Answers
Answer:
x² = \(\frac{256}{9}\)
Step-by-step explanation:
\(\frac{3x}{4}\) - 13 = - 9 ( add 13 to both sides )
\(\frac{3x}{4}\) = 4 ( multiply both sides by 4 to clear the fraction )
3x = 16 ( divide both sides by 3 )
x = \(\frac{16}{