Probability and statistics


The median of a random variable X to a continuous probability distribution is a

constant m such that P(X ≤m) = 1/2

Find the median of a random variable having pdf f(x) = 3x−4 for x ≥1 (and 0

otherwise).

If someone asked me how it could be possible for an annual interest rate to be higher than 100%, I would explain that it is actually quite common in certain situations, particularly in the case of loans with very short terms or loans with high fees.

For example, let's say you borrowed $100 from a lender and agreed to pay back $110 in one week. The lender is essentially charging you 10% interest for the one-week loan period, but if you annualize that rate, it comes out to over 520%. This is because the lender is charging you a very high interest rate for a very short period of time.

Another example would be if you took out a payday loan, which typically have very high fees attached to them. For instance, you might borrow $500 and have to pay back $575 in two weeks. The interest rate on this loan might be calculated as the $75 fee divided by the $500 borrowed, which comes out to 15%. However, if you annualize that rate, it comes out to over 390%.

In both of these examples, the interest rate is very high because the loan term is very short and/or the fees are very high. It's important to note that borrowing at such high interest rates can be extremely costly and can lead to a cycle of debt, so it's generally recommended to avoid loans with high interest rates whenever possible.

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Using the intersection of sets we find that A ∩ B = {1, 3}

A set is a collection of well ordered items.

Givent hat set A = {x| x is an od number greater than 0 and less than 10} and set B = {1, 2, 3, 4}, we desire to find A ∩ B. We proceed as follows.

First since set A = {x| x is an odd number greater than 0 and less than 10}, its elements are A = {1, 3, 5, 7, 9}

Also, set B =  {1, 2, 3, 4}

Since we require A ∩ B which is the intersection of both sets and is  the elements that both sets have in common. Since both sets have element 1 and 3 in common, we have that

A ∩ B = {1, 3}

So, A ∩ B = {1, 3}

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The time till you hit the water, given your height above the water, would be 0.76 seconds.

The maximum height above the pool you would get is 15.39 feet.

We are given h ( t ) = - 16 t ² + 5t + 15

You hit the water at 0 so the formula is:

0 = - 16 t ² + 5t + 15

Using the quadratic equation, we can solve:

t = ( - b ± √ ( b ² - 4  ac ) ) / 2a

t = ( - 5 ± √ ( 5 ² - 4 ( - 16 ) ( 15 ) )) / 2 (- 16)

t = (- 5 ± √ 985 ) / -32

t = 0. 76 seconds

The vertex of the parabolic function would be:

= - b / 2a

= - 5 / ( 2 x - 16 )

=  0. 15625 seconds

Maximum height is therefore:

= -16 ( 0. 15625 ) ² + 5 ( 0. 15625 ) + 15

=  15.39 feet

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Polar coordinates for rectangular coordinates if x<0: r=√(x²+y²) and θ=tan⁻¹(y/x)+π if y≥0 or θ=tan⁻¹(y/x)−π if y<0, For x≥0: r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.

The polar coordinates of a point with rectangular coordinates (x,y) depend on the sign of x.

If x<0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2. If x≥0, the polar coordinates are (r,θ) where r≥0 and −π/2≤θ<3π/2.

If x<0, t

hen r=√(x²+y²) and

θ=tan⁻¹(y/x)+π if y≥0

or θ=tan⁻¹(y/x)−π if y<0.

The value of r is the distance from the origin to the point and θ is the angle between the positive x-axis and the line segment from the origin to the point.

If x≥0, then r=√(x²+y²) and θ=tan⁻¹(y/x) if y≥0 or θ=tan⁻¹(y/x)+2π if y<0.

In this case, θ is the angle between the positive x-axis and the line segment from the origin to the point, measured counterclockwise.

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By setting up an inequality, the maximum number of vans that are needed to ferry 80 people are 7 vans.

To find the maximum number of vans needed to ferry 80 people using the given terms, let's set up an inequality. Let's use the variable "v" to represent the number of vans.

