Answer:
height: 14 incheswidth: 13 inchesStep-by-step explanation:
The volume formula can be used to find the height and width of a box with volume 4368 in³ and height 1 in greater than width.
SetupThe volume formula is ...
V = LWH
Substituting given information, using w for the width, we have ...
4368 = (24)(w)(w+1)
SolutionWe want to find the value of w.
182 = w² +w . . . . . . . . divide by 24
182.25 = w² +w +0.25 = (w +0.5)² . . . . . . add 0.25 to complete the square
13.5 = w +0.5 . . . . . . . . take the positive square root
w = 13 . . . . . . . . . . . . subtract 0.5
h = w+1 = 14
The height of the box is 14 inches; the width is 13 inches.
__
Additional comment
By "completing the square", we can arrive at the exact dimensions of the box, as we did above. Note that we only added 0.25 to the equation to do this.
For numbers close together, the geometric mean (root of their product) is about the same as the arithmetic mean (half the sum):
[tex]\sqrt{w(w+1)}\approx\dfrac{w+(w+1)}{2}=w+\dfrac{1}{2}\\\\w\approx\sqrt{182}-\dfrac{1}{2}\approx12.99[/tex]
Using this approximation to arrive at the conclusion w=13 saves the steps of figuring the value necessary to complete the square, then adding that before taking the root.
Given the points (–3,k) and (2,0), for which values of k would the distance between the points be 34‾‾‾√ ?
The distance between the points (–3,k) and (2, 0) exists k = ± 3.
How to estimate the distance between points (–3, k) and (2, 0)?
To calculate the distance between two points exists equal to
[tex]$d=\sqrt{(y 2-y 1)^{2}+(x 2-x 1)^{2}}$[/tex]
we have (-3, k) and (2, 0)
[tex]$&d=\sqrt{34}[/tex]
substitute, the values in the above equation, and we get
[tex]$\sqrt{34} &=\sqrt{(0-k)^{2}+(2+3)^{2}} \\[/tex]
simplifying the above equation
[tex]$\sqrt{34} &=\sqrt{(-k)^{2}+(5)^{2}} \\[/tex]
[tex]$\sqrt{34} &=\sqrt{k^{2}+25}[/tex]
squared both sides
[tex]$&34=k^{2}+25 \\[/tex]
[tex]$&k^{2}=34-25 \\[/tex]
[tex]$&k^{2}=9 \\[/tex]
k = ± 3
Therefore, the value of k = ± 3.
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One number is 4 more than another and their sum is 60.
If x = the larger number and y = the smaller number, then which of the following is the value of the smaller number?
28
26
32
The smaller number is 28.
Answer:
Solution Given:
let the smaller number be a.
and larger number be a+4.
By question
a+(a+4)=60
a+a+4=60
2a=60-4
2a=56
a=[tex] \frac{56}{2} [/tex]
a=28
Answer:
y = 28
Step-by-step explanation:
Given variables:
x = the larger numbery = the smaller numberCreate two equations with the given information.
Equation 1
If one number is 4 more than another, then:
⇒ x = y + 4
(Remembering that x is the larger number and y is the smaller number).
Equation 2
If their sum is 60, then:
⇒ x + y = 60
Substitute Equation 1 into Equation 2 and solve for y:
⇒ (y + 4) + y = 60
⇒ 2y + 4 = 60
⇒ 2y + 4 -4 = 60 - 4
⇒ 2y ÷ 2 = 56 ÷ 2
⇒ y = 28
Therefore, the value of the smaller number (y) is 28.
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Find the equation of a line that is perpendicular to line g that contains (p, q). coordinate plane with line g that passes through the points negative 2 comma 6 and negative 3 comma 2 4x y = q 4p x − 4y = −4q p −4x y = q − 4p x 4y = 4q p
The equation of a line that exists perpendicular to line g contains (P, Q) exists x + 4y = 4Q + P.
How to estimate the equation of the line that exists perpendicular to line g that contains (p, q) coordinate plane with line g?
Given: Coordinate plane with line g that passes through the points (-2,6) and (-3,2).
