S$ $=$ $1.195(201)-35.419$ Substitute $H=201$

$S$ $=$ $204.776$

$S=204.776 \text(cm)$
Question 11 of 13

11. Question
The height and weight of $10$ girls were listed in a table of values. Draw the least square line of best fit given that its equation is $W=0.86H-72.22$ .
- 1.
- 2.
- 3.
- 4.
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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

First, create a table of values illustrating the values of Height $(x)$ and Weight $(y)$

We can use $145$ , $160$ and $180$ as test values of $x$ .

$x$ $145$ $160$ $180$

$y$

Next, recall that $W=y$ and $H=x$

Substitute the values of $x$ to $H$ to solve for the $y$ values.

$x=145$

$W$ $=$ $0.86H-72.22$

$W$ $=$ $0.86(145)-72.22$ Substitute $x=145$

$W$ $=$ $52.48$

$x=160$

$W$ $=$ $0.86H-72.22$

$W$ $=$ $0.86(160)-72.22$ Substitute $x=160$

$W$ $=$ $65.30$

$x=180$

$W$ $=$ $0.86H-72.22$

$W$ $=$ $0.86(180)-72.22$ Substitute $x=180$

$W$ $=$ $82.58$

$x$ $145$ $160$ $180$

$y$ $52.48$ $65.30$ $82.58$

Finally, plot these three points and draw a line connecting them.

$W=0.86H-72.22$

$x$	$145$	$160$	$180$
$y$

$x$	$145$	$160$	$180$
$y$	$52.48$	$65.30$	$82.58$

Question 12 of 13

Solve for the gradient given the following values.

$barx=28.5$

$bary=53.4$

$b=-3.6$

$m=$ (2)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

Use the gradient-intercept form and substitute the known values to solve for

$m$

Use

$bary$ and

$barx$ as substitute for

$y$ and

$x$ , respectively.

$b=-3.6$

$barx=28.5$

$bary=53.4$

$bary$	$=$	$m$ $barx+$ $b$	Gradient-Intercept Form
$53.4$	$=$	$($ $m$ $)28.5-$ $3.6$	Substitute values
$53.4$ $+3.6$	$=$	$28.5m-3.6$ $+3.6$	Add $3.6$ to both sides
$57$ $divide28.5$	$=$	$28.5m$ $divide28.5$	Divide both sides by $28.5$
$2$	$=$	$m$
$m$	$=$	$2$

$m=2$

Question 13 of 13

Solve for the

$y$ intercept given the following values.

$barx=40.5$

$bary=51.2$

$m=2/5$

$b=$ (35)

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The Least Squares Line of Best Fit is a more accurate interpretation of a correlation.

Gradient Formula

$\color{#007DDC}{m}=r\cdot\frac{\color{#9a00c7}{\sigma_y}}{\color{#00880A}{\sigma_x}}$

$y$ intercept Formula

$\color{#e65021}{b}=\bar{y}-\color{#007DDC}{m}\bar{x}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

Substitute the known values to the

$y$ intercept formula.

$m=2/5$

$barx=40.5$

$bary=51.2$

$b$	$=$	$\bar{y}-\color{#007DDC}{m}\bar{x}$	$y$ intercept Formula

	$=$	$51.2-\left(\color{#007DDC}{\frac{2}{5}}\right)40.5$	Substitute values

	$=$	$51.2-16.2$	Evaluate
$b$	$=$	$35$

$b=35$

$W$	$=$	$0.86H-72.22$
$W$	$=$	$0.86(175)-72.22$	Substitute $H=175$
$W$	$=$	$77.73$

$S$	$=$	$1.195H-35.419$
$S$	$=$	$1.195(201)-35.419$	Substitute $H=201$
$S$	$=$	$204.776$

$W$	$=$	$0.86H-72.22$
$W$	$=$	$0.86(145)-72.22$	Substitute $x=145$
$W$	$=$	$52.48$

$W$	$=$	$0.86H-72.22$
$W$	$=$	$0.86(160)-72.22$	Substitute $x=160$
$W$	$=$	$65.30$

$W$	$=$	$0.86H-72.22$
$W$	$=$	$0.86(180)-72.22$	Substitute $x=180$
$W$	$=$	$82.58$

Least Squares Line of Best Fit

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yyy intercept Formula

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yy intercept Formula

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yy intercept Formula

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yy intercept Formula

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yy intercept Formula

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Gradient Formula

$y$ $y$ intercept Formula

$y$ $y$ intercept Formula

$y$ $y$ intercept Formula

$y$ $y$ intercept Formula

$y$ $y$ intercept Formula

$y$ $y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula

$y$ intercept Formula