I’m rectangle LAGF, OT and GS intersect at point L

Answer: 4080

Step-by-step explanation:

First you have to find the area of the triangle. 24*10 = 240. 240/2 = 120. Then you multiply the area of the triangle and multiply it by 34. 120 * 34 = 4080. This means the answer is 4080

The ball that weighs 30g is ball A.

The degree to which a measured value resembles the true or recognised value is known as accuracy. On the other hand, precision describes how closely two measurements of the same quantity agree. In other words, precision is a measure of consistency, whereas accuracy is a measure of correctness. The values obtained are consistent but not always close to the genuine value when a measurement is precise but not exact. On the other hand, a measurement might be accurate without being exact, which means that the result is close to the correct value but produces erratic or dispersed results when repeated.

From the given weights corresponding to different balls, 30 g corresponds to ball A.

Hence, the ball that weighs 30g is ball A.

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Just use the pythagorean theorem to solve the hypotenuse!

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

9+4=13^2

[tex]\sqrt{13}[/tex] = [tex]\sqrt{x}[/tex]

[tex]13^{2}[/tex] km

Hope this helps <3

The terms arranged in order from smallest to biggest are: (-2)³, -√25, √11, 10, and 4² after comparing the values of the final numbers.

We shall first simplify the numbers to get their final values and then compare to which is smaller as follows:

4² = 4 × 4 = 16

-√25 = -5

10 = 10

√11 = 3.3166

(-2)³ = -2 × -2 × -2 = -8

In conclusion, we have by comparing the final values of the numbers the terms arranged from smallest to the biggest as: (-2)³, -√25, √11, 10, and 4².

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Tim was driving for 5 hours before he caught up to Dora.

The answer is (a) 5 hours.

To solve this problem, we can use the formula:
distance = rate × time
Let's denote the time Tim drove as t hours.

Since Dora started driving one hour earlier, her driving time would be (t + 1) hours.
Dora's distance: 75 kph × (t + 1)
Tim's distance: 90 kph × t
Since Tim catches up to Dora, their distances will be equal:
75(t + 1) = 90t
Now we can solve for t:
75t + 75 = 90t
75 = 15t
t = 5.

The answer is (a) 5 hours.

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The points on the surface z² = xy + 16 closest to the origin are: (-4,4,0) and (4, -4, 0)

We know that the distance between an arbitrary point on the surface and the origin is d(x, y, z) = √(x² + y² + z²)

Using Lagrange multipliers,

L(x, y, z, λ) = x² + y² + z² + λ(z² - xy - 16)

We have partial derivatives.

[tex]L_x[/tex] = 2x - λy

[tex]L_y[/tex] = 2y - λx

[tex]L_z[/tex] = 2z + 2zλ

[tex]L_\lambda[/tex] = z² - xy - 16

Now we set each partial derivative to zero to find critical points.

[tex]L_x[/tex] = 0

2x - λy = 0

[tex]L_y[/tex] = 0

2y - λx = 0

After solving above equations simultaneously we get (x + y)(x - y) = 0

i.e., x = -y   OR   x = y

[tex]L_z[/tex] = 0

2z + 2zλ = 0

z = 0  OR  λ = 0

Consider [tex]L_\lambda[/tex] = 0

z² - xy - 16 = 0

-xy = 16                  ............(as z = 0)

when x = y then -y² = 16 which is not true.

So, consider x = -y

-(-y)y = 16

y² = 16

y = ±4

when y = 4 then we get x = -4

and when y = -4 then we get x = 4

Therefore, the closest points are:(-4,4,0) and (4, -4, 0)

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The value of angle GFH is 18°

A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.

A theorem in circle geometry starts that angle in the same segment are equal. In triangle EFG, angle F and G are on the same segment, this means that angle F and G are equal.

Represent angle F as x

therefore 144+2x = 180° ( sum of angle in a triangle)

2x = 180-144

2x = 36

x = 36/2 = 18°

Therefore the measure of angle GFH is 18°

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The value of x in the intersecting chords that extend outside circle is 5

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

intersecting chords that extend outside circle

Using the theorem of intersecting chords, we have

4 * (x + 6 + 4) = 6 * (x - 1 + 6)

Evaluate the like terms

So, we have

4 * (x + 10) = 6 * (x + 5)

Using a graphing tool, we have

x = 5

Hence. the value of x is 5

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the simplified  form  of expression is: -(x² + 2x - 2)/((x+2)*(x+4))

what is expression  ?