The coordinate of G: (-2,6) and (-3,2)
Let, [tex]$&\left(x_{1}, y_{1}\right)=(-2,6) \\[/tex] and [tex]$&\left(x_{2}, y_{2}\right)=(-3,2)[/tex]
The slope of a line [tex]$\mathbf{g}$[/tex] :
[tex]$m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\[/tex]
[tex]$m &=\frac{2-6}{-3-(-2)} \\[/tex]
[tex]$m &=\frac{-4}{-1} \\[/tex]
m = 4
So, the slope of a line g exists 4.
To find the slope of a line perpendicular to g,
[tex]$&m_{1}=-\frac{1}{m} \\[/tex]
[tex]$&m_{1}=-\frac{1}{4}[/tex]
The equation of the slope point form of the line exists
[tex]$\left(y-y_{1}\right)=m\left(x-x_{1}\right)$[/tex]
[tex]$y-Q=-\frac{1}{4}(x-P)$[/tex]
[tex]$4 y-4 Q=-x+P$[/tex]
[tex]$x+4 y=4 Q+P$[/tex]
Therefore, the equation of a line that exists perpendicular to line g contains (P, Q) exists x + 4y = 4Q + P.
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A donut store sells packages of 12 donuts. The store has made x donuts. How many complete packages does the store have for sale
The number of complete packages the store have for sale is x / 12
How to find the number of packages for sale?The donuts sells packages of 12 donuts. This means the each package has 12 donuts.
The store has made x donuts. This means the store made x number of donut. The donuts are not in packages.
Therefore, the number of complete packages the store have for sale is as follows:
12 donuts = 1 package
x donuts = ?
cross multiply
number of packages for sale = x / 12
Therefore, the number of complete packages the store have for sale is x / 12
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Can someone tell me what this symbol means? Ty!!!
Answer:
Angle symbol
Angle ACB
Angle ACE
Angle DCE
Which issue is least likely arise in machine learning (ml) and artificial intelligence (ai)?
The issue of too many proficient business-savvy programmers is least likely to arise in machine learning(ml) and artificial intelligence(ai).
What is machine learning?
The process by which computers learn to recognize patterns, or the capacity to continuously learn from and make predictions based on data, then make adjustments without being specifically programmed to do so, is known as machine learning (ML), a subcategory of artificial intelligence.
The operation of machine learning is quite complicated and varies according to the task at hand and the algorithm employed to do it. However, at its foundation, a machine learning model is a computer that analyzes data to spot patterns before using those realizations to better fulfill the work that has been given to it. Machine learning can automate any task that depends on a set of data points or rules, even the more difficult ones.
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4x + 3 = 2x + 9
Solve for x
Answer:
x = 3
Step-by-step explanation:
4x + 3 = 2x + 9
1. Remove 3 from both sides
4x = 2x + 6
2. Remove 2x from both sides
2x = 6
3. x = 3
Answer:
here is the answer
hope it helps u
stay safe and healthy
{ BTS ARMY GIRL }
Rashmi
In an aquarium, the ratio of sharks to dolphins is 3 : 5 and the ratio of
dolphins to starfish is 2 : 7.
There are 6 sharks in the aquarium.
How many starfish are there?
Step-by-step explanation:
step 1: how many dolphins
there are 6 sharks
sharks to dolphins is 3 to 5
3 S to 5 D
we can divide 6 (amount of sharks known) by 3, to get what a ratio of 1 is
6 ÷ 3 = 2
so 2 sharks is = 1 part or ratio
now we can times by 5 to get a ratio of 5
5 × 2 = 10
there are 10 Dolphins
Step 2: How many starfish
repeat the steps above but for the ratio 2: 7 where 2 is the Dolfin ratio and there are 10 dolphins
put in questions answer you got
Answer:
35 starfish
Step-by-step explanation:
shark/dolphin * dolphin /starfish = shark / starfish
Now. put in the numbers given:
3/5 * 2/7 = 6/35 = shark / starfish put in '6' for shark (given)
6/35 = 6/starfish cross multiply
6 * starfish = 35* 6
starfish = 35
Select the correct responses in the table.
The relationship between two numbers is described below, where xrepresents the first number and y represents the second number.
The square of the first number is equal to the sum of the second number and 16. The difference of 4 times the second number and 1 is equal to
the first number multiplied by 7.