Since a van can ferry a maximum of 12 people, we can write the inequality as:

12v ≥ 80

Now, let's solve for "v":

Divide both sides of the inequality by 12.
v ≥ 80/12

Simplify the inequality.
v ≥ 6.67

Since we cannot have a fraction of a van, we need to round up to the nearest whole number:

v ≥ 7

Therefore, the maximum number of vans needed to ferry 80 people is 7 vans.

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The man needs to pay a total tax of Rs 69780 (3000 for social security and 66780 for yearly tax).

To find the tax, let's find the man's yearly income:

Yearly income = Monthly income x 12

Yearly income = 53000 x 12 = 636000

Next, let's find how much he deposits in civil investment fund and charity:

Amount deposited in civil investment fund = Yearly income x 20%

Amount deposited in civil investment fund = 636000 x 0.2 = 127200

Amount deposited in charity = Yearly income x 10%

Amount deposited in charity = 636000 x 0.1 = 63600

Now, let's calculate the total taxable income:

Total taxable income = Yearly income - Amount deposited in civil investment fund - Amount deposited in charity

Total taxable income = 636000 - 127200 - 63600 = 445200

Since the man's taxable income is above Rs 300000, he needs to pay 1% social security tax on Rs 300000:

Social security tax = 1% of 300000 = 3000

Now, let's calculate the yearly tax imposed at a rate of 15%:

Yearly tax = Total taxable income x 15%

Yearly tax = 445200 x 0.15 = 66780

Therefore, the man needs to pay a total tax of Rs 69780 (3000 for social security and 66780 for yearly tax).

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The rate of change of revenue per unit is $8 when the unit cost is changing by $8 and the revenue is $4,000.

We are given that the unit cost (C) and unit revenue (R) are related by the equation C = R + 10621.

We want to find the rate of change of revenue per unit, dR/dx, when the unit cost (C) is changing by $8 per unit and the revenue (R) is $4,000.

1. Differentiate the given equation with respect to x: dC/dx = dR/dx

2. Plug in the given values: dC/dx = $8 (cost per unit is changing by $8) R = $4,000 (revenue per unit)

3. Solve for the cost per unit (C) using the equation

C = R + 10621

C = $4,000 + 10621

C = $14,621

4. Since dC/dx = dR/dx, and we know dC/dx = $8,

we can find the rate of change of revenue per unit (dR/dx) when the cost per unit is changing by $8: dR/dx = $8

Thus, the rate of change of revenue per unit is $8 when the unit cost is changing by $8 and the revenue is $4,000.

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(a) The cost of City Bus for driving 10 miles = $3.

(b) The cost of City Bus for driving 25 miles = $7.5.

(c) The cost of City Bus for driving 42 miles = $12.6.

(d) The cost of City Bus for driving 68 miles = $20.4.

We previously choose City Bus as our mode transport since the per mile cost for City Bus is less.

Let the model for City Bus be f(x) = cx + d, where f(x) is total cost and x is number of miles.

From the table of Taxi we get, f(2) = 0.60; f(4) = 1.20; f(6) = 1.80 and f(8) = 2.40.

So, 2a + b = 0.60 and 4a + b = 1.20

(4a + b) - (2a + b) = 1.20 - 0.60

2a = 0.60

a = 0.60/2 = 0.30

Now, f(8) = 2.40

8*0.30 + b = 2.40

2.40 + b = 2.40

b = 2.40 - 2.40 = 0

So the function rule for City Bus is, f(x) = 0.3x.

(a) Total cost to drive 10 miles is,

f(10) = 0.3*10 = 3

(b) Total cost to drive 25 miles is,

f(25) = 0.3*25 = 7.5

(c) Total cost to drive 42 miles is,

f(42) = 0.3*42 = 12.6

(d) Total cost to drive 68 miles is,

f(68) = 0.3*68 = 20.4

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The tent has a volume of 86 cubic feet.

The tent owned by Dante is in the shape of a triangular prism, which means it has two identical equilateral triangle bases that measure 5 feet each. The tent's length is 8 feet, and its height is 4.3 feet.

To calculate the tent's volume, we can use the formula for the volume of a triangular prism, which is [tex]V = (1/2) * b * h * l[/tex], where b is the base, h is the height, and l is the length of the prism.