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that represents a mathematical quantity or relationship. Expressions can be simple or complex

In the given question,

To evaluate the expression 1/(x+2) - (x+1)/(x+4), we need to find a common denominator for the two terms. The least common multiple of (x+2) and (x+4) is (x+2)(x+4).

So, we can rewrite the expression as:

(1*(x+4) - (x+1)(x+2))/((x+2)(x+4))

Expanding the brackets, we get:

(x+4 - x² - 3x - 2)/((x+2)*(x+4))

Simplifying the numerator, we get:

(-x² - 2x + 2)/((x+2)*(x+4))

Therefore, the simplified expression is:

-(x² + 2x - 2)/((x+2)*(x+4))

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Part 1- The actual margin of error (in percent) for the large-sample confidence interval from this sample is c) 2.26%.

Part 2- The margin of error (in percent) would be 3.1% if the sample had resulted in p-hat = 0.5.

Part 3- The reason why Harris announces a ±3% margin of error for all samples of about this size is c) because the

margin of error for samples of about this size is no more than the announced margin of error.

Part 1: The actual margin of error for the large-sample confidence interval from this sample can be calculated using the

formula: Margin of error = (z-score)*(standard error),

where the z-score for 95% confidence level is 1.96 and the standard error is [tex]\sqrt{[(p-hat \times(1-p-hat))/n]}[/tex]

Here, p-hat (proportion of smokers who agreed that smoking would probably shorten their lives) = 848/1010 = 0.84 and

n (sample size) = 1010.

Plugging these values into the formula, we get

Margin of error =[tex]1.96\times(sqrt[(0.84\times(1-0.84))/1010])[/tex] = 0.026 = 2.6%.

Part 2: The margin of error is largest when p-hat = 0.5, which means that the sample is equally split between two

options.

In this case, the standard error becomes [tex]\sqrt{[(0.5\times(1-0.5))/1010] } = 0.0158.[/tex]

Using the same formula as in Part 1, we get

Margin of error = [tex]1.96\times0.0158 = 0.031 = 3.1%.[/tex]

Part 3: The reason why Harris announces a ±3% margin of error for all samples of about this size is c) because the

margin of error for samples of about this size is no more than the announced margin of error.

This means that Harris is being conservative in their estimate of the margin of error and is ensuring that they do not

overstate the precision of their results. Additionally, they may want to account for other sources of error such as non-

response bias or sampling bias, which can also contribute to the overall margin of error.

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Answer:

2

Step-by-step explanation:

i know

The expression for the area of the rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b is ad + bd + ac + bc.

The rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b can be visualized as a rectangle with width d+c and height a+b.

Area is a measurement of the size of a two-dimensional surface, such as a plane or a flat shape.

Therefore, the area of the rectangular region is given by

Area = Width x Height

Substitute the values in the equation and find the expression of area

= (d + c) x (a+b)

= ad + bd + ac + bc

So the expression for the area of the rectangular region is

ad + bd + ac + bc.

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The given question is incomplete, the complete question is:

Let a, b, c, and d be positive numbers. Find the expression for the area of the rectangular region whose boundary is formed by the equations x=−c, x=+d, y=+a, y=−b?

The sides and the angle of the right triangle are a = 10√2, b = 10√2 and B = π / 4.

In this problem we need to determine the values of two sides and an angle of the right triangle. This can be done by means of the following properties:

A + B + C = π

sin A = a / c

cos A = b / c

tan A = a / b

Where:

If we know that A = π / 4, C = π / 2 and c = 20, then the missing angle and missing sides are, respectively:

B = π - π / 4 - π / 2

B = π / 4

cos (π / 4) = b / 20

b = 20 · cos (π / 4)

b = 10√2

sin (π / 4) = a / 20

a = 20 · sin (π / 4)

a = 10√2

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The chance that the number of heads will be in the range 4850 to 5150 is approximately 0.9973, or about 99.73%.