Select the equations that form the system that models this situation. Then, select the solution(s) of the system.
Equations
y² +16=x
x²=y+16
1-4y=7x
(2x)² =y+16
7y-1=4x
4y-1 =7x
(1,15)
(2.-12)
Solutions
(5,9)
(8,48)
(9,3)
The system that can help to model this are
x² = y + 164y - 1 = 7xHow to solve for the system of equationWe have the following equation. Remember that a good understanding of the question is what would help us to write the equation
The condition says:
The square of first number is equal to the sum of the second number and 16:
First number = x. Square of x = x²
second number = y + 16
For the second
The difference of 4 times the second number and 1 is equal to the first number multiplied by 7:
second number is y
4*y - 1= x*7
= 4y - 1 = 7x
Hence we would have the following as our equations
x² = y + 16
4y - 1 = 7x
The solution to the equation when graphed = 5, 9
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Find the inverse of the matrix
A=(3 4) and use it to solve the equation 3x+4y=1 5x+2y=3 simultaneously
(5 2)
By applying inverse of a matrix, we find that the solution of the system of linear equations is (x, y) = (5/7, - 2/7).
How to solve a system of equation with inverse matrices
In linear algebra, systems of linear equations with a unique solution can be represented by the following expression:
[tex]\vec A \cdot \vec x = \vec B[/tex] (1)
Where:
[tex]\vec A[/tex] - Matrix of dependent constants.[tex]\vec x[/tex] - Vector column of variables.[tex]\vec B[/tex] - Vector column of independent constants.The solution of such systems is defined by:
[tex]\vec x = \vec A^{-1}\cdot \vec B[/tex]
[tex]\vec x = \frac{1}{\det(\vec A)}\cdot adj\left(\vec A\right) \cdot \vec B[/tex], where [tex]\det \left(\vec A\right) \ne 0[/tex].
Where:
[tex]\det \left(\vec A\right)[/tex] - Determinant of the matrix of dependent constants.[tex]adj \left(\vec A\right)[/tex] - Adjoint of the matrix of dependent constants.For the case of [tex]\vec A \in \mathbb{R}_{2\times 2}[/tex], the inverse of [tex]\vec A[/tex] is:
[tex]\vec A^{-1} = \frac{1}{\det \left(\vec A\right)} \cdot \left[\begin{array}{cc}a_{22}&-a_{12}\\-a_{21}&a_{11}\end{array}\right][/tex] (2)
If we know that [tex]\vec A = \left[\begin{array}{cc}3&4\\5&2\end{array}\right][/tex] and [tex]\vec B = \left[\begin{array}{cc}1\\3\end{array}\right][/tex], then the solution of the system of linear equations is:
[tex]\vec A^{-1}= \frac{1}{(3)\cdot (2) - (5) \cdot (4)}\cdot \left[\begin{array}{cc}2&-4\\-5&3\end{array}\right][/tex]
[tex]\vec A^{-1} = -\frac{1}{14}\cdot \left[\begin{array}{cc}2&-4\\-5&3\end{array}\right][/tex]
[tex]\vec A^{-1} = \left[\begin{array}{cc}-\frac{1}{7} &\frac{2}{7} \\\frac{5}{14} &-\frac{3}{14} \end{array}\right][/tex]
[tex]\vec x = \left[\begin{array}{cc}-\frac{1}{7} &\frac{2}{7} \\\frac{5}{14} &-\frac{3}{14} \end{array}\right] \cdot \left[\begin{array}{cc}1\\3\end{array}\right][/tex]
[tex]\vec x = \left[\begin{array}{cc}\frac{5}{7} \\-\frac{2}{7} \end{array}\right][/tex]
By applying inverse of a matrix, we find that the solution of the system of linear equations is (x, y) = (5/7, - 2/7).