Plugging in the given values, we get [tex]V = (1/2) * 5 * 4.3 * 8[/tex] = 86 cubic feet. The volume of a tent is an important consideration when deciding which one to purchase or use for a particular activity, as it determines how much space is available inside for people and belongings.

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Answer:

m=2

Step-by-step explanation:

The areas of the sector in the circle are 18π, 3π, 10π and 5π

The orange sector

This is calculated as

Sector area = central angle/360 * πr²

Where

So, we have

Sector area = 180/360 * π * 6²

Sector area = 18π

The yellow sector

This is calculated as

Sector area = central angle/360 * πr²

Where

So, we have

Sector area = 30/360 * π * 6²

Sector area = 3π

The green sector

This is calculated as

Sector area = central angle/360 * πr²

Where

So, we have

Sector area = 100/360 * π * 6²

Sector area = 10π

The purple sector

This is calculated as

Sector area = central angle/360 * πr²

Where

So, we have

Sector area = 50/360 * π * 6²

Sector area = 5π

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Answer: 161.2

Step-by-step explanation:

Plug into formula

6.9² = 13²+17.8²-2(17.8)(13) cos C      >simplify numbers

-438.23 = -2(17.8)(13) cos C               >Divide both sides by  -2(17.8)(13)

cos C=.947                                         > use   [tex]cos^{-1}[/tex] C  to solve for angle

<C=180-18.75 = 161.2                          > neded to subtract from 180 for this                    

                                                                one

Answer:

  [tex]\displaystyle\textsf{a) }\binom{8}{4}\\\\\textsf{b) }\binom{-8}{4}[/tex]

Step-by-step explanation:

Given a translation vector ...

  [tex]\displaystyle \binom{g}{h}[/tex]

moves g to the right and h up, you want the vectors for 8 right, 4 up, and for 8 left, 4 up.

When we have g = units to the right, and we want 8 units to the right, we know that g = 8. Similarly, h = units up, and we want 4 units up, so h = 4.

Putting these values in the vector form, we have ...

a)   8 right, 4 up matches vector ...

  [tex]\displaystyle \boxed{\binom{8}{4}}[/tex]

b)  Left is the opposite of right, so 8 units left will be represented by ...

  g = -8

As before, 4 units up means h = 4.

  [tex]\displaystyle \boxed{\binom{-8}{4}}[/tex]

<95141404393>

The solution of the exponents is shown below.

Exponents are mathematical shorthand for multiplying a number by itself a certain number

We have that;

5^n = 1

5^n = 5^0

n = 0

2) 2^-7/2^n = 2^2

2^-7 - n = 2^2

-7 - n = 2

-n = 2 + 7

n = -9

3) 6^5 * 6^n = 6^1

6^ 5 + n = 6^1

5 + n = 1

n = 1 - 5

n = -4

4) (8^n)^7 = 8^21

8^7n = 8^21

7n = 21

n = 3

5) 4^n = (1/4)

4^n = 4^-1

n = -1

In each of the cases, we have applied the laws of the exponents as we know them.

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The equation of the forms are matched as;

2⁻⁷/2ⁿ = 2², n = -9

6⁵ * 6ⁿ = 6. n = -4

(8ⁿ)⁷ = 8²¹, n = 3

4ⁿ = 1/4 , n = -1

Index forms are simply described as mathematical forms that are used to represent numbers of variables that are too large or small.

To multiply index forms, you need to add the exponents of the same bases.

To divide index forms, you need to subtract the exponents of the same bases.

From the information given, we have that;

2⁻⁷/2ⁿ = 2²

cross multiply the values

2⁻⁷ = 2²⁺ⁿ

Then,

-7 = 2 + n

n = -9

6⁵ * 6ⁿ = 6

take the exponents

5 + n= 1

n =- 4

(8ⁿ)⁷ = 8²¹

We have;

7n = 21

Make 'n' the subject

n = 3

4ⁿ = 1/4

4ⁿ = 4⁻¹

n =-1

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The expression that represents the volume of the cylinder, in cubic units, is:

[tex]$$V = 2\pi x^3$$[/tex]

The expression that represents the volume of a cylinder in cubic units is given by the formula:

[tex]$$V = \pi r^2h$$[/tex]

where [tex]$r$[/tex] is the radius of the base of the cylinder and [tex]$h$[/tex] is the height of the cylinder.

