The sum of two numbers is 96. If the ratio between them is 11:1, find the difference between the two numbers.

The information for a 95% confidence interval for the mean cholesterol are:Sample mean = x = 161.5 Standard Deviation = σ = 19.5 Sample size = n = 169 Z or t = Z Standard Error (Rounded to nearest tenth)= 1.5 Margin of Error (Rounded to nearest tenth) = 2.83 Lower limit (Rounded to nearest tenth) = 158.67 Upper limit (Rounded to nearest tenth) = 164.33

Given that the sample size, n = 169, sample mean, x = 161.5, and population standard deviation, σ = 19.5 are to be used to compute the confidence interval for the mean cholesterol. We are to find the following information for a 95% confidence interval for the mean cholesterol.

We know that if the population standard deviation is known and the sample size is greater than 30, then we use the z-value instead of the t-value. Since the sample size is n = 169, we can use the z-value. Z or t = Z For a 95% confidence level, α = 0.05/2 = 0.025 Zα/2 = Z 0.025 (from the standard normal distribution table)Z 0.025 = 1.96 The formula to calculate the standard error of the mean cholesterol is:Standard error = σ/√n=19.5/√169= 1.5

The margin of error is given by Margin of error = Zα/2 × (σ/√n)Margin of error = 1.96 × (19.5/√169)= 2.83 (rounded to the nearest tenth)The lower limit and upper limit of the confidence interval are given by the formulas:Lower limit = x - Margin of error Upper limit = x + Margin of error Lower limit = 161.5 - 2.83 = 158.67 (rounded to the nearest tenth)Upper limit = 161.5 + 2.83 = 164.33 (rounded to the nearest tenth)

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For the first equation with y = 3x + 2 and y = -4x + 2, the lines have the same slope, but a different y-intercept. This means that the lines are parallel and they will never intersect. Therefore, the system of equations has no solution.

For the second equation with y = 2x + 7 and y = 2x - 4, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations has one solution.

For the third equation with y = 6x + 3 and y = -x - 4, the lines have a different slope and a different y-intercept. This means that the lines are not parallel and they will intersect at one point. Therefore, the system of equations has one solution.

For the fourth equation with y = 4x + 3 and y = 4x - 1, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations has one solution.

For the fifth equation with y = 3x + 6 and y = 3x + 6, the lines have the same slope and the same y-intercept. This means that the lines are coincident and they will intersect at one point. Therefore, the system of equations

Answer:

  y = -1/3x +6

Step-by-step explanation:

You want the equation of the line through the point (3, 5) and perpendicular to y = 3x +2.

The slope-intercept form of the equation of a line is ...

  y = mx +b

where m is the slope, and b is the y-intercept.

Comparing this to the given equation, we see that m=3 for the given line.

The slopes of perpendicular lines are opposite reciprocals of one another. This means the slope of the line we want is ...

  desired slope = -1/m = -1/3

The slope-intercept equation above can be solved for b to give ...

  b = y -mx

Then the y-intercept for the line we want is ...

  b = 5 -(-1/3)(3) = 5 +1 = 6

The equation of the desired line is y = -1/3x +6.

__

Additional comment

Once you understand how to find the slope of the given line and of the desired line, you can write down the desired equation in point-slope form.

Given slope = 3; perpendicular slope = -1/3

Point-slope equation: y -k = m(x -h) . . . . line through (h, k) with slope m

  y -5 = -1/3(x -3) . . . . . line through (3, 5) with slope -1/3

The only "work" required is to rearrange this equation to whatever form you may want. In standard form it is x +3y = 18.

Answer:

Let's start by defining our variables:

Now, let's write the system of inequalities to represent the constraints:

The company has 260 boards of mahogany, so x ≤ 260.

The company has 320 boards of black walnut, so y ≤ 320.

The company expects to sell at most 380 boards, so x + y ≤ 380.

We cannot sell a negative number of boards, so x ≥ 0 and y ≥ 0.

Graphically, these constraints represent a feasible region in the first quadrant of the xy-plane bounded by the lines x = 260, y = 320, and x + y = 380, as well as the x and y axes.