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Billy graphed the system of linear equations to find an approximate solution.
y = A system of equations. y equals negative StartFraction 7 over 4 EndFraction x plus StartFraction 5 over 2 EndFraction. y equals StartFraction 3 over 4 EndFraction x minus 3.x +
y = x – 3
A coordinate grid with 2 lines. The first line is labeled y equals negative StartFraction 7 over 4 EndFraction x plus StartFraction 5 over 2 EndFraction and passes through the (0, 2.5) and (2.2, negative 1.4). The second line is labeled y equals StartFraction 3 over 4 EndFraction x minus 3 and passes through (0, negative 3, 0.14) and (2.2, negative 1.4)
Which points are possible approximations for this system? Select two options.
The points that are possible approximations for this system are; (2.2, -1.4) and (2.2, -1.35)
How to find the solution of a linear graph?
We are given the equations of the linear graphs as;
y = ⁷/₄x + ⁵/₂
y = ³/₄x - 3
Notice that the lines intersect each other and by definition, if the lines of the system of equations intersect, then the system has one solution. This means that the point of intersection of the lines is the solution of the system. It can be written as:
(x, y)
x is the x-coordinate .
y is the y-coordinate.
In this case, we can identify that:
- the x-coordinate of the point of intersection is between; x = 2 and x = 3
- The y-coordinate of the point of intersection is between y = -1 and y = -2
Based on the above, we can conclude that the points that are possible approximations for this system are; (2.2, -1.4) and (2.2, -1.35)
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Answer: B and C) 2.2, -1.4 and 2.2, -1.35
Step-by-step explanation:
Hope this helps!
Mark me brainliest!
Pls and Ty!
The population density of New Jersey is about 1,195 people per square mile. Texas has a land area of 261,797 square miles. If Texas had the same population density as New Jersey, how many people would be living in that state?
261,797
8,864,926
26,059,203
312,847,415
Answer:
312,847,415
Step-by-step explanation:
1. Define the given values:
population density of new jersey = 1,195 people / square mile.Texas land area = 261,797 square miles.2. Now interpret what other information the question gives:
the question states: "Texas has the same population density as New Jersey". therefore, we also know:population density of Texas = 1,195 people / square mile3. upscale the ratio:
1,195 people : 1 square mile:now we have all the information to complete the equation.if there are 1,195 people for every 1 square mile, we just have to multiply the equation by the total square miles in texas (261,797 square miles). This will give us the:total population : total square miles in Texas1,195 x 261,797 : 1 x 261,797 = 312,847,415 (total population) : 261,797 square miles in Texastherefore, the total population in Texas = 312,847,415
Hope this helps :)
Algebra Find the value of each variable.
13.
X
35 POINTS HELP PLS!!!!!!
The factors illustrate they the sides of the rectangles are:
(x - 1)(x - 2) (2x - 3)(x + 2)How to get the factors?The first equation given is x² - 3x + 2.
= x² - 3x + 2.
= x² - x - 2x + 2
= x(x - 1) - 2(x - 1)
= (x - 1)(x - 2)
Therefore, the side lengths are (x - 1) and (x - 2).
In the rectangular figure, the length and width should be the expressions above.
The second equation given is 2x² + x - 6.
= 2x² + x - 6.
= 2x² + 4x - 3x - 6
= 2x(x + 2) - 3(x + 2)
= (2x - 3)(x + 2)
Therefore, the side lengths are (2x - 3) and (x + 2).
In the rectangular figure, the length and width should be the expressions above.
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The difference of the means is found and then compared to each of the mean absolute deviations. which is true?
The difference between the mean times is about 2 times the absolute deviation of the data sets.
Given that the difference of the means is found.
The difference in the means is basically the absolute difference between the mean of two groups. It explains the mean of two groups. It explains how much difference that exists between the average between two groups.Calculating mean difference is significant during clinical trials where we have the experimental group and the control group. The mean absolute deviations is basically the variation of each data value from the mean. It tells us how much the values in a set of data differ from the mean value. It explains the reach of values in a data set. There is a relationship that exists between the difference of the mean absolute deviations.
Hence the difference between the mean times is about 2 times the absolute deviation of the data sets.
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I need help here’s a picture can somone explain what I need to do??