Now, let's consider each option provided:

[tex]1. $4\pi x^2$[/tex]

This expression only includes the radius, but it does not include the height of the cylinder, so it cannot be the correct answer.

[tex]2. $2\pi x^3$[/tex]

This expression includes both the radius and the height of the cylinder, but it does not include the squared term for the radius, so it cannot be the correct answer.

[tex]3. $\pi x^2$[/tex]

This expression includes the squared term for the radius, but it does not include the height of the cylinder, so it cannot be the correct answer.

[tex]4. $2x$[/tex]

This expression only includes a single variable, which is neither the radius nor the height of the cylinder, so it cannot be the correct answer.

[tex]5. $2\pi x^3$[/tex]

This expression includes both the squared term for the radius and the height of the cylinder, so it is the correct answer.

Therefore, the expression that represents the volume of the cylinder, in cubic units, is:

[tex]$$V = 2\pi x^3$$[/tex]

This formula can be used to calculate the volume of a cylinder given the value of its radius and height.

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The missing numbers on the grids are given as follows:

2 and -6.

The amounts are obtained by a system of equations, for which the variables are given as follows:

x and y.

The top grid represents the sum of the measures, hence:

x + y = -4.

The bottom grid represents the subtraction of the measures, hence:

x - y = 8.

Adding the two equations, the value of x, representing the left grid, is given as follows:

2x = 4

x = 2.

Then the value of y, representing the right grid, is given as follows:

2 + y = -4

y = -6.

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The expression that is equivalent to the one Regina wrote is y + 27/4

From the question, we have the following parameters that can be used in our computation:

y + 9 x 3/4

This means that

Expression = y + 9 x 3/4

When expanded, we have

Expression = y + 27/4

Using the above as a guide, we have the following:

The expression that is equivalent to the one Regina wrote is y + 27/4

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Answer:

El coeficiente del término 23x^5 es 23.

Step-by-step explanation:

Answer:

x = -6

Step-by-step explanation:

5x + 3 = 2x - 15

3x + 3 = -15

3x = -18

x = -6

Let's Check

5(-6) + 3 = 2(-6) - 15

-30 + 3 = -12 - 15

-27 = -27

So, x = -6 is the correct answer.

Answer:

x = -6

Step-by-step explanation:

Equation is 5x + 3 = 2x - 15

First, we can subtract the constants

5x = 2x - 18

Then, we can subtract 2x on both sides

3x = -18

Divide both sides by 3 to isolate x

x = -6

The cost of the stock for the investor would be calculated as follows:

Cost of 1 share = $98.93
Number of shares purchased = 500

Total cost of the stock = Cost per share x Number of shares
Total cost of the stock = $98.93 x 500
Total cost of the stock = $49,465

The broker charges 2% of the cost of the stock, so the broker's fee would be calculated as:

Broker's fee = 2% x Total cost of the stock
Broker's fee = 2% x $49,465
Broker's fee = $989.30

Therefore, the total cost of the stock for the investor including the broker's fee would be:

Total cost of the stock + Broker's fee = $49,465 + $989.30
Total cost of the stock + Broker's fee = $50,454.30

So the cost of the stock for the investor including the broker's fee is $50,454.30.

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The probability that out of three exactly one car will require a minor repair in the first year = 33%

P(E ) = no. of favorable outcome/total no. of outcome

E here represents exactly one car that requires a minor repair.

out of three cars, exactly one car will need a minor repair

No. of favorable outcome = 1

Total no. of outcome = 3

Now, putting value in P(E) we get

P(E) = 1/3

P(E) = 0.333

To get percentage we multiply by 100

P(E) = 0.333 × 100

P(E) = 33.3

The probability to the nearest percent = 33%

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In general, the cost of the visit will depend on the number of cavities found by the dentist as: Cost = $50 + $100 * n.

