Question 2
Which of the following quadratic functions has a vertex of (2,4)?
O A
B
C
f(x) = 4(x − 2)² + 4
f(x) = 3(x + 2)² + 4
f(x) = 2(x-4)² + 2
f(x) = 2(x + 4)² + 2

The value of the given double integral is 1008/3.

We are given the region R with vertices (0,0), (1,4), (3,-1), and (4,3).

To evaluate the given double integral using the transformation u = x + 3y and v = 4u - y, we need to first find the Jacobian J of the transformation. The Jacobian is given by:

J = ∂(x,y) / ∂(u,v) = [∂x/∂u ∂x/∂v; ∂y/∂u ∂y/∂v]

Now, u = x + 3y and v = 4u - y, so we can solve for x and y in terms of u and v to get:

x = (4v - 9u) / 7

y = (3u - v) / 7

Differentiating with respect to u and v, we get:

∂x/∂u = -9/7

∂x/∂v = 4/7

∂y/∂u = 3/7

∂y/∂v = -1/7

Therefore, the Jacobian is:

J = [∂x/∂u ∂x/∂v; ∂y/∂u ∂y/∂v] = [-9/7 4/7; 3/7 -1/7]

Now we can evaluate the double integral using the transformed variables u and v:

S = ∫∫R (4x + 3y) dA

= ∫∫R (4u - v)(7/9) dudv (since J = det(Jacobian)

= (-9/7)(-1/7) - (4/7)(3/7)

= 1/9)

= (7/9) ∫∫R (4[tex]u^2[/tex] - uv) dudv

The limits of integration for u and v are determined by the region R. The vertices of R in terms of u and v are:

(0,0) → u = 0, v = 0

(1,4) → u = 1, v = 3

(3,-1) → u = 3, v = -13

(4,3) → u = 4, v = 7

So the limits of integration are:

0 ≤ u ≤ 4

(4u - 13) ≤ v ≤ (4u + 7)

Substituting these limits of integration, we get:

S = (7/9) ∫[0 to 4] ∫[4u - 13 to 4u + 7] (4[tex]u^2[/tex] - uv) dvdu

= (7/9) ∫[0 to 4] [(16[tex]u^3[/tex] + 390u - 253) / 3] du

= (7/9) [ (64/3)([tex]4^4[/tex]) + (390/2)([tex]4^2[/tex]) - (253/3) ]

= 1008/3

For similar question on double integral:

https://brainly.com/question/30217024

#SPJ11

The number of boxes required to pack 2004 Saturn chocolate is 167 boxes of 12 each.

Saturn chocolate bars are either packed in 5's or 12's.

For finding the number of boxes, we have to divide the total number of chocolate available for packing by the number of chocolate per each box.

Since, we have minimise the number of boxes, we will select 12 pieces per box so that the number of boxes can be as less as possible.

In other to arrive at smallest number of boxes, then we need to pack primarily in 12's

Number of boxes required for 12 packing :

2004 /12 = 167 boxes.

Hence, smallest number of boxes to pack 2004 Saturn chocolate bars is 167 boxes.

The given question is incomplete, complete question is given below:

‘Saturn' chocolate bars are packed either in boxes of 5 or boxes of 12. What is the smallest number of full boxes required to pack exactly 2004 ‘Saturn' bars?

To know more about saturn chocolate

https://brainly.com/question/20698048

#SPJ4

Answer:

Step-by-step explanation:

(6x+2)

a. A function of x. [tex]F(x) = 100 sec^2(x).[/tex]

b. The average force exerted  by the press over the interval [tex][0, \pi/3][/tex] is approximately 173.2 N.

(a) Since the force F is proportional to the square of sec x, we can write:

[tex]F(x) = k sec^2(x)[/tex]

where k is the constant of proportionality.