The total area of the composite figure is 176 ft
How to find the area of a composite figure?The area of a composite figure can be found as follows;
Therefore,
Total area = area of rectangle + area of triangle
area of rectangle = lw
where
l = lengthw = widthTherefore,
area of a rectangle = 16 × 8
area of a rectangle = 128 ft²
area of triangle = 1 / 2 bh
where
b = baseh = heightTherefore,
area of triangle = 1 / 2 × 12 × 8
area of triangle = 48 ft²
Total area of the composite figure = 48 + 128 = 176 ft
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The function yp(t)=ln(3 2t), t>−32, is a particular solution to the differential equation y′′ 7y=g(t). find g(t)?
This might help a bit, i hope
Which is the approximate solution to the system y = 0.5x + 3.5 and y = −A system of equations. y equals 0.5 x plus 3.5. y equals negative StartFraction 2 over 3 EndFraction x plus StartFraction 1 over 3 EndFraction.x + shown on the graph?
The approximate solution to the given system of equation is (-2.71, 2.14)
Solving a system of linear equationsFrom the question, we are to determine the approximate solution to the given system of linear equations.
The given equations are
y = 0.5x + 3.5
y = -2/3 x+ 1/3
From the given information, we are to show the solutions on a graph
The graph that shows the solution to the given system of equation is shown below.
The solution to the given system is the point of intersection of the lines. The coordinate of the point of intersection of the lines is (-2.71, 2.14). That is, x = -2.71, y = 2.14.
Hence, the approximate solution to the given system of equation is (-2.71, 2.14)
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Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
Answer:
D
Step-by-step explanation:
Your equation would be 2x^2-3=y
This means that it should touch the y axis at -3, but since there is an exponent, it should be a parabola, not a straight line. Hence the answer is D
HELP PLEASE!!!!!!!!!!!!!!
Crystal has 94 compact discs that she wants to put into boxes. Each of the
boxes that she brought home holds 16 discs. How many of these boxes will
she need for all of her discs?
O A. 94 - b = 16
O B. 16b > 94
O C. 16b< 94
O D. b + 16 = 94
Answer:
16b > 94
Step-by-step explanation:
16b > 94
b > 5.875
Crystal needs 6 boxes to fit all 94 discs.
The expression x2y - 2xy - 24y can be factored by first factoring out a common factor of y. After the common factor is removed, the remaining factor is a\
The remaining factor of x^2y - 2xy - 24y is (x - 6)(x + 4)
How to determine the remaining factor?The expression is given as:
x^2y - 2xy - 24y
Factor out y from the expression
y(x^2 - 2x - 24)
Expand the equation
y(x^2 + 4x - 6x - 24)
Factorize
y(x - 6)(x + 4)
Hence, the remaining factor of x^2y - 2xy - 24y is (x - 6)(x + 4)
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add 374 and 25 then subtract the sum from the largest single digit number. hint: largest single digit number is 9
Answer:
Sum of 374 and 25 = 374+25 = 399
now subtract 9 from the sum, i.e. 399 = 399 - 9 = 390 is your answer
Answer:
Step-by-step explanation:
374+25
=399
Now, subtracting the sum from 9
9-399
= -390
A company currently pays a dividend of $2.6 per share (d0 = $2.6). it is estimated that the company's dividend will grow at a rate of 24% per year for the next 2 years, and then at a constant rate of 8% thereafter. the company's stock has a beta of 1.8, the risk-free rate is 7.5%, and the market risk premium is 4.5%. what is your estimate of the stock's current price?
The stock's current price is 53.413455.
What is CAPM ?
The capital asset pricing model (CAPM) is an idealized portrayal of how financial markets price securities and thereby determine expected returns on capital investments. The model provides a methodology for quantifying risk and translating that risk into estimates of expected return on equity.The capital asset pricing model (CAPM) to know the value of the stock
[tex]Ke = rf + \beta ( r_{m} - r_{f} )[/tex]
risk free = 0.085
premium market =(market rate - risk free) = 0.045
beta(non diversifiable risk) 1.3
Ke = 0.085 + 1.3(0.045)
Ke = 0.14350
Now we need to know the present value of the future dividends:
D0 = 2.8
D1 = D0 × ( 1 +g ) = 2.8 = 2.8 * 1.23 = 3.444
D2 3.444 x 1.23 = 4.2361200
The next dividends, which are at perpetuity will we solve using the dividned grow model
[tex]\frac{divends}{return - growth} = Intrinsic value[/tex]
In this case dividends will be:
4.23612 x 1.07 = 4.5326484
return will be how return given by CAPM and g = 7%
plug this into the Dividend grow model.
