MULTIVARIATE STATISTICS
(a) What are the two ways in which the exploratory factor
analysis (EFA) can be useful?
(b) Differentiate between principal component method
and factor analysis

In the figure angle5 and angle 6 are vertical angles

Vertical angles are a pair of angles formed by two intersecting lines or rays. They are opposite to each other and have equal measures. In other words, if two lines intersect at a point, the angles that are across from each other, or directly opposite each other, are vertical angles.

Vertical angles are always congruent, which means that they have the same angle measurement. This can be proven using the properties of parallel lines and transversals, which state that when a transversal intersects two parallel lines, the alternate interior angles are congruent, and the corresponding angles are congruent.

Since vertical angles are formed by intersecting lines, they are neither adjacent nor supplementary.

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Answer

Choice C: (1 1/4 + 2 3/8) / 3/4

Step-by-step explanation:

His salad consists of 1 1/4 cups of purple cabbage and 2 3/8 cups of lettuce. First, you would add those two together to get how much the salad is. THEN divide by how much he eats each day which is 3/4.

Answer:

18

Step-by-step explanation:

We know that for every 4 green marbles, there are 3 blue marbles.
so if there are 8 green marbles, there are 6 blue marbles.
This continues on until we get 24 blue marbles.

Expression : Taking Green Marbles as 'x' and Blue Marbles as 'y'

4x = 3y

so if there are 24 Blue marbles , this means:

4x = 3 x 24

4x = 72

x = 72/4 = 18 Marbles are green

After using trig identities for all three given functions are a). [tex]x(t) = √29cos(4t - 111.8°)[/tex] , b). [tex]x(t) = √29cos(4t - 68.2°),[/tex] c). [tex]x(t) = e^-2t(√29cos(4t - 111.8°))[/tex].

Using the trig identity [tex]cos(A - B) = cos(A)cos(B) + sin(A)sin(B)[/tex], we can rewrite [tex]x(t) = -2cos(4t) + 5sin(4t)[/tex] using only one cosine function as follows : [tex]x(t) = Rcos(4t - α)[/tex] where [tex]R = √((-2)^2 + 5^2) = √29[/tex] and [tex]α = tan^-1(5/-2) = 111.8°[/tex]. Therefore, x(t) = √29cos(4t - 111.8°)

Similarly, using the same trig identity, we can rewrite [tex]x(t) = 2cos(4t) + 5sin(4t)[/tex] using only one cosine function as follows : [tex]x(t) = Rcos(4t - α)[/tex] where [tex]R = √(2^2 + 5^2) = √29[/tex] and [tex]α = tan^-1(5/2)[/tex] = 68.2°. Therefore, [tex]x(t) = √29cos(4t - 68.2°)[/tex].

Using the same trignometric identity, we can rewrite[tex]x(t) = e^-2t(-2cos(4t) + 5sin(4t))[/tex] using only one cosine function as follows: [tex]x(t) = e^-2t(Rcos(4t - α))[/tex] where R = [tex]√((-2)^2 + 5^2) = √29[/tex] and [tex]α = tan^-1(5/-2) = 111.8°[/tex]. Therefore, x(t) =[tex]e^-2t(√29cos(4t - 111.8°))[/tex]

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Answer:

The model train can travel 2 feet per second.

Answer:13.5

Step-by-step explanation:

use sinuses theorem:

[tex]\frac{x}{sina}=\frac{y}{sinb} \\\frac{12}{sin63}=\frac{x}{sin90} \\x=\frac{12}{sin63}=13.5[/tex]

Yes, this is a statistical question as the answer involves gathering and analyzing data about college football and tennis coaches' salaries. It's important to remember that a variety of factors could influence these salaries.

Sylvie's question, 'Do college tennis coaches generally get paid more than college football coaches?' is considered a statistical question because it anticipates a response based on data analysis. To answer this, one would typically gather data on the salaries of both college tennis and football coaches, process this data, and then perform a comparative analysis. It's important to understand that salaries can vary widely, influenced by factors including the size of the college, the success of the program, and the region of the country.

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Homer would have taken approximately 8.11 hours to do the job by himself.