To find k, we use the fact that F(0) = 100:

[tex]100 = k sec^2(0)[/tex]

100 = k

So, [tex]F(x) = 100 sec^2(x).[/tex]

(b) The average force exerted by the press over the interval[tex][0, \pi/3][/tex] is given by:

[tex]F = (1/(\pi/3 - 0)) * ∫[0, \pi/3] F(x) dx.[/tex]

Using the expression for F(x) found in part (a), we have:

[tex]F = (3/\pi) * \int[0, \pi/3] 100 sec^2(x) dx[/tex]

We can simplify this integral using the trigonometric identity [tex]sec^2(x) = 1 + tan^2(x):[/tex]

[tex]F = (3/\pi) * \int[0, \pi/3] 100 (1 + tan^2(x)) dx[/tex]

[tex]F = (300/\pi) * [x + (1/3) tan^3(x)]|[0, \pi/3][/tex]

Evaluating this expression at [tex]x = \pi/3[/tex]  and x = 0, we get:

[tex]F = (300/\pi) * [(\pi/3) + (1/3) tan^3(\pi/3) - 0 - (1/3) tan^3(0)][/tex]

[tex]F = (300/\pi) * [(\pi/3) + (1/3) * (\sqrt{(3)} /3)^3][/tex]

[tex]F = (300/\pi) * [(\pi/3) + (1/9) * \sqrt{(3)} ][/tex]

F ≈ 173.2 N.

For similar question on function.

https://brainly.com/question/2328150

#SPJ11

The width of the sidewalk is approximately 3.21 feet (rounded to two decimal places).

Let's use the given information to solve for the width (x) of the sidewalk. The area of the rectangular swimming pool is 20 feet by 30 feet, which equals 600 square feet (20*30 = 600).

The total area of the pool and sidewalk is given as 825 square feet. To find the area of just the sidewalk, we subtract the area of the pool from the total area: 825 - 600 = 225 square feet.

Now, let's express the area of the sidewalk using the dimensions of the pool and the width of the sidewalk (x). Since the sidewalk is on 3 sides, we can represent the area as follows:

2(20x) + 30x = 225

Simplify the equation:

40x + 30x = 225

Combine the terms:

70x = 225

Now, solve for x:

x = 225 / 70

x ≈ 3.21

The width of the sidewalk is approximately 3.21 feet (rounded to two decimal places).

To learn more about width here;

brainly.com/question/12298654#

#SPJ11

A Z-score represents how many standard deviations an individual data point is from the mean of a dataset. The formula for calculating a Z-score is:

Z = (X - μ) / σ

Where:
- Z is the Z-score
- X is the individual data point
- μ (mu) is the mean of the dataset
- σ (sigma) is the standard deviation of the dataset

If you can provide the data or statistics needed, I'd be more than happy to help you calculate the Z-scores for the given problems.

To learn more about mean, refer below:

https://brainly.com/question/31101410

#SPJ11

The antiderivative of f(x) = ³√x² + x√x is [tex]1/3 x^3/2 (2x^1/3 + 3x^1/2)[/tex] + C.

The antiderivative of a capability is the converse of the subsidiary. As such, assuming that we have a capability f(x) and we take its subordinate, we get another capability that lets us know how the first capability is changing regarding x. The antiderivative of f(x) is a capability that lets us know how the first capability changes as for x the other way. It is additionally called the endless vital of f(x).

Presently, we should view as the antiderivative of the given capability f(x) = ³√x² + x√x. We can separate it into two sections:

f(x) = ³√x² + x√x

=[tex]x^(2/3) + x^(3/2)[/tex]

To see as the antiderivative of [tex]x^(2/3)[/tex], we want to add 1 to the example and separation by the new type:

∫[tex]x^(2/3)[/tex] dx = (3/5)[tex]x^(5/3)[/tex] + C

where C is the steady of incorporation. Also, to view as the antiderivative of[tex]x^(3/2)[/tex], we add 1 to the example and separation by the new type:

∫[tex]x^(3/2)[/tex] dx = (2/5)[tex]x^(5/2)[/tex] + C

where C is the steady of incorporation.

Accordingly, the antiderivative of f(x) = ³√x² + x√x is:

∫f(x) dx = ∫[tex]x^(2/3)[/tex] dx + ∫[tex]x^(3/2)[/tex] dx

= (3/5)[tex]x^(5/3)[/tex] + (2/5)[tex]x^(5/2)[/tex] + C

where C is the steady of incorporation.

To learn more about antiderivative, refer:

https://brainly.com/question/14011803

#SPJ4

On solving the query we can say that The trapezium is 16 cm tall as a of result.