To maximize profit, we need to write a function that represents the objective. The profit for selling one board of mahogany is $20, and the profit for selling one board of black walnut is $6. Therefore, the total profit P can be calculated as:

To maximize P, we need to find the values of x and y that satisfy the constraints and make P as large as possible. This is an optimization problem that can be solved using linear programming techniques.

The solution to this problem can be found by graphing the feasible region and identifying the corner point that maximizes the objective function P. However, since we cannot draw a graph here, we will use a table of values to find the maximum profit.

Let's consider the corner points of the feasible region:

Corner point (0, 0):

P = 20(0) + 6(0) = 0

Corner point (260, 0):

P = 20(260) + 6(0) = 5200

Corner point (0, 320):

P = 20(0) + 6(320) = 1920

Corner point (100, 280):

P = 20(100) + 6(280) = 3160

Corner point (200, 180):

P = 20(200) + 6(180) = 5520

Corner point (380, 0):

P = 20(380) + 6(0) = 7600

The maximum profit is $7600, which occurs when the company sells 380 boards of wood, all of which are mahogany.

Answer:51

Step-by-step explanation:

Answer:

Step-by-step explanation:

(3x+1)^2 = 3x+3

9x^2 +6x +1=3x+3

9x^2+3x-2=0

finally we got a trinomial quadratic equation solve by factorizing

9x^2 -6x+3x-2=0

3x(3x-2)+(3x-2)=0

3x-2 = 0 or 3x+1=0

x= 2/3 or x= -1/3

Therefore , the solution of the given problem of percentage comes out to be were tuned in to a specific program or show at a given moment.

A number or figure stated as a fraction of 100 is referred to as "a%" in statistics. The versions that begin with "pct," "pct," and "pc" are also uncommon. The common way to indicate it is with the numeral "%," though. Furthermore, there are no indicators and a flat ratio of every single thing to the total number. Percentages are basically integers because they frequently add up to 100.

Here,

The TV ratings, also known as the TV audience share, are a measurement of the proportion of all television-owning households that are watching the same program at the same moment.

Networks and marketers use it as a gauge of a TV show's popularity to decide how successful a program will be and how much to charge for advertising during it.

Companies like Nielsen, which use a sample of homes with televisions to estimate the audience size for a given program or show, are usually in charge of gathering the TV ratings.

TV ratings are expressed as a proportion of all households with televisions that were tuned in to a specific program or show at a given moment.

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The test has demonstrated a good level of test-retest reliability.

If a student took an assessment on their math ability at one time point and then the same assessment a month later, with no substantial changes such as getting a tutor, and their math ability was the same between time 1 and time 2, then the assessment has achieved Test-Retest Reliability.Test-Retest Reliability: Test-Retest reliability is the measure of consistency of a test over time. A test has test-retest reliability if a person performs similarly on the same test taken at two different times.A reliable test must always provide consistent results. Therefore, if the math ability was the same between time 1 and time 2, and no substantial changes occurred during that time, then the test has demonstrated a good level of test-retest reliability.

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Step-by-step explanation:

remember, the sum of all angles in a triangle is always 180°.

the law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

with a, b, c being the sides, and A, B, C being the three corresponding opposite angles.

so, the angle at Q is

180 = 48 + 48 + angle Q = 96 + angle Q

84° = angle Q

5mm/sin(48) = PR/sin(84)

PR = 5×sin(84)/sin(48) = 6.691306064... mm

The length of PR is approximately 10.33 mm.

what is isosceles triangle ?

An isosceles triangle is a triangle with at least two sides that have equal length, and thus two corresponding angles that are also equal in measure. The third side and angle of an isosceles triangle may or may not be of different length or measure. The two sides that are equal in length are called the legs, and the third side is called the base. The angle opposite the base is called the vertex angle, while the angles adjacent to the legs are called the base angles. In an isosceles triangle, the two base angles are equal in measure.

Since the sum of the angles in a triangle is 180 degrees, we can find the measure of angle PQR as follows:

PQR = 180 - QPR - QRP

PQR = 180 - 48 - 48

PQR = 84 degrees

Since angles QRP and QPR have the same measure, we know that sides OP and OR have equal length (they are opposite those angles). Therefore, triangle POR is an isosceles triangle.