Let h be the time it would take Homer to do the job by himself, and let m be the time it would take Mike to do the job by himself. We know from the problem that:

m = h - 1 (Mike can do the job one hour quicker than Homer)

We also know that they worked together for 4 hours, so they completed 4/h + 4/m of the job. We can set up an equation based on this:

4/h + 4/m = 1 (they completed the entire job)

Substituting m = h - 1, we get:

4/h + 4/(h-1) = 1

Multiplying both sides by h(h-1), we get:

4(h-1) + 4h = h(h-1)

Simplifying:

8h - 4 =

[tex]h^2 - h[/tex]

Rearranging:

[tex]h^2 - 9h + 4 = 0[/tex]

Using the quadratic formula, we get:

[tex]h = (9 ± \sqrt{(9^2 - 414)) / 2} [/tex]

Since we are looking for a positive value of h, we take the positive square root:

h ≈ 8.11

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Therefore , the solution of the given problem of expressions comes out to be a) 10⁻4.10⁻⁶ = 10⁻¹⁰ , b) -5⁸/-5⁴ = -5⁴ and c) y⁶/y⁷ = 1/y.

Calculations that involve joining, random subdivision variable changing multipliers, and removal are required. They could accomplish the following if they united: An algorithm, some information, and a mathematical challenge. Formulas, factors, and arithmetic operations like additions, deductions, errors, and groupings are all present in a declaration of truth. Words and phrases can be evaluated and analysed.

Here,

a) 10⁻4.10⁻⁶

b) -5⁸/-5⁴

c) y⁶/y⁷

and

To simplify this expression :

a) 10⁻4.10⁻⁶ = 10⁻¹⁰

b) -5⁸/-5⁴ = -5⁴

c) y⁶/y⁷ = 1/y

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The value of s is 107° for this we have know the definition and property of triangle.

A triangle is a polygon that has three(3) edges and three(3) vertices. It is one of the fundamental(basic) shapes in geometry.

Vertices, A vertex in geometry is a location where two(2) or more curves, lines, or edges come together.

s-37° + s-38° + x = 180° (Sum of three angle is 180°)

2s + x = 180° + 75°

2s + x = 255° ( Equation 1 )

s+32° + x = 180°

s + x = 180° - 32°

s + x = 148° ( Equation 2)

After solving equation 1 and 2 we get,

s = 107°

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Accοrding tο the given infοrmatiοn, nοne οf the given vectοrs οr pairs οf vectοrs fοrm a basis fοr R2.

A vectοr space is a cοllectiοn οf οbjects called vectοrs, which can be added tοgether and multiplied by scalars (usually real numbers), in a way that satisfies certain axiοms. These axiοms define the prοperties that the additiοn and scalar multiplicatiοn οperatiοns must have in οrder fοr the set οf vectοrs tο be a vectοr space.

Tο determine if a set οf vectοrs fοrms a basis fοr a vectοr space, we need tο check if the vectοrs are linearly independent and if they span the entire space.

Let's cοnsider the given set οf vectοrs:

v1 = (1, -1)

v2 = (2, -2)

v3 = (3, -2, 0)

v4 = (1, 4, 1)

v5 = (1, -1)

Nοte that these vectοrs have different dimensiοns, sο they cannοt fοrm a basis fοr the same vectοr space.

Vectοrs v1 and v2 are in R2, and they are linearly dependent, since v2 is a scalar multiple οf v1. Therefοre, {v1, v2} cannοt fοrm a basis fοr R2.

Vectοr v3 is in R3, and it is nοt pοssible fοr it tο fοrm a basis fοr R2.

Vectοr v4 is in R3, and it cannοt fοrm a basis fοr R2 either.

Vectοrs v1 and v5 are in R2, and they are alsο linearly dependent, since v5 is a scalar multiple οf v1. Therefοre, {v1, v5} cannοt fοrm a basis fοr R2.

In cοnclusiοn, nοne οf the given vectοrs οr pairs οf vectοrs fοrm a basis fοr R2.

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Complete question:

The expression which could be used to determine the area of the triangle is: One-half (2) (three-fourths) (StartFraction 7 Over 8 EndFraction).

Expression in math is a combination of numbers, operations, variables, and/or other mathematical symbols that represent a quantity or an action. It is a statement that can be evaluated to a single value, or a set of values. Expressions in math can be used to solve equations, calculate values, and form models of real-world scenarios.

This expression multiplies together the base, two and three-fourths, and the height, StartFraction 7 Over 8 EndFraction, and then divides the result by two to give the area of the triangle.

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Answer:

By the Rational Root Theorem, the possible rational roots of a polynomial are of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

The constant term of P(x) is 4, whose factors are ±1 and ±4. The leading coefficient is 10, whose factors are ±1, ±2, ±5, and ±10.