Mathematics is concerned with numbers and their variations, equations and related structures, shapes and their places, and possible placements for them. The relationship between a collection of inputs, each of which has an associated output, is referred to as a "function". An relationship between inputs and outputs, where each input yields a single, distinct output, is called a function. Each function has a domain and a codomain, often known as a scope. The letter f is frequently used to represent functions (x). X is the input. The four main types of functions that are offered are on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

The formula for a trapezoid's area is:

Area is equal to (b1+b2)*h/2.

where h is the height of the trapezium, and b1 and b2 are the lengths of the two bases.

The trapezoid's size is 48 cm2, and its bases (b1) and (b2) are each assigned lengths of 5 cm and 7 cm, respectively. In order to solve for the height (h), we may enter these values into the formula as follows:

48 = (5 + 7) * h / 2

When we simplify the equation, we obtain:

48 = 6h / 2

48 = 3h

h = 48 / 3

h = 16

The trapezium is 16 cm tall as a result.

To know more about function visit:

https://brainly.com/question/28193995

#SPJ1

The volume of the triangular prism is 360 cubic inches

What is volume of triangular prism?

Volume = Area × Height

Here given, the base is a right triangle with a leg of 8 in. and hypotenuse of 10 in. the height of the prism is 15 in.

We want to find volume of the triangular prism.

We can find the length of the other leg of the triangle,

Height ² + Base² = Hypotenuse ²

[tex]a^2 + b^2 = c^2 \\ 8^2 + b^2 = 10^2 \\ 64 + b^2 = 100 \\ b^2 = 36 \\ b = 6[/tex]

So the base triangle is 6 in.

Area of a triangle = 1/2 × base × height

A = 1/2 × 8 in. × 6 in.

A = 24 in²

Now volume of the prism,

V = A × height

V = 24 in² × 15 in

V = 360 in³

Therefore, the volume of the triangular prism is 360 cubic inches (to the nearest tenth).

Learn more about triangular prism here,

https://brainly.com/question/31342575

#SPJ1

a) First, let's define the null hypothesis (H0) and alternative hypothesis (H1).

H0: The average customer age is 35 years or less (μ ≤ 35)
H1: The average customer age is greater than 35 years (μ > 35)

b) Now, we need to perform a hypothesis test to determine if there's enough evidence to refute the store's claim. We'll use a one-sample z-test with an alpha level of 0.05.

Step 1: Calculate the test statistic:
z = (sample mean - population mean) / (standard deviation / √sample size)
z = (38.5 - 35) / (9 / √43)
z ≈ 2.26

Step 2: Determine the critical z-value for a one-tailed test with alpha = 0.05:
For a one-tailed test with an alpha level of 0.05, the critical z-value is 1.645.

Step 3: Compare the test statistic to the critical z-value:
Since our calculated z-value (2.26) is greater than the critical z-value (1.645), we reject the null hypothesis.

Based on this test, there is enough evidence to refute the age claim made by the sporting goods store. The sample suggests that the average customer age is greater than 35 years.

Learn more about Statistical Tests here: brainly.in/question/9402988

#SPJ11

The Marshall Plan was an economically strategic maneuver designed to rebuild Western European capitalism. It aimed to support the recovery and stability of European countries after World War II and prevent the spread of communism.

The Marshall Plan violated the philosophy of containment by providing economic aid to European countries, which some saw as indirectly aiding the spread of communism. It was also an economically strategic maneuver to rebuild Western European capitalism and prevent the spread of communism. Additionally, it was an offshoot of the Displaced Persons Plan, which aimed to help refugees and displaced persons in Europe after World War II.

Know more about Marshall Plan here;

https://brainly.com/question/22602380

#SPJ11

(a) The probability that a quart of this brand of milk chosen at random will contain between 35 and 39 g of butterfat is 0.3413.

(b) The probability that a quart of this brand of milk chosen at random will contain between 33 and 35 g of butterfat is 0.1915.

We are given

Mean, μ = 35 g

Standard deviation, σ = 4 g

(a) Probability of a quart containing between 35 and 39 g of butterfat

We need to find the probability P(35 ≤ X ≤ 39) where X is the butterfat content of a quart of milk

To solve this, we first standardize the values using the standard normal distribution formula

Z = (X - μ) / σ

For X = 35

Z = (35 - 35) / 4 = 0

For X = 39

Z = (39 - 35) / 4 = 1

Using a standard normal distribution table or calculator, we can find the probabilities

P(35 ≤ X ≤ 39) = P(0 ≤ Z ≤ 1) = 0.3413

Therefore, the probability that a quart of this brand of milk chosen at random will contain between 35 and 39 g of butterfat is 0.3413.