To find the length of PR, we can use the Law of Cosines:

PR^2 = OP^2 + OR^2 - 2(OP)(OR)cos(POR)

Since OP and OR are equal in length, we can simplify this equation to:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

We know that POR is 180 - PQR = 96 degrees. We also know that OP = OR, and that QP = 5 mm. Using the Law of Cosines, we can find the length of OP:

OP^2 = QP^2 + OR^2 - 2(QP)(OR)cos(QPR)

OP^2 = 5^2 + OR^2 - 2(5)(OR)cos(48)

OP^2 = OR^2 - 5ORcos(48) + 25

Since OP = OR, we can substitute OP for OR in the above equation:

OP^2 = OP^2 - 5OPcos(48) + 25

5OPcos(48) = 25

OP = 25/(5cos(48))

OP ≈ 6.25 mm

Now we can substitute this value into the equation we derived earlier to find PR:

PR^2 = 2(OP^2) - 2(OP^2)cos(POR)

PR^2 = 2(6.25^2) - 2(6.25^2)cos(96)

PR ≈ 10.33 mm

Therefore, the length of PR is approximately 10.33 mm.  

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The answer of the given question based on the the number of students using the cafeteria's healthy lunch line can be found by solving the inequality 1/5s - 51 < 20, the correct answer is (1) 142 , (1) or more.

An equation is  mathematical statement that shows  equality of two expressions, typically separated by  equal sign. Equations are used to express relationships between variables and to solve problems in various fields of mathematics, science, and engineering. In essence, an equation is representation of  real-world scenario or problem in mathematical terms.

The variables in equation can take on different values, and  equation remains true regardless of the values chosen for  variables. Solving  equation means finding  values of the variables that make  equation true.

Equations can be linear, quadratic, polynomial, exponential, logarithmic, or trigonometric, among other types. They are fundamental tool in mathematics and have numerous applications in fields like physics, engineering, economics, and finance.

There are 155 students or more students that use this lunch line.

The correct answer is 1. or more.

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The company should sell their phones for $250 each to get the maximum revenue.

What do you mean by maximum revenue?

Maximum revenue refers to the highest possible amount of income that can be generated from a particular product or service. In the context of the given problem, it means finding the price at which the company can sell its cell phones to earn the highest amount of revenue.

Finding the price at which the company should sell their phones to get the maximum revenue:

We need to find the vertex of the parabolic function [tex]r(x)=x(1,000-2x)[/tex], which represents the revenue as a function of the selling price.

To find the vertex of the function r(x), we need to first rewrite it in standard form by expanding the product:

[tex]r(x) = 1000x - 2x^2[/tex]

Now we can see that the function is a quadratic polynomial in standard form, with [tex]a=-2, b=1000[/tex], and [tex]c=0[/tex]. To find the x-coordinate of the vertex, we can use the formula:

[tex]x = -b / (2a)[/tex]

Substituting the values of a and b, we get:

[tex]x = -1000 / (2\times(-2)) = 250[/tex]

Therefore, the company should sell their phones for $250 each to get the maximum revenue. To find the maximum revenue, we can substitute this value of x into the function r(x):

[tex]r(250) = 250\times(1000-2\times250) = $125,000[/tex]

So the maximum revenue the company can expect to earn is $125,000 if they sell their phones for $250 each.

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One characteristic of all exponential functions is that they change by a constant factor at each step, which means that they exhibit exponential growth or decay.

The constant factor by which an exponential function changes is called the base, which is usually denoted by the symbol "b". If b is greater than 1, the function exhibits exponential growth, and if b is between 0 and 1, the function exhibits exponential decay.

For example, the function f(x) = 2^x is an exponential function with a base of 2. At each step, the function increases by a factor of 2. For instance, f(0) = 1, f(1) = 2, f(2) = 4, f(3) = 8, and so on.

On the other hand, the function g(x) = (1/2)^x is an exponential function with a base of 1/2. At each step, the function decreases by a factor of 1/2. For instance, g(0) = 1, g(1) = 1/2, g(2) = 1/4, g(3) = 1/8, and so on.