The number of heads in 10,000 coin tosses follows a binomial distribution with parameters n = 10,000 (the number of trials) and p = 0.5 (the probability of heads on a single toss).

We can approximate this binomial distribution using the normal distribution, with mean μ = np = 5000 and variance σ² = np(1-p) = 2500.

To find the probability that the number of heads is in the range 4850 to 5150, we can use the normal distribution and standardize the range using the z-score formula:

z = (x - μ) / σ

where x is the number of heads in the range we're interested in.

For the lower bound of 4850, we have:

[tex]z_lower = (4850 - 5000) / \sqrt{(2500)}[/tex]

= -3

For the upper bound of 5150, we have:

[tex]z_upper = (5150 - 5000) / \sqrt{(2500)} = 3[/tex]

Using a standard normal distribution table or calculator, we can find the probability of being within 3 standard deviations of the mean:

P([tex]z_lower[/tex]  < Z < [tex]z_upper[/tex] ) ≈ P(-3 < Z < 3)

= 0.9973.

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To form a suitable quadratic equation using the values from the table given in the question also  considering the event of forming a equation that represents f(x) is Option A.
In order to find the equation that is  represented by f(x), we have to implement the standard form of a quadratic function
f(x) = ax² + bx + c
here a, b and c = constants.
We can utilize the given table of values to evaluate these constants.

Now, we have to  place each x value into f(x) to get the concerning y value. Then we can utilize these points to create three equations with three undetermined (a, b and c).
Evaluating these equations will give us the values of a, b and c.

Now, the table of values given in the question is

f(-2) = 48 = 4a - 4b + c
f(-1) = 50 = a - b + c
f(0) = 48 = c
f(1) = 42 = a + b + c
f(2) = 32 = 4a + 4b + c
f(3) = 18 = 9a + 3b + c
f(4) = 0 = 16a + 4b + c


Calculating these equations

a = -2
b = -4
c = 48

Hence, the equation that represents f(x) is f(x) = -2x² - 4x + 48.

The correct option for the given question after considering the given conditions is Option A.

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The complete question is
The table of values forms a quadratic function f(x). X f(x)−2 48
f(−1) = 50
f(0) = 48
f(1) = 42
f(2) = 32
f(3) = 18
f(4) = 0

What is the equation that represents f(x)?
a) f(x) = –2x² – 4x + 48
b) f(x) = 2x² + 4x – 48
c) f(x) = x² + 2x – 24
d) f(x) = –x² – 2x + 24

Without knowing which specific function you're referring to, the answer to this question may depend on the type of function and the nature of the iterative process applied to it. In some cases, the function value may converge towards a limiting value as the number of iterations increases, while in other cases it may oscillate or diverge.

For example, in the case of the fixed-point iteration method used to find the root of a function, the value of the function typically converges towards the root as the number of iterations increases. More specifically, if we have a function f(x) and a starting guess x0 for its root, we can use the iterative formula x(+1)=g(x()), where g(x) is some function that we set based on f(x), to generate a sequence of increasingly accurate approximations to the root. As the number of iterations increases, this sequence of approximations typically converges towards the root of the function, unless some conditions are not met (e.g., the method is not well-suited for some functions, or the iteration formula is not properly set.)

In the case of other types of iterative methods or other functions, however, the behavior of the function value as the number of iterations increases may differ. For instance, in some cases, the function value may oscillate between two or more values or diverge to infinity as the number of iterations increases.

Therefore, the specific behavior of the function value as the number of iterations increases may depend on the specific function being evaluated and the iterative method used.

The area of the triangle is: 15 square feet

The area of a rectangle is: 96 square feet

The formula for the area of a triangle is:

Area = ¹/₂ * base * height

This triangle has the dimensions:

base = 12 ft

height = 2.5 ft

Thus:

Area = ¹/₂ * 12 * 2.5

Area = 15 square feet

The area of a rectangle is:

A = length * width

A = 12 * 8

A = 96 square feet

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Answer:

x = -5, x = 1

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

Step-by-step explanation:

The x-intercepts are the x-values of the points at which the curve crosses the x-axis, so when y = 0.

From inspection of the given graph, we can see that the parabola crosses the x-axis at x = -5 and x = 1.