(b) Probability of a quart containing between 33 and 35 g of butterfat

We need to find the probability P(33 ≤ X ≤ 35) where X is the butterfat content of a quart of milk.

Using the same standardization process as above

For X = 33

Z = (33 - 35) / 4 = -0.5

For X = 35

Z = (35 - 35) / 4 = 0

Using a standard normal distribution table or calculator, we can find the probabilities

P(33 ≤ X ≤ 35) = P(-0.5 ≤ Z ≤ 0) = 0.1915

Learn more about probability here

brainly.com/question/11234923

#SPJ4

The percentage of absent employees that are on the night shift is  56.25%.

To find the percent of absent employees on the night shift, we'll first calculate the percentage of employees who are absent from both shifts, and then use that information to find the desired percentage.

1. We know that 70% of the employees work the day shift, so the remaining 30% work the night shift.

2. On any given day, 1% of day-shift employees are absent, and 3% of night-shift employees are absent.

3. Calculate the percentage of absent employees from both shifts:

(1% of 70%) + (3% of 30%).

This would be (0.01 * 70) + (0.03 * 30) = 0.7 + 0.9 = 1.6%.

4. Now, we'll find the percent of absent employees on the night shift out of the total absent employees. Since 3% of night-shift employees are absent, and there are 30% night-shift employees, the percentage of absent night-shift employees out of total employees is (0.03 * 30) = 0.9%.

5. Finally, divide the percentage of absent night-shift employees by the total percentage of absent employees and multiply by 100 to get the answer: (0.9 / 1.6) * 100 = 56.25%.

So, 56.25% of absent employees are on the night shift.

Learn more about Percentage:

https://brainly.com/question/24877689

#SPJ11

The particular solution determined by the initial condition.

[tex]f(x) = 18/5x^{5/3}  - 5/4x^4 - 19/10[/tex]

The first step to finding the particular solution determined by the initial condition is to integrate f'(x) to obtain the general solution f(x).

Assuming that f'(x) is given, we can integrate it as follows:

∫f'(x) dx = f(x) + C

where C is the constant of integration.

To find the particular solution determined by the initial condition.
[tex]f'(x) = 6x^{2/3} - 5x^3[/tex]
Next, we need to use the initial condition f(1) = -9 to find the value of C. Substituting x=1 and f(1)=-9 into the general solution f(x) + C, we get:

Initial condition: f(1) = -9.
Integrate f'(x) to find the general solution, f(x).
[tex]\int(6x^{2/3} - 5x^3) dx = 6\int x^{2/3} dx - 5\int x^3 dx[/tex]
Perform the integration.
[tex]6[(3/5)x^{5/3}] - 5[(1/4)x^4] + C = 18/5x^{5/3} - 5/4x^4 + C[/tex]
Now, we need to substitute the value of C into the general solution to obtain the particular solution. So, the particular solution determined by the initial condition f(1)=-9 is:

Use the initial condition f(1) = -9 to find the constant C.
[tex]-9 = 18/5(1)^{5/3} - 5/4(1)^4 + C[/tex]
Solve for C.
C = -9 - 18/5 + 5/4

= (-36 + 18 - 20)/20

= -38/20

= -19/10
Write the particular solution, f(x), using the constant C.

The particular solution determined by the initial condition f(1)=-9 is:
[tex]f(x) = 18/5x^{5/3} - 5/4x^4 - 19/10[/tex]

For similar question on  particular solution.

https://brainly.com/question/31401412

#SPJ11

Answer:

D

Step-by-step explanation:

the volume (V) of a sphere is calculated as

V = [tex]\frac{4}{3}[/tex] πr³ ( r is the radius )

here diameter = 15 , then r = 15 ÷ 2 = 7.5 , so

V = [tex]\frac{4}{3}[/tex] × π × 7.5³

  = [tex]\frac{4}{3}[/tex] × π × 421.875

  = [tex]\frac{4\pi (421.875)}{3}[/tex]

  ≈ 1767.1 in³ ( to the nearest tenth )

The value of the integral from 0 to 1 of e⁻⁴ˣ with respect to x is approximately 0.284.