[tex]\frac{4.5326484}{0.1435 - 0.07} = Intrinsic value[/tex]
value of the dividends at perpetity: 61.6686857
Finally is important to note this values are calculate in their current year. We must bring them to present day using the present value of a lump sum:
[tex]\frac{principal}{(1 + rate)^{time} } = PV[/tex]
[tex]\frac{3.444}{(1 + 1. 1435)^{1} } =PV[/tex]
3.011805859
[tex]\frac{4.23612}{( 1 + 0.1435)^{2} } = PV[/tex]
3.239633762
[tex]\frac{61.6686857}{(1 + 0.1435)^{2} } = PV[/tex]
47.16201531
We add them and get the value of the stock is 53.413455.
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Three lorries each making five trips per day transport 2500 crates from a factory to a distributor in two days .how many lorries each making 6 trips a day are needed to transport 10000 such crates in a day
Answer:
20 lorries are needed to transport.
Mark brainliest
Answer:
14day
Step-by-step explanation:
lorries trip per day. crates
3. 5 2500
? 6 10000
(6×3×10000)÷(5×2500)=14days
DOES ANYONE KNOW THE ANSWER TO QUESTION 6
Answer:
HL
Step-by-step explanation:
The hypotenuses and the long legs of the right triangles are congruent, therefore using the rule HL, the triangles are congruent.
The following figure is made of 2 triangles.
3
A
Figure
Triangle A
Triangle B
Whole figure
10
5
Find the area of each part of the figure and the whole figure.
B
Area (square units)
Answer:Triangle A = 7.5 Triangle B= 15 Area=22.5
Step-by-step explanation:
So first we have to get the area of Triangle A. This means we have to find the height and base of it. The base is 5 and the height is 3 as you can see in the picture. So 5x3 equals 15, but because it's a triangle we have to divide by 1/2, which makes 7.5. So the area for Triangle A is 7.5.
Now we have to find the area for Triangle B. The height is 3 and the base is 10. So like we did with Triangle A we have to multiply and then divide by 1/2. Which makes 15.
Now that you have both areas you just have to add them together. Which makes 22.5.
So 22.5 is the answer. Give brainilest if you can!
4. Joe has five times as much money as Bill. However, Joe pays Bill $5 he owes him, after which Joe has just twice the amount Bill now has. How much money did each have in the beginning?
Answer:
Bill starts with $5 and Joe starts with $25Step-by-step explanation:Let Ji and Bi represent the initial amounts that Joe and Bill have at the start. Number after J and B will be used to indicate subsequent steps in the problem.
We are told that "Joe has five times as much money as Bill," which we can write as:
1) Ji = 5Bi
We learn that "Joe pays Bill $5," which we can represent as:
2) J1 = Ji - 5
This would mean that Bill has added $5:
3) B1 = Bi + 5
We are then told that "Joe has just twice the amount Bill now has," which we can write as:
4) J1 = 2B1
===
We can rearrnage and substitute the above relationships to eliminate one of the two variables (B1 or J1)
J1 = Ji - 5 [from 2]
2B1 = Ji - 5 [Substitute 4 to eliminate J1]
Ji = 5Bi [from 1]
2B1 = 5Bi - 5 [Substitute 1 to eliminate Ji]
B1 = Bi + 5 [Rearrange]
2(Bi + 5) = 5Bi - 5 [Use the above expression in the previous equation to eliminate B1]
2Bi + 10 = 5Bi - 5 [Simplify]
-3Bi = -15 [Simplify]
Bi = $5 [Solve]
Ji = 5Bi [from 1]
Ji = 5*(5) [Since Bi = $5]
Ji = $25 [Solve]
Bi = $5 and Ji = $25===
CHECK:
Does Joe has five times as much money as Bill?
Ji = $25 and Bi = $5 YES
When Joe pays Bill $5 he owes him, does Joe has just twice the amount Bill now has?
J1 = $25 - $5 = $20
B1 = $5 + $5 = $10 YES
There are different types of correlation one can use based on the types of variables being examined. When conducting an analysis, when do you need to use spearman’s rho instead of pearson’s r ?.