Therefore, the x-intercepts of the parabola are:

As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).

the average speed of the plane in still air is s + 30.

Let's call the average speed of the plane in still air "s" (in miles per hour).

We can use the formula:

time = distance / speed

to find the time it takes the plane to travel from Salt Lake City to Oakland with the wind and against the wind.

With the wind:

time with wind = [tex]600 / (s + 30)[/tex]

Against the wind:

time against wind =[tex]600 / (s - 30)[/tex]

time against wind > time with wind

So we can set up an inequality:

[tex]600 / (s - 30) > 600 / (s + 30)[/tex]

Multiplying both sides by [tex](s - 30)(s + 30)[/tex], we get:

[tex]600(s + 30) > 600(s - 30)[/tex]

Expanding and simplifying, we get:

[tex]600s + 18000 > 600s - 18000[/tex]

Subtracting 600s from both sides, we get:

[tex]18000 > -18000[/tex]

This inequality is true for all values of s. In other words, there are no restrictions on the value of s that would make the return trip take longer than the trip with the wind.

Therefore, we can use the average of the two speeds (with and against the wind) to find the average speed of the plane in still air:

Average speed = [tex]2s(s + 30) / (s + 30 + s - 30)[/tex]

Simplifying, we get:

Average speed = [tex]2s(s + 30) / (2s)[/tex]

Canceling the common factor of 2s, we get:

Average speed = s + 30

We know that the distance from Salt Lake City to Oakland is 600 miles, and we can use the formula:

time = distance / speed

to find the time it takes the plane to travel this distance:

time = [tex]600 / (s + 30)[/tex]

We also know that the return trip (against the wind) takes longer, so we can set up another equation:

time return trip =[tex]600 / (s - 30)[/tex]

We can use these two equations to solve for s:

[tex]600 / (s + 30) = 600 / (s - 30)[/tex]

Cross-multiplying, we get:

[tex]600(s - 30) = 600(s + 30)[/tex]

Expanding and simplifying, we get:

[tex]600s - 18000 = 600s + 18000[/tex]

Subtracting 600s from both sides, we get:

[tex]-18000 = 18000[/tex]

This is not a valid equation, so there must be no solution.

However, we can still find the average speed of the plane in still air by using the equation we derived earlier:

Average speed = s + 30

So the average speed of the plane in still air is s + 30. We don't have a specific value for s, but we can say that the average speed is equal to the speed with the wind plus 30 (which is the speed of the wind).

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The least number of arc measures needed to determine the measures of each angle in the inscribed quadrilateral is 2.

To determine the measures of each angle in the quadrilateral, we need to find the central angles of the arcs that intersect the quadrilateral's vertices. Since the quadrilateral is not a rectangle, it is not a cyclic quadrilateral, which means that its opposite angles do not add up to 180 degrees.

Therefore, we need to use the fact that the sum of the measures of the opposite angles in an inscribed quadrilateral is 360 degrees. Let the angles of the quadrilateral be A, B, C, and D, with opposite angles A and C, and B and D. We can find the measure of arc AC by drawing a chord connecting the endpoints of AC and finding the central angle that intercepts it. Similarly, we can find the measure of arc BD.

Now, we can use the fact that the sum of the central angles that intercept arcs AC and BD is equal to 360 degrees. Let these angles be x and y, respectively. Then, we have:

x + y = 360

We can solve for one of the variables, say y, in terms of the other:

y = 360 - x

Substituting this into the equation for arc BD, we have:

2x + 2(360 - x) = arc BD

Simplifying this equation, we get:

arc BD = 720 - 2x

Now, we can use the fact that the sum of the measures of angles A and C is equal to the measure of arc AC, and the sum of the measures of angles B and D is equal to the measure of arc BD. Therefore, we have:

A + C = arc AC
B + D = arc BD = 720 - 2x

We need to find the least number of arc measures needed to determine the measures of A, B, C, and D. Since we have two equations and two variables (x and A), we can solve for both variables. Then, we can use the equations for B and D to find their measures.

Solving for A in terms of x, we have:

A = arc AC - C
A = 360 - x - C

Substituting this into the equation for B + D, we have:

(360 - x - C) + B + D = 720 - 2x

Simplifying this equation, we get:

B + D = 360 + x - C

Now, we have three equations and three variables (x, A, and C). We can solve for each variable in terms of x, and then use the equation for B + D to find their measures.