To begin, we need to understand what the integrand represents. The function e⁻⁴ˣ is an exponential function, where the base of the exponent is e, a mathematical constant approximately equal to 2.718. The exponent is a function of x, meaning that as x changes, the value of the exponent changes accordingly.

To evaluate this integral, we can use the formula for integrating exponential functions. The integral of eⁿˣ with respect to x is equal to (1/n)eⁿˣ + C, where C is a constant of integration. Using this formula, we can integrate e⁻⁴ˣ with respect to x to get:

∫e⁻⁴ˣ dx = (-1/4)e⁻⁴ˣ + C

Next, we can evaluate this expression at the upper and lower limits of integration, which are 0 and 1, respectively:

(-1/4)e⁻⁴ˣ evaluated from 0 to 1 = (-1/4)(e⁰ - e⁻⁴) = (-1/4)(1 - 1/e⁴) ≈ 0.284

To know more about integral here

https://brainly.com/question/18125359

#SPJ4

Complete Question:

Evaluate the Integral integral from 0 to 1 of e⁻⁴ˣ with respect to x

The following mathematical operation must be carried out in order to determine a cylinder's volume: V = h r².

How can I calculate a cylinder's volume?

We must work out the following mathematical equation in order to determine a cylinder's volume:

V = Πhr² (h = height of cylinder, r= radius)

Let's use an instance.

The size of our example cylinder is 6 centimetres in diameter and 10 centimetres height. What is the size of it?

We substitute the values as follows to determine the volume:

Volume = 3.1415 x 10 cm x 3 cm²

= 307.35 cm³

Therefore, The following mathematical operation must be carried out in order to determine a cylinder's volume: V = h r².

To know more about volume check the below link:

https://brainly.com/question/9554871

#SPJ1

that's so hard I'm in year 6

Step-by-step explanation:

I don't know I'm in year 6, that's literally impossible :(

False.

Mutually exclusive means that the occurrence of event A and the occurrence of event B cannot happen at the same time.

In this case, the occurrence of event A does affect the probability of the occurrence of event B because if event A occurs, then event B cannot occur.

Independent means that the occurrence of event A has no effect on the probability of the occurrence of event B. In this case, the occurrence of event A does not prevent the occurrence of event B.

To learn more about probability, refer below:

https://brainly.com/question/30034780

#SPJ11

The total number of visitors to the website in the first 10 weeks are 40960.

What are numbers?

An arithmetic value used for representing the quantity and used in making calculations.

Here given that the number of visitors each week is twice the number of visitors of the previous week.

So it is clear that the number of visitors to the website in the first ten weeks will be

[tex]Week \: 1: \: 40 \: visitors \\ Week \: 2: \: 240 = \: 80 \: visitors \\ Week 3: \: 280 = 160 \: visitors \\ Week \: 4: 2160 = 320 \: visitors \\ Week \: 5: 2320 = 640 \: visitors \\ Week \: 6: 2640 = 1280 \: visitors \\ Week \: 7: 21280 = 2560 \: visitors \\ Week \: 8: 22560 = 5120 \: visitors \\ Week \: 9: 25120 = 10240 \: visitors \\ Week \: 10: 2 \times 10240 = 20480 \: visitors[/tex]

Now we need to add up the number of visitors from each week:

Total number of visitors

[tex]= 40 + 80 + 160 + 320 + 640 + 1280 + 2560 + 5120 + 10240 + 20480[/tex]

Total number of visitors = 40960

Therefore, the total number of visitors to the website in the first 10 weeks is 40,960.

Learn more about numbers here,

https://brainly.com/question/31583261

#SPJ1

a)  the determinant of J is 1xT - 0x2 = 0, which means that the area of O(R) is 0.
b)  the determinant of J is 1x1 - 0x2 = 1, which means that the area of Q(R) is the same as the area of R, which is (13-1) x (18-6) = 144.

To find the area of O(R) using the Jacobian, we need to calculate the determinant of the Jacobian matrix of Q(u,v):

J = [∂(u+30)/∂u   ∂(u+30)/∂v  ]
    [∂(2u+Tv)/∂u  ∂(2u+Tv)/∂v]

= [1  0]
  [2  T]

(a) For R = [0,9] x [0,7], we have T = 0 since there is no v-dependence in the range of R. Therefore, the determinant of J is 1xT - 0x2 = 0, which means that the area of O(R) is 0.