An equation is a mathematical statement that says that two expressions are equal. It consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=). An equation expresses a relationship between the quantities involved and represents a balance or equality between the two sides. Equations are used in various fields of mathematics, science, and engineering to model real-world situations and solve problems. They can be linear or nonlinear, simple or complex, and involve different types of variables, functions, and operators. Solving equations is an essential skill in mathematics and involves applying various techniques and strategies to find the solution or solutions to the equation.

Here,

If the dentist finds n cavities, the cost of the visit will be:

Cost = $50 + $100 * n

So, if the dentist finds, for example, 3 cavities, the cost of the visit will be:

Cost = $50 + $100 * 3 = $350

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Complete question:

The price of a visit to the dentist is $50 if the dentist feels any cavities in additional charge of $100 per cavity get added to the bill. If the dentist finds n cavities What will the cost of the visit be?

Answer:

  (c)  x ≤ 5 or x ≥ 7

Step-by-step explanation:

You want the solution to |x -6| ≥ 1.

The absolute value relation represents two relations, one for the domain x < 6, and one for the domain x ≥ 6.

In this domain, the inequality becomes ...

  -1 ≥ x -6

  5 ≥ x . . . . . . add 6

  x ≤ 5 . . . . . . . put x on the left

In this domain, the inequality is ...

  x -6 ≥ 1

  x ≥ 7

The disjoint solution sets are x ≤ 5 or x ≥ 7.

__

Additional comment

For |x -a| ≤ b, we can "unfold" this to the compound inequality ...

  -b ≤ (x -a) ≤ b

copying the inequality symbol to the left side, and writing the opposite of the constant there.

We can do the same thing with the inequality ...

  |x -a| ≥ b

but it doesn't really make sense as a compound inequality.

Instead, we have to write it as ...

  -b ≥ (x -a)   or   (x -a) ≥ b

in recognition of the fact that the solution spaces are disjoint.

Answer:

Step-by-step explanation:

as we alderdy have value of y,we can substuite it in the place of y

4x+8(2)=3x+7(2)+14

4x+16=3x+14+14

4x-3x=28-16

x=12

We can see here that the amount he or she will pay in taxes for their entire lifetime is: $344,000

Tax is a financial obligation that all people, businesses, and other types of entities must fulfill for a government organization.

Let us say that an average American makes $40,000  for his or her lifetime and works from age 22 to 65, and pays a combined federal and state income tax rate of 20%, the amount of taxes paid per year would be:

$40,000 x 0.20 = $8,000

Over a 43-year period (from age 22 to 65), the total amount of taxes paid would be: $8,000 x 43 = $344,000

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[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{0})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~5 - (-3)~~)^2 + (~~0 - (-6)~~)^2} \implies d=\sqrt{(5 +3)^2 + (0 +6)^2} \\\\\\ d=\sqrt{( 8 )^2 + ( 6 )^2} \implies d=\sqrt{ 64 + 36 } \implies d=\sqrt{ 100 }\implies d=10[/tex]

The distance is negative because it's a fall, so the cat falls 27.36 ft between t = 0.5 s and t = 1.4 s.

To find the distance the cat falls between t = 0.5 s and t = 1.4 s, we need to use the formula for velocity and distance.
we first need to find the position at each of these times using the given formula u(t) = -32t ft/s.

The formula for distance fallen is:
distance(t) = initial position + initial velocity × t + (1/2) × acceleration × t²

Since the cat falls with zero initial velocity and starts from the tree, we can simplify the formula:
distance(t) = (1/2) × acceleration × t²

First, let's find the velocity of the cat at t = 0.5 s and t = 1.4 s using Galileo's formula:
u(0.5) = -32(0.5) = -16 ft/s
and, u(1.4) = -32(1.4) = -44.8 ft/s

Now, we can use the formula for distance:
distance = (velocity at t = 0.5 s + velocity at t = 1.4 s) / 2 x (t = 1.4 s - t = 0.5 s)
⇒ distance = (-16 ft/s + (-44.8 ft/s)) / 2 x (1.4 s - 0.5 s)
⇒ distance = (-60.8 ft/s) / 2 x (0.9 s)
⇒ distance = -27.36 ft/s x s

Therefore, the cat falls 27.36 feet between t = 0.5 s and t = 1.4 s.