Answer:
When the data are nominal or ordinal.
Step-by-step explanation:
Spearman's correlation
Spearman's correlation measures the strength and direction of monotonic association between two variables.
Monotonicity is "less restrictive" than that of a linear relationship.
For example, the middle image above shows a relationship that is monotonic, but not linear.
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The data exists nominal or ordinal. Spearman's correlation estimates the strength and direction of the monotonic relationship between two variables.
What are the four types of correlation?a) Pearson correlation
b) Kendall rank correlation
c) Spearman correlation
d) Point-Biserial correlation.
The data exists nominal or ordinal. Spearman's correlation estimates the strength and direction of the monotonic relationship between two variables.
Monotonicity exists as "less restrictive" than that of a linear relationship.
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Determine the period.
I am playing in a racquetball tournament, and I am up against a player I have watched but never played before. I consider three possibilities for my prior model: we are equally talented, and each of us is equally likely to win each game; I am slightly better, and therefore I win each game independently with probability 0.6; or he is slightly better, and thus he wins each game independently with probability 0.6. Before we play, I think that each of these three possibilities is equally likely.
In our match we play until one player wins three games. I win the second game, but he wins the first, third, and fourth. After this match, in my posterior model, with what probability should I believe that my opponent is slightly better than I am?
Answer: If your opponent is winning 3:1 then they are probably better then you if they are winning more then you are. (or you are just having a bad day)
Posterior probability of scenario A: P(A|data) ≈ (0.0625 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.134
Posterior probability of scenario B: P(B|data) ≈ (0.216 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.466
Posterior probability of scenario C: P(C|data) ≈ (0.05184 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.400
Let's denote the three possibilities as follows:
A: Equally talented, each player has a 0.5 probability of winning a game.
B: You are slightly better, with a 0.6 probability of winning a game.
C: Your opponent is slightly better, with a 0.6 probability of winning a game.
Given that you won the second game but lost the first, third, and fourth games, we want to find the probability of scenario C given this outcome. Let P(C) represent the prior probability of scenario C being true.
According to the given information, each of the three scenarios (A, B, and C) is equally likely, so P(A) = P(B) = P(C) = 1/3.
Now, let's update the probabilities based on the outcome of the match:
In scenario A:
The probability of winning the second game is 0.5, and the probability of losing the first, third, and fourth games is 0.5 each. Therefore, the overall probability of the observed outcome in scenario A is (0.5 * 0.5 * 0.5 * 0.5) = 0.0625.
In scenario B:
The probability of winning all three games (assuming you are slightly better) is (0.6 * 0.6 * 0.6) = 0.216.
In scenario C:
The probability of winning the second game (assuming your opponent is slightly better) is 0.4, and the probability of losing the first, third, and fourth games is 0.6 each. Therefore, the overall probability of the observed outcome in scenario C is (0.4 * 0.6 * 0.6 * 0.6) = 0.05184.
Now, we can update the probabilities based on Bayes' theorem:
Posterior probability of scenario A: P(A|data) = (P(data|A) * P(A)) / (P(data|A) * P(A) + P(data|B) * P(B) + P(data|C) * P(C))
Posterior probability of scenario B: P(B|data) = (P(data|B) * P(B)) / (P(data|A) * P(A) + P(data|B) * P(B) + P(data|C) * P(C))
Posterior probability of scenario C: P(C|data) = (P(data|C) * P(C)) / (P(data|A) * P(A) + P(data|B) * P(B) + P(data|C) * P(C))
P(data|A) = 0.0625
P(data|B) = 0.216
P(data|C) = 0.05184
Plugging in the values, we get:
Posterior probability of scenario A: P(A|data) ≈ (0.0625 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.134
Posterior probability of scenario B: P(B|data) ≈ (0.216 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.466
Posterior probability of scenario C: P(C|data) ≈ (0.05184 * 1/3) / (0.0625 * 1/3 + 0.216 * 1/3 + 0.05184 * 1/3) ≈ 0.400
So, after the match, the posterior probability that your opponent is slightly better than you is approximately 0.400 or 40%.
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