(b) For R = [1,13] x [6,18], we have T = 1 since v ranges from 6 to 18. Therefore, the determinant of J is 1x1 - 0x2 = 1, which means that the area of Q(R) is the same as the area of R, which is (13-1) x (18-6) = 144.

learn more about the Jacobian

https://brainly.com/question/29855578

#SPJ11

Answer:

10x(x^2 - 2)

Step-by-step explanation:

We can factor out the greatest common factor of the terms in the binomial, which is 10x:

10x(x^2 - 2)

Therefore, the factored form of the binomial 10x^3 - 20x is:

10x(x^2 - 2)

I can check this.

To do so, we can use the distributive property to expand the factored form and see if it matches the original binomial:

10x(x^2 - 2) = 10x * x^2 - 10x * 2

= 10x^3 - 20x

As you can see, expanding the factored form using the distributive property gives us the original binomial, which confirms that the factored form is correct.

The correct interpretation of the function is that it depicts the exponential rise of the number of Movie Bank worldwide locations x years since 2004, with a growth rate of 0.62.

What is the function?

A function is a rule that pairs up every element of one set, known as the domain, with a specific aspect of another set, known as the range or codomain. In other words, a function describes a relationship between two sets in which each input from the domain corresponds to exactly one output from the range.

To interpret the function correctly, we need to consider the values of x.

If x = 0, then the year 2004 is indicated because the function is expressed in terms of x years since 2004.

The base, 0.62, falls between 0 and 1, making the function an exponential growth function. In other words, the function will approach but never reach 0.

The function, for example, becomes m(1) = [tex]9,000(0.62)^{1}[/tex] = 5,580 if x = 1. This indicates that around 5,580 Movie Bank locations exist in the world as of 2004, one year later.

The function will continue to increase if x is raised further but at a diminishing rate. The function, for instance, becomes m(5) = [tex]9,000(0.62)^{5}[/tex] = 1,673 if x = 5. This indicates that roughly 1,673 Movie Bank locations are located throughout the world, which is significantly fewer than the previous figure, five years following the year 2004.

Therefore, the correct interpretation of the function is that it shows the exponential growth of the number of worldwide locations of Movie Bank x years since 2004, where the growth rate is 0.62.

Learn more about Functions from the given link.

https://brainly.com/question/11624077

#SPJ1

Answer:

Movie Bank locations have decreased at a rate of 38% each year since 2004.

The company should hire students for 10 hours of labor per day to maximize profit.

To determine the number of hours (L) of labor the company should use per day to maximize profit, we need to consider the revenue, cost, and profit functions, and then find the critical points.

1. Revenue: The company sells 3L^(2/3) Kg of wild mushrooms at $15.00 per Kg.

So, the revenue function R(L) = 15 * 3L^(2/3) = 45L^(2/3).

2. Cost: The company pays $6.00 per hour for L hours of labor.

So, the cost function C(L) = 6L.

3. Profit: The profit function P(L) is the difference between revenue and cost:

P(L) = R(L) - C(L) = 45L^(2/3) - 6L.

4. To maximize profit, we need to find the critical points by taking the derivative of the profit function and setting it equal to zero:

P'(L) = d(45L^(2/3) - 6L) / dL = 0.

5. Derivative:

P'(L) = (2/3)*45L^(-1/3) - 6.

6. Solve for L:

Set P'(L) = 0 and solve for L:

(2/3)*45L^(-1/3) - 6 = 0.

By solving this equation, we find that L ≈ 9.58 hours.

Since the company cannot hire a fraction of an hour, it should hire students for 10 hours of labor per day to maximize profit.

Learn more about Profit:

https://brainly.com/question/19104371

#SPJ11

Answer:

217 days.

Step-by-step explanation:

The 99% confidence interval for the mean score of all students is constructed as (85.497, 96.5026).

Given,

Sample size, n = 30

Sample mean, x = 91

Standard deviation, s = 11.7

The 99% confidence interval for the mean score of all students can be calculated as,

x ± z [tex]\frac{s}{\sqrt{n} }[/tex]

z value for 99% confidence interval = 2.576

Confidence interval = (91 ± (2.576 × 11.7/√30)

                                 = (91 ± 5.5026)

                                 = (85.497, 96.5026)

Hence the confidence interval is (85.497, 96.5026).