So, the possible rational roots of P(x) are: ±1/1, ±2/1, ±4/1, ±1/10, ±2/10, ±5/10, and ±10/10. This gives us a total of 8 possible rational roots.

Therefore, the answer is a. 8.

Step-by-step explanation:

If Ricky walks οne lap arοund the perimeter οf the pοnd then he walks 443.14 feet far.

The British imperial and American custοmary systems οf measurement bοth use the unit οf length knοwn as the fοοt (plural: feet), standard symbοl: ft. Usually used as an alternate symbοl is the prime symbοl, One yard is equal tο three feet in bοth custοmary and imperial measures; οne fοοt is equal tο 12 inches.

Thrοughοut histοry, the "fοοt" was a cοmpοnent οf a number οf regiοnal systems οf measurement, including the Greek, Rοman, Chinese, French, and English systems. Frοm οne natiοn tο the next, frοm οne city tο the next, and perhaps frοm οne trade tο the next, it varied in length. Its length was οften divided intο 12 inches οr 16 numbers and ranged frοm 250 tο 335 mm in length, thοugh this was nοt always the case.

Hence, The diameter οf pοnd 141 feet

The radius οf pοnd = 141/2 feet

Area οf pοnd = 2πr

Or,  Area οf pοnd = 2×(22/7)×(141/2)

Or, Area οf pοnd = 443.14 feet

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The vertex of the absolute function for the paddle boards is at x = 10.

A function that wraps an algebraic expression in absolute value symbols is known as an absolute value function.

So,

lengths of paddle boards is defined by :

V(x) = |x - 10|.

Thus, the vertex of the absolute function for the paddle boards is at x = 10.

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Answer:

Step-by-step explanation:

v(x)=||x-10||

Answer:

∠OPM

Step-by-step explanation:

∠QPR = ∠3 = ∠1 = ∠OPM

Answer:

∠3

Step-by-step explanation:

We can see that the sequence  1, 3, 9, 27 is not arithmetic.

An arithmetic sequence is a sequence where the difference between any pair of consecutive terms is a constant knowed as the common difference d.

Generally, the recursive relation for these sequences is:

a(n) = a(n - 1) + d

Here we have the sequence:

1, 3, 9, 27, ...

Taking the differences between consecutive terms of the sequence we get:

3 - 1 = 2

9 - 3 = 6

27 - 9 = 18

The differences are different, so there  is no common difference, meaning that this sequence is not arithmetic.

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The initial velocity required in the roller coaster design is also 26 m/s.

what is an algebraic expression?

An algebraic expression is a mathematical phrase consisting of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

To determine the initial velocity required in a roller coaster design, we can use the conservation of energy principle. At the top of the drop, the total energy of the roller coaster is potential energy, given by mgh, where m is the mass of the roller coaster, g is the acceleration due to gravity, and h is the height of the drop.

At the bottom of the drop, the total energy of the roller coaster is kinetic energy, given by (1/2)mv^2, where v is the velocity of the roller coaster at the bottom.

By equating the two energies, we get:

mgh = (1/2)mv^2

Solving for the initial velocity, we get:

v = sqrt(2gh)

where h is the height of the drop.

Substituting the given values, we get:

v = sqrt(2 x 9.8 m/s^2 x 34 m) = 26 m/s

Therefore, the initial velocity required in the roller coaster design is also 26 m/s.

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Answer:

measure of arc JL is 170°

Step-by-step explanation:

The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Inscribed angles that intercept the same arc are congruent.

so 85*2 is 170°

The standard deviation is the square root of the variance and is used to measure how spread out the data is. To calculate the standard deviation, take the square root of the variance.

Variance is the measure of how far a set of numbers are spread out from their average value. It is a measure of how much variation or dispersion exists from the average. Variance is represented as the square of the standard deviation, which is a measure of the spread of numbers in a set.

The mean of the data is 18.25. The variance is a measure of the spread of the data away from the mean, and is calculated by taking the average of the squared differences from the mean. To calculate the variance, subtract the mean from each data point, square this result and add up the values. Then divide this sum by the number of data points minus one.

For example, if the data set contains the values 10, 15, 20, 25 and 30 then the mean is 18.25. Subtracting the mean from each data point gives -8.25, -3.25, 1.75, 6.75, 11.75. Squaring each of the differences gives 68.0625, 10.5625, 3.0625, 45.5625, 137.0625. Adding these results together gives 264.3125. Dividing this by 4 (the number of data points minus one) gives 66.075. Taking the square root of this gives 8.1 as the standard deviation.