Therefore, exponential functions exhibit a characteristic change by a constant factor at each step, which leads to either exponential growth or decay.

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The probability of getting an ace and a three is (4/52) × (3/51) = 12/2652 which simplifies to 1/221.

There are 4 aces and 4 threes in a deck of 52 standard cards.

The probability of getting an ace on your first draw is 4/52.

Once you have the ace, there are 51 cards left in the deck, 3 of which are threes.

Therefore, the probability of drawing a three is 3/51.

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To catch the top 1% of all cars on the freeway, the speed of the car needs to be greater than 99 mph. This is because 99 mph is three standard deviations above the mean, which would put it in the top 1% of speeds.

The question asks for the z-scores and probabilities of cars going 60 mph and 86 mph on the freeway. It also asks for the speed at which the highway patrol woman needs to catch the top 1% of cars.

Let xx be the speed for a randomly selected car on the freeway, with a mean of 66 mph and standard deviation of 10 mph.

The z-score for a car that is caught going 60 mph is -2.0000. This means that the car is two standard deviations below the mean. The probability of randomly choosing a car on the freeway and it going below 60 mph is 0.0228.

The z-score for a car that is caught going 86 mph is 2.0000. This means that the car is two standard deviations above the mean. The probability of randomly choosing a car on the freeway and it going above 86 mph is 0.0228.

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When the radius of circle 2 is twice the radius of circle 1, the size of circle 2 is larger than circle 1.

In geometry, a triangle is a polygon with three sides and three angles. The sum of the angles in a triangle is always 180 degrees. Triangles are one of the most basic and fundamental shapes in geometry and are used in many mathematical and real-world applications, such as in architecture, engineering, and physics. There are different types of triangles based on the length of their sides and the measures of their angles, such as equilateral triangles, isosceles triangles, scalene triangles, acute triangles, obtuse triangles, and right triangles.

Here,

2. This is because the circumference and area of a circle are directly proportional to the radius.

3. To determine if triangles ABC and DEF are similar, we need to check if their corresponding angles are congruent and if their corresponding sides are in proportion. From the diagram, we can see that angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F. This satisfies the angle-angle (AA) similarity criterion. Additionally, we can use the side-side-side (SSS) similarity criterion to determine if the corresponding sides are in proportion. From the diagram, we can see that side AB is parallel to side DE, side AC is parallel to side DF, and side BC is parallel to side EF. Therefore, we can conclude that triangles ABC and DEF are similar.

4. To find the coordinates of D using the coordinates of A, we need to determine the translation from A to D. From the diagram, we can see that A is translated two units to the right and three units down to get to D. Therefore, we can find the coordinates of D by adding two to the x-coordinate of A and subtracting three from the y-coordinate of A. If the coordinates of A are (x1, y1), then the coordinates of D would be (x1 + 2, y1 - 3).

A= (-0.87,0.5)

D=(-0.87 + 2, 0.5 - 3)

D=(1.13,-2.5)

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Therefore, the volume of the prism is 200 cubic centimeters.

Volume is a measurement of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet. The volume of an object can be calculated using a variety of formulas, depending on its shape. The concept of volume is used in various fields, including physics, engineering, architecture, and manufacturing.

Here,

Identify the base of the prism. In this case, the base is a right triangle with legs of 5 cm and 8 cm, and a hypotenuse of 11 cm.

Use the formula for the area of a triangle to find the area of the base. The formula for the area of a right triangle is A = (1/2)bh, where b and h are the lengths of the base and height, respectively. In this case, we can use the legs of the triangle as the base and height, since they are perpendicular. So, the area of the base is:

A = (1/2)(5 cm)(8 cm) = 20 cm²

Multiply the area of the base by the height of the prism to find the volume. The height of the prism is given as 10 cm, so:

Volume = (Area of base) x (Height)

= (20 cm²) x (10 cm)

= 200 cm³

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The number of 50 cents in the container is 380 fifty cents

Since Thabang save money by putting coins in a money box. The money box has 600 coins that consist of 20 cents and 50 cents

To calculate how many 50 cents pieces are in the container if there are 220 pieces of 20 cents, we proceed as follows.