Learn more about Confidence Interval here :

https://brainly.com/question/2598134

#SPJ4

a. â converges in probability to a as n approaches infinity, and is consistent.

b. A converges in probability to [tex]a_true[/tex]as n approaches infinity, and is consistent, a = Y bar.

c. B converges in probability to [tex]B_true[/tex] as n approaches infinity, and is consistent.   B [tex]= B_true.[/tex]

(a) To show that the ordinary least squares estimators â and B are consistent, we need to show that they converge in probability to the true values of a and B as the sample size increases.

Using the properties of the OLS estimators, we have:

â = Y bar - B X bar,

[tex]B = \sum(Xi - X \bar)(Yi - Y \bar) / ∑(Xi - X \bar)^2[/tex]

where Y bar and X bar are the sample means of Y and X, respectively.

To show consistency, we need to show that as n approaches infinity, â and B converge in probability to a and B, respectively.

First, consider â. We have:

â [tex]= Y\bar - B X \bar = (1/n) \sum Yi - B (1/n) \sum Xi[/tex]

Taking the limit as n approaches infinity, we have:

lim(n→∞) â = lim(n→∞) [(1/n) ∑Yi - B (1/n) ∑Xi]

= E(Y) - B E(X)

= a + B E(X) - B E(X)

= a

Therefore, â converges in probability to a as n approaches infinity, and is consistent.

Next, consider B. We have:

[tex]B = \sum(Xi - X \bar)(Yi - Y\bar) / \sum(Xi - X bar)^2[/tex]

[tex]= (\sum XiYi - n X \bar Y \bar) / (\sum Xi^2 - n X \bar^2)[/tex]

Taking the limit as n approaches infinity, we have:

lim(n→∞) B = lim(n→∞) [tex](\sum XiYi - n X \bar Y \bar) / (\sum Xi^2 - n X \bar^2)[/tex]

= Cov(X,Y) / Var(X)

[tex]= B_true.[/tex]

Therefore, B converges in probability to [tex]B_true[/tex] as n approaches infinity, and is consistent.

(b) Suppose B = 0. Then, the restricted sum of squared errors is [tex]Li = \sum(Yi - a)^2.[/tex]

To find the value of a that minimizes Li, we take the derivative of Li with respect to a and set it equal to 0:

dLi/da = -2∑(Yi - a) = 0

Solving for a, we get:

a = Y bar

To show consistency, we need to show that as n approaches infinity, a converges in probability to [tex]a_true.[/tex]

Using the law of large numbers, we have:

lim(n→∞) a = lim(n→∞) Y bar

= E(Y)

[tex]= a_true[/tex]

Therefore, a converges in probability to [tex]a_true[/tex]as n approaches infinity, and is consistent.

(c) Suppose a = 0.

Then, the restricted sum of squared errors is [tex]21 = \sum(Yi - BXi)^2.[/tex]

To find the value of B that minimizes 21, we take the derivative of 21 with respect to B and set it equal to 0:

d21/dB = -2∑Xi(Yi - BXi) = 0

Solving for B, we get:

[tex]B = \sum XiYi / \sum Xi^2[/tex]

To show consistency, we need to show that as n approaches infinity, B converges in probability to[tex]B_true.[/tex]

Using the law of large numbers, we have:

lim(n→∞) B = lim(n→∞) [tex](\sum XiYi / \sum Xi^2)[/tex]

= Cov(X,Y) / Var(X)

[tex]= B_true.[/tex]

Therefore, B converges in probability to [tex]B_true[/tex] as n approaches infinity, and is consistent.

For similar question on probability.

https://brainly.com/question/28213251

#SPJ11

The value of y for the function y = cos(-60°) is y = -1/2, option A is correct.

An equation with trigonometric functions that holds true for all of the variables in it is known as a trigonometric identity. A few normal geometrical characters incorporate the Pythagorean personality, the total and distinction characters, and the twofold point characters.

Using the unit circle and the trigonometric identity for cosine, we know that:

cos(-60°) = cos(360° - 60°)

               = cos(300°)

               = cos(180° + 120°)

               = -cos(120°)

               = -1/2

Therefore, the value of y for the function y = cos(-60°) is y = -1/2.

To know more about trigonometric functions, visit:
https://brainly.com/question/25618616
#SPJ1