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The variance of the given data is calculated to be 118.63 and the standard deviation is 10.89.

The variance of the data is a measure of how far the data points are spread out from the mean. It is calculated by taking the differences between each data point and the mean, squaring the differences, and then taking the average of those squared differences.

For our data, the mean is 18.25. To find the variance, we can subtract 18.25 from each of the data points and square the differences. Then we can add up all of the squared differences and divide by the number of data points.

Variance = (2 - 18.25)² + (3 - 18.25)² + (4 - 18.25)² + (18 - 18.25)² + (20 - 18.25)² + (25 - 18.25)²

= (15.752 + 14.752 + 13.752 + 0.752 + 1.752 + 6.752) ÷ 6

= (255.2 + 217.2 + 190.2 + 0.6 + 3.0 + 45.8) ÷ 6

= 711.8 ÷ 6

Variance = 118.63

To find the standard deviation, we take the square root of the variance.

Standard deviation = √118.63

Standard deviation = 10.89

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Question:

The mean of the data is 18.25. Find the variance. Find the standard deviation.

Data: 2, 3, 4, 18, 20, 25

Therefore , the solution of the given problem of coordinates comes out to be y = (66/49)x and  y = (-11/6)x + 110.67 .

By utilising one or more variables or coordinates, a coordinate system can precisely find coordinates or other algebraic objects on such an area, including Euclidean space. Coordinates, that appears to be couples of numbers, can be used to find a point or object on a double plane. The y and x vectors on even a two-dimensional surface indicate the position of a point. a set of pictures that serve as markers for particular places.

Here,

Find the slope and equation of the line that goes through points B and D using their coordinates.

BD's slope is:

=>  m = (y2 - y1) / (x2 - x1) (x2 - x1)

=>  m = (75 - 86) / (44 - 38) (44 - 38)

=>  m = -11 / 6

Using the slope and position B:

=>  y - y1 = m(x - x1) (x - x1)

=>  y - 86 = (-11/6)(x - 38) (x - 38)

Simplifying:

=>  y = (-11/6)x + 110.67

The line that best matches points B and D has this equation as its equation.

Connect the starting point and position E with a line. Write the equation y = mx after determining the slope, m.

OE's slope is:

=>  m = (y2 - y1) / (x2 - x1) (x2 - x1)

=>  m = (66 - 0) / (49 - 0) (49 - 0)

=>  m = 66 / 49

Using the slope and position E:

=>  y - y1 = m(x - x1) (x - x1)

=>  y - 66 = (66/49)x

Simplifying:

=> y = (66/49)x

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The twenty-third term of the given arithmetic sequence is 192.

The formula to find the nth term of an arithmetic sequence with a common difference (d) is Tn = a + (n - 1)d, where a is the first term of the sequence. To solve for the twenty-third term of the given sequence, we can substitute the given values into the formula. The common difference (d) is 10 and the eighth term (a) is 52, so the formula becomes T23 = 52 + (23 - 1)10. When we solve this equation, we get T23 = 192. Therefore, the twenty-third term of this arithmetic sequence is 192.

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The autograph of the professional basketball player's was worth $250.00 a year ago.

Given that, in the last year, the value of a professional basketball player's autograph rose 40% and it is now worth $350.00.

To determine it worth a year ago, let's call the original value of the autograph "x".

We know that the value increased by 40%, which means it became:

100% + 40% = 140% of its original value.

We can set up the following equation:

140% × x = 350

(140/100)x = 350

1.4x = 350

To solve for x, we can divide both sides by 1.4:

x = 350 ÷ 1.4

x = 250

Therefore, the autograph was worth $250.00 a year ago.

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Julie Spiros's basic premium is [tex]740[/tex] per year, and her premium is [tex]1,067.20[/tex] per year.

The basic price factor is multiplied by the regular premium to determine the basic premium. The converted loss is computed by dividing the losses suffered by the loss conversion factor. Due to the basic premium component, the basic premium is lower than the standard premium.