Let

Since the total number of cents in the container is 600, we have that

x + y = 600

So, making y subject of the formula, we have that

y = 600 - x

Since x = 220

y = 600 - 220

= 380

So, there are 380 fifty cents

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The correct order of steps is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

To find the inverse of 55 modulo 89 using the Euclidean algorithm, the steps are:
1. 89 = 1.55 +34
2. 55 = 1.34 +21
3. 34 = 1.21 + 13
4. 21 = 1.13 + 8
5. 13 = 1.8 +5
6. 8 = 1.5 +3
7. 5 = 1.3 +2
8. 3 = 1.2 +1
9. 2 = 1.2
10. The Bézout coefficient of 55 is 34, and it is an inverse of 55 modulo 89.
11. The Bézout coefficients of 55 and 89 are 34 and -21, respectively.
12. The gcd in terms of 55 and 89 is written as 1 = 34-21.89 = 34-21. (55-1.34) = 13.34-21.55 = 13. (21-1.13) - 3.13 = 5.21-8. 13 = 5. (8-1.5) - 1.5 = 2.8-3.5 = 2. (3-1.2) = 2.3-1.5 = 2. (2-1) = 3-1.2 = 3-1. (1-0) = 34-21.89
The correct order of steps is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

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Answer:

Step-by-step explanation:

Let x₁ and x₂ be two independent random variabIes, each with a mean of 10 and a variance of 5.y has a mean of 203 and a variance of 85.

A function, in mathematics, is a reIationship between a set of possibIe inputs and an equaIIy IikeIy set of outputs, where each input is associated to exactIy one outcome. Functions are commonIy represented as equations or graphs, and they are used to modeI many reaI-worId processes in domains such as physics, engineering, and economics.

Function types incIude Iinear, quadratic, trigonometric, and exponentiaI functions, among others. CaIcuIus, a fieId of mathematics that investigates how quantities change over time or space, heaviIy reIies on functions.

given

The foIIowing is the MacIaurin series for the function g(x) = (1+x)e(-x):

g(x) = ∑[n=0 to ∞] ((-1)ⁿ*xⁿ) / n!

We may simpIify and pIug in the first few vaIues of n to determine the first four nonzero terms of this series:

n = 0: ((-1)⁰*x⁰) / 0! = 1

n = 1: ((-1)¹*x¹) / 1! = -x

n = 2: ((-1)²*x²) / 2! = x²/2

n = 3: ((-1)³*x³) / 3! = -x³/6

The MacIaurin series for g(x) therefore has the foIIowing first four nonzero terms:

1 - x + x²/2 - x³/6

Let x₁ and x₂ be two independent random variabIes, each with a mean of 10 and a variance of 5.  y has a mean of 203 and a variance of 85.

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last one is the answer

Step-by-step explanation:

not a function because every input has more than 1 output

The given lengths of 3 feet, 3 feet, and 7 feet cannot form a triangle because they do not satisfy the Triangle Inequality Theorem, which is the sum of the lengths of any two sides is greater than the length of the third side.

To form a triangle, the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.

Let's apply this theorem to the given lengths of 3 feet, 3 feet, and 7 feet:

The sum of the first two sides is 3 + 3 = 6 feet, which is less than the length of the third side of 7 feet. So, the first two sides cannot form a triangle.

The sum of the first and third sides is 3 + 7 = 10 feet, which is greater than the length of the second side of 3 feet. However, the sum of the second and third sides is 3 + 7 = 10 feet, which is also greater than the length of the first side of 3 feet.

Therefore, neither of the two combinations of sides satisfy the Triangle Inequality Theorem, and so it is impossible to form a triangle with sides of 3 feet, 3 feet, and 7 feet.

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In response to the stated question, we may state that As a result, the figure's pairs of congruent angles are: ∠BAC ≅ ∠EDF; ∠ABC ≅ ∠DEF; ∠ACB ≅ ∠DFE

An angle is a form in Euclidean geometry that is constructed from two rays, known as the tone's sides, that connect at a central location known as angle's vertex.