Therefore, the least number of arc measures needed to determine the measures of each angle in the quadrilateral is two: arc AC and arc BD.

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the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.

To calculate the likelihood of Erin getting at least one bullseye, we need to first calculate the probability of her not getting a bullseye in a single throw. Since she scores a bullseye 15% of the time, the probability of her not getting a bullseye in a single throw is 85% (100% - 15%).

Using the probability of not getting a bullseye in a single throw, we can use the following formula to calculate the probability of not getting a bullseye in all 5 throws:

0.85 x 0.85 x 0.85 x 0.85 x 0.85 = 0.4437

Therefore, the probability of Erin not getting a bullseye in all 5 throws is 0.4437 or 44.37%.

To calculate the probability of Erin getting at least one bullseye in 5 throws, we can subtract the probability of her not getting a bullseye in all 5 throws from 1:

1 - 0.4437 = 0.5563

Therefore, the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.

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The probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.

To find the probability that she will get at least one bullseye in 5 throws, we can use the complementary probability.

This means we will first find the probability of her not getting a bullseye in all 5 throws, and then subtract that from 1.

Find the probability of not getting a bullseye (1 - bullseye probability)

1 - 0.15 = 0.85

Calculate the probability of not getting a bullseye in all 5 throws

0.85^5 ≈ 0.4437

Find the complementary probability (probability of at least one bullseye)

1 - 0.4437 ≈ 0.5563

So, the probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.

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The Cube root of the resulting number is 8.

The smallest number that 40000 can be divided by so that the result is a perfect cube, we need to factorize 40000 into its prime factors:

[tex]40000 = 2^6 \times 5^4[/tex]

To make this a perfect cube, we need to ensure that the powers of each prime factor are multiples of 3.

The smallest number we can divide 40000 by so that the result is a perfect cube is:

[tex]40000 = 2^6 \times 5^4[/tex]

Now we can find the cube root of the resulting number:

[tex]3\sqrt (40000 \div 100) = 3\sqrt400 = 8.[/tex]

Factories 40000 into its prime components in order to determine.

The least number that the result may be divided by while still producing a perfect cube.

The powers of each prime factor must be multiples of three in order for this to be a perfect cube.

The least number that 40000 may be divided by to produce a perfect cube is:

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Answer:

[tex]h(x)=2x^2-12x+9[/tex],  [tex]p(x)=-5x^2-20x-5[/tex]

Step-by-step explanation:

To get to the standard form of a quadratic equation, we need to expand and simplify. Recall that standard form is written like so:

[tex]ax^2+bx+c[/tex]

Where a, b, and c are constants.

Let's expand and simplify h(x).

[tex]2(x-3)^2-9=\\2(x^2+9-6x)-9=\\2x^2+18-12x-9=\\2x^2+9-12x=\\2x^2-12x+9[/tex]

Thus, [tex]h(x)=2x^2-12x+9[/tex]

Let's do the same for p(x).

[tex]-5(x+2)^2+15=\\-5(x^2+4+4x)+15=\\-5x^2-20-20x+15=\\-5x^2-5-20x=\\-5x^2-20x-5[/tex]

Thus, [tex]p(x)=-5x^2-20x-5[/tex]

The 'footprint' of CO2 emissions for a person in 1830 would be 818,199 tons of CO2 emissions per person.

To find the 'footprint' of CO2 emissions for a person in 1830, we need to substitute the value of x = 1830 - 1800 = 30 into the given function C(x) = 0.0365 (1.758)^x.

Plugging in x = 30 into the function, we get:

C(30) = 0.0365 * (1.758)^30

Substituting this value back into the function, we get:

C(30) = 0.0365 * 22416413.1381

C(30) = 818199.079541

C(30) ≈ 818,199.08

Answered question "Scientists studying the 'footprint' of carbon dioxide (CO2) emissions attributed to the average person for each decade from 1800 to 1910 used the function C(x) = 0.0365 (1.758)*, where x is the number of decades since 1800 and C is the number of tons of CO2 emissions per person. What is the 'footprint' of CO2 emissions for a person in 1830??"