By dividing the amount insured by sum assured, the insurance rate is determined. This indicates that your monthly price would be 10% if you had an insurance coverage of Rs. 10,000 and an adequate amounts of Rs. One of the most important steps in the insurance buying process is figuring out the insurance premium rate.

annual base premium = (vehicle factor) x (driver-rating factor) x (base rate)

[tex]= 1 * 1.85 * 400[/tex]

[tex]= 740[/tex]

Property damage coverage [tex]\frac{25000}{1000}*0.8=20[/tex]

Bodily injury coverage [tex]\frac{50}{100}*2*300=300[/tex]

Comprehensive coverage [tex]\frac{50}{1000}*0.6*12=3.60[/tex]

Collision coverage [tex]\frac{50}{1000}*0.6*12=3.60[/tex]

Adding these costs to the annual base premium, we get:

annual premium = annual base premium + property damage + bodily injury + comprehensive + collision

[tex]= 740 + 20 + 300 + 3.60 + 3.60[/tex]

[tex]= 1,067.20[/tex]

Therefore, Julie's annual premium is [tex]1067.20[/tex]

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The total area of the stickers is (1/2) x (11/16) x 4 = 11/8 inches squared.

To calculate the total area of the stickers that were used to decorate the notebook, we need to know the dimensions of each sticker. In this case, the stickers were rectangular and each sticker was 1/2 inches wide and 11/16 inches long. To find the total area, we need to multiply the width of each sticker by the length of each sticker and then multiply this number by the number of stickers used. In this example, we have 4 stickers, so we must multiply (1/2) x (11/16) x 4. When this calculation is done, we end up with 11/8 inches squared as the total area of the stickers.

the complete question is :

Maria decorated her notebook with 4 rectangular stickers. Each sticker was 1/2 inches wide and 11/16 inches long

What was the area of each sticker?

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According to the information in the graph Carissa used the motor for the first twelve minutes, then rowed for the next 12 minutes. During the next 48 minutes she dropped the ship's anchor and finally, during the last 18 minutes she returned to the starting point using the motor.

To identify the moments in which Carissa used the motor, the oars or the anchor we must look at the graph. Additionally, we must be clear that when she uses the motor she is faster than when she uses the oar and that when she uses the anchor she is totally stopped.

Based on the above, we can infer that:

the first twelve minutes, then rowed for the next twelve minutes. During the next forty-eight minutes he dropped the ship's anchor and finally, during the last eighteen minutes, he returned to the starting point using the motor.

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Two boxes containing candies were placed on a table, labelled B1 and B2. Box B1 contained 0.0125d of candies while Box B2 contained 0.014d. Therefore, Box B2 contained a greater amount of candies than Box B1, but only by a small amount.

The question asks about two boxes containing candies placed on a table. The boxes are labelled B1 and B2. To answer this question, we must first consider what is contained in each box. Box B1 contains 0.0125d of candies while Box B2 contains 0.014d. Therefore, Box B2 contains a greater amount of candies than Box B1.
It is important to note that the difference between the two boxes is minimal, with Box B2 containing only 0.0015d more candies than Box B1. Thus, both boxes contain similar amounts of candies.

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Answer:

The probability is 4/52

What is Probability?

the extent to which something is probable; the likelihood of something happening or being the case.

"the rain will make the probability of their arrival even greater"

Similar:

likelihood

likeliness

prospect

expectation

chance

chances

odds

possibility

a probable or the most probable event.

plural noun: probabilities

"for a time revolution was a strong probability"

Similar:

probable event

prospect

possibility

good/fair/reasonable bet

the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

"the area under the curve represents probability"

Answer:

x = 3 cm

Step-by-step explanation:

This is a right-angled triangle where:

hyp = Hypotenuse = 5 cm

adj = Adjacent = 4 cm

opp = Opposite = [tex]x[/tex] cm

To determine the Length of any of the three sides of a right-angled triangle mentioned above, applying the Pythagorean Theorem:

[tex]hyp^{2} = opp^{2} + adj^{2}[/tex]

[tex]5^{2} = x^{2} + 4^{2}[/tex]

[tex]25 = x^{2} + 16[/tex]

[tex]x^{2} = 25 - 16[/tex]

[tex]x^{2} = 9[/tex]

[tex]\sqrt{x^{2}} = \sqrt{9}[/tex]

[tex]x = 3[/tex] cm

The solutions of the given equation as required to be determined are as given in; +3, -3, +1, and -1.

As evident from the task content; it follows that the solutions of the given equation are to be determined.

Given; 4x⁴ - 40x² + 36 = 0

Therefore, we have;

x⁴ - 10x² + 9 = 0

Let y = x² so that we have;

y² - 10y + 9 = 0

(y - 9) (y -1) = 0

y = 9 Or y = 1

Therefore, since y = x²;

x = ± 3 or ±1

Hence, the solutions for the equation as required are; +3, -3, +1, and -1.

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