Two beams may combine to generate an inclination in the plane in which they are located. They are known as dihedral angles. In plane geometry, an angle is a potential arrangement of two rays or planes that meet a termination.

The English term “angle” derives from the Latin phrase "angulus," which means "horn." The vertex is the point at where the twin rays, also termed as the angle's sides, converge.

The congruent angles in the illustration are:

BAC and EDF are vertically opposed angles with the same measurement.

ABC and DEF are equivalent angles with the same measure.

ACB and DFE are equivalent angles with the same measure.

Therefore, the figure's pairs of congruent angles are:

∠BAC ≅ ∠EDF

∠ABC ≅ ∠DEF

∠ACB ≅ ∠DFE

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The number of ways of doing both things is N = m × n

Since there are m ways of doing one thing and n ways of doing another, to find how many ways are there to do both, we proceed as follows.

Since there are m ways of doing one thing and n ways of doing another, to find the number of many ways to do both things,we multiply both numbers together.

So, then number of ways of doing both things is N = m × n

So, there are m × n ways of doing both things

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The absolute maximum value is attained: (0, 0)

The given function is, f(x, y) = xy - 7y - 49x + 343The region is on or above y = x^2 and on or below y = 50. To find the absolute minimum and absolute maximum value of the function, f(x, y), first we will find the critical points of the function.f(x, y) = xy - 7y - 49x + 343 ⇒ ∂f/∂x = y - 49 = 0 ⇒ y = 49 ⇒ ∂f/∂y = x - 7 = 0 ⇒ x = 7Thus, the critical point is (7, 49).Next, we will check for the boundary points. The boundary of the region is y = x^2 and y = 50. The points of intersection are:x^2 = 50 ⇒ x = ±√50 (not in the region)x = ±1.58 ⇒ y = x^2 = 2.50 (not in the region)Also, x = 0 ⇒ y = 0, and x = 0 ⇒ y = 50Thus, the critical points are (7, 49) and (0, 0).f(7, 49) = 7(49) - 7(49) - 49(7) + 343 = -7f(0, 0) = 0 - 7(0) - 49(0) + 343 = 343f(0, 50) = 0 - 7(50) - 49(0) + 343 = -357f(±1.58, 2.50) = ±1.58(2.50) - 7(2.50) - 49(±1.58) + 343 = ∓36.97The absolute minimum value is -7. The points at which the absolute minimum value is attained are (7, 49) and (0, 50).The absolute maximum value is 343. The point at which the absolute maximum value is attained is (0, 0).Hence, the required points are as follows:Points at which the absolute minimum value is attained: (7, 49) and (0, 50)Points at which the absolute maximum value is attained: (0, 0)

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Answer:

The Answer is -5 (negative five)

A fair coin is flipped 40 times. The probability of getting a head when the coin is tossed is 0.5. If the same coin is flipped 40 times, the probability will remain 0.5. That is, there are equal chances of getting heads and tails when a fair coin is flipped.

Based on the confidence interval, if the actual proportion of heads falls within the range of the confidence interval, it can be said that the coin used was fair. If the actual proportion is outside the confidence interval, it may be an indication that the coin was not fair.

The level of confidence is typically 95% or 99%. If a confidence interval is constructed for the proportion of heads based on a sample of 40 flips and the interval includes the expected proportion of 50%, it can be said that the coin used was fair. If the interval does not include 50%, there is evidence that the coin may not be fair.

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Step-by-step explanation:

x is the number.

the equation is

7x + 6 = 3

The cost of playing Ratio Rage is provided by the equation "Cost(s) of 1 expression Game: 5.5-3 RATIO RAGE!", but we don't know the value of the game's ratio.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.

The equation "Cost (s) of 1 Game: 12.70 - n +9" gives the cost of playing Decimal Decisions, but we don't know the number of n, thus we can't determine the overall cost for a group of n persons.

Similarly, the cost of playing Ratio Rage is provided by the equation "Cost(s) of 1 Game: 5.5-3 RATIO RAGE!", but we don't know the value of the game's ratio.

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