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Therefore, the roots of the polynomial equation x² + x - 72 = 0 are -9 and 8.

A quadratic equation is a type of polynomial equation of the second degree, which means it has one or more terms in which the variable is raised to the power of two, but no higher powers.Quadratic equations can have zero, one, or two real solutions, depending on the values of a, b, and c. These solutions are also called the roots or zeros of the equation.

Here,

To find the roots of the polynomial equation x² + x - 72 = 0, we can set y = 0 and solve for x. This is equivalent to finding the x-intercepts of the graph of the function f(x) = x² + x - 72.

We can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

In this case, a = 1, b = 1, and c = -72, so we have:

x = (-1 ± √(1² - 4(1)(-72))) / (2(1))

x = (-1 ± √(1 + 288)) / 2

x = (-1 ± √(289)) / 2

x = (-1 ± 17) / 2

Therefore, the roots of the polynomial equation x² + x - 72 = 0 are:

x = -9 or x = 8

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Answer:

y = x - 5

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

Slope = rise/run or (y2 - y1) / (x2 - x1)

Points (-1, -6) and (6, 1)

We see the y increase by 7, and the x increase by 7, so the slope is

m = 7/7 = 1

Y-intercept is located at (0, -5)

So, the equation is y = x - 5

[tex](\stackrel{x_1}{-1}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{(-6)}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{(-1)}}} \implies \cfrac{1 +6}{6 +1} \implies \cfrac{ 7 }{ 7 } \implies 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-6)}=\stackrel{m}{ 1}(x-\stackrel{x_1}{(-1)}) \implies y +6 = 1 ( x +1) \\\\\\ y+6=x+1\implies {\Large \begin{array}{llll} y=x-5 \end{array}}[/tex]

The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is approximately [-6.62, 22.02].

The critical value that should be used in constructing the confidence interval.

Since we are looking for a 90% confidence interval, we need to find the critical value associated with a 5% level of significance in a two-tailed test.

Using a t-distribution with (n1-1) + (n2-1) degrees of freedom and a significance level of 0.05, we find the critical value to be:

t-critical = 1.717 (using a t-distribution table or a calculator)

Step 2 of 3:

Next, we need to find the standard error of the sampling distribution to be used in constructing the confidence interval.

Since the population variances are not equal, we need to use the Welch-Satterthwaite equation to calculate the standard error:

SE = sqrt[([tex]s1^2[/tex]/n1) + ([tex]s2^2[/tex]/n2)]

where s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Substituting the given values, we get:

SE = sqrt[([tex]19.25^2[/tex]/17) + ([tex]26.25^2[/tex]/12)]

SE ≈ 8.35

Step 3 of 3:

To construct the 90% confidence interval, we can use the formula:

(mean1 - mean2) ± t-critical * SE

where mean1 and mean2 are the sample means, and t-critical and SE are the values calculated in steps 1 and 2.

Substituting the given values, we get:

= (84.80 - 77.10) ± 1.717 x 8.35

= 7.70 ± 14.32

Therefore,

The 90% confidence interval for the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases is (approx) [-6.62, 22.02].

We can be 90% confident that the true mean difference between the mean amount of mail-order purchases and the mean amount of internet purchases falls within this interval.

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The probability of getting 'p' success when a die rolled 12 times with 'x' success that has binomial distribution is equal to ¹²Cₓ pˣ ( 1 - p )¹²⁻ˣ.

Number of times die to be rolled = 12

In this scenario, a trial refers to a single roll of the 20-sided die.

Here , 'x' represents the the number times the die lands on a green square.

A success would be defined as landing on a green square,

And a failure would be landing on any other color.

Since the die will be rolled 12 times, there are 12 trials in total.

This implies, the number of times the die lands on a green square, x, has a binomial distribution.

With parameters n = 12 the number of trials and p the probability of success  which is landing on a green square.

Probability = ⁿCₓ pˣ ( 1 - p )ⁿ⁻ˣ

Therefore, the probability of rolling a die 12 times with 'x' success which has binomial distribution and 'p' probability of success is equal to ¹²Cₓ pˣ ( 1 - p )¹²⁻ˣ.

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