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3 years ago

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For a set of data, x is the explanatory variable. Its mean is 8.2, and its standard deviation is 1.92. For the same set of data,

y is the response variable. Its mean is 13.8, and its standard deviation is 3.03. The correlation was found to be 0.223.
Select the correct slope and y-intercept for the least-squares line.
a. Slope = -0.35
y-intercept = -10.9
b. Slope = -0.35
y-intercept = 10.9
c. Slope = 0.35
y-intercept = -10.9
d. Slope = 0.35
y-intercept = 10.9

1 answer:

aniked [119]3 years ago

4 0

Answer:

d. Slope = 0.35

y-intercept = 10.9

Step-by-step explanation:

The computation is shown below:

Data given in the question

Mean $\bar X$ = 8.2

The standard deviation of x = $\sigma$ = 1.92

Mean $\bar Y$ = 13.8

The standard deviation of y = $\sigma$ = 3.03

The correlation = r = 0.223

Based on the above information,

As we know that

The slope is

$b = r \frac{\sigma y }{\sigma x} \\\\ = (0.223) (\frac{3.03}{1.92} \\\\ = (0.223) (1.578125)$

= 0.3519

Now the y-intercept is

$a = \bar Y - b \bar X \\\\$

= 13.8 - (0.351922) 8.2

= 13.8 - 2.88579

= 10.91

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Answer:

Part 1) The equation of the perpendicular bisector side AB is $y=-\frac{1}{4}x+\frac{15}{4}$

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Part 3) The equation of the perpendicular bisector side AC is $y=-3x+4$

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Step-by-step explanation:

Part 1) Find the equation of the perpendicular bisector side AB

we have

A(–2, 0), B(0, 8)

<em>step 1</em>

Find the slope AB

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

substitute the values

$m=\frac{8-0}{0+2}$

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<em>step 2</em>

Find the slope of the perpendicular line to side AB

Remember that

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therefore

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<em>step 3</em>

Find the midpoint AB

The formula to calculate the midpoint between two points is equal to

$M(\frac{x1+x2}{2},\frac{y1+y2}{2})$

substitute the values

$M(\frac{-2+0}{2},\frac{0+8}{2})$

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<em>step 4</em>

Find the equation of the perpendicular bisectors of AB

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The equation in slope intercept form is equal to

$y=mx+b$

substitute

$4=(-\frac{1}{4})(-1)+b$

solve for b

$b=4-\frac{1}{4}$

$b=\frac{15}{4}$

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Part 2) Find the equation of the perpendicular bisector side BC

we have

B(0, 8) and C(4, 2)

<em>step 1</em>

Find the slope BC

The formula to calculate the slope between two points is equal to

$m=\frac{y2-y1}{x2-x1}$

substitute the values

$m=\frac{2-8}{4-0}$

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<em>step 2</em>

Find the slope of the perpendicular line to side BC

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

therefore

The slope is equal to

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<em>step 3</em>

Find the midpoint BC

The formula to calculate the midpoint between two points is equal to

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substitute the values

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<em>step 4</em>

Find the equation of the perpendicular bisectors of BC

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A(–2, 0) and C(4, 2)

<em>step 1</em>

Find the slope AC

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$m=\frac{y2-y1}{x2-x1}$

substitute the values

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<em>step 2</em>

Find the slope of the perpendicular line to side AC

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If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)

therefore

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so

$y=-3x+4$

Part 4) Find the coordinates of the point of concurrency of the perpendicular bisectors (P)

we know that

The point of concurrency of the perpendicular bisectors is called the circumcenter.

Solve by graphing

using a graphing tool

the point of concurrency of the perpendicular bisectors is P(0.091,3.727)

see the attached figure

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2 x+ 5y= 20-------- equation 1

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2x + 5y= 20 x (3)

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6x + 15y=60-------equation 3

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