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As a result, the function's absolute maximum and minimum values are 5 and -45, respectively, at x = -5 and x = 2, respectively.

Every element in a set (referred to as the domain) in mathematics is connected by a rule known as a function to exactly one component in some other set (called the range or codomain). In other terms, a role is a connection among 2 sets where every element in the domain matches exactly one member in the range. Using a formula or equation with a variable input, function notation is a common way to represent functions. As an illustration, the formula f(x) = 2x + 1 gives each true figure x the value 2x + 1.

given

We can apply the second derivative test to determine whether this critical point is a maximum or minimum. By taking f'(xderivative, )'s we arrive at:

f''(x) = -4

The critical point at x = 0 is a local maximum since f"(0) = -4 is a negative value.

Secondly, we must determine whether the interval's endpoints of -5 x 2 provide values that are higher or lower than the crucial point. By entering x = -5 and x = 2, we obtain:

[tex]f(-5) = 5-2(-5) (-5)^2 = 5 – 50 = -45[/tex]

[tex]f(2) = 5 - 2(2) (2)^2 = -3[/tex]

As a result, the function's absolute maximum and minimum values are 5 and -45, respectively, at x = -5 and x = 2, respectively.

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The ratio of the corresponding dimensions of the smaller pyramid to the larger pyramid is 1/3.

Dimension is the measure of the distance or length of an obeject.

To calculate the ratio of the corresponding dimensions of the smaller pyramid to the larger pyramid, we use the formula below

Formula:

l/L = √(a/A) ........................ Equation 1

Where:

From the question,

Given:

Substitute these values into equation 1

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Therefore, (70+72)/2 = **71 minutes** is the median time. B)42.5 minutes as a result. C,D)The range is determined by deducting the dataset's smallest value from highest value.

A)When the data are organized in order of magnitude, the median time is the middle value. The median in this situation is the average of the 14th and 15th values, which are 70 and 72, respectively. There are 28 data points in this situation. Therefore, (70+72)/2 = **71 minutes** is the median time.

b) The median of the lowest half of the data constitutes the lower quartile (Q1). We must arrange the data in descending order of magnitude before determining the median of the first half of the data in order to determine Q1.  is the average of the seventh and eighth values, which are 40 and 45, respectively, in the first half of the data, which consists of 14 values. Q1 = (40+45)/2 = **42.5 minutes as a result.

b) The median of the upper half of the data constitutes the upper quartile (C). We must first organise the data in descending order of magnitude before determining the median of the remaining data in order to determine Q3. Q3 is the average of the seventh and eighth values from the last, which are 90 and 105, respectively, in the second half of the data, which consists of 14 values. Q3 = (90+105)/2 = **97.5 minutes**3 as a result.

d) The range is determined by deducting the dataset's smallest value from highest value.

In this instance, Ben's practise time can be anywhere from **10 minutes** to **180 minutes**. Range then equals maximum value - minimum value, which in this case is 180 - 10 = **170 minutes**

e) Extreme values that are beyond the typical range of a dataset's values are known as outliers. We can use a criterion that states that any value that sits more than 1.5 times the interquartile range (IQR) below Q1 or above Q3 is regarded as an outlier to ascertain whether there are outliers in this dataset. When Q1 is subtracted from Q3, the result is the IQR: Q3 - Q1 = 97.5 - 42.5 = **55 minutes**3. By using this rule, we can see that the dataset contains the outliers **180** and **120** minutes.

The term "interquartile range" is IQR. It is a measure of variability that is based on quartilizing a dataset. The first quartile (Q1) is subtracted from the third quartile to determine the IQR. (Q3). It is a representation of the middle 50% of the data's range.

f) A box-and-whisker plot illustrates a dataset's quartiles, outliers, and range1. For Ben's practice, here's how to create a box-and-whisker plot:

- Create a number line with all the values Ben practised with.

- Draw a box spanning Q1 through Q3.

- Inside the box, at the location of Q2, draw a vertical line. (the median).

Draw whiskers from the box's two ends to all values that are not outliers.

- Place every outlier on the graph as a separate point, outside of any whiskers.

To know more about IQR visit:

brainly.com/question/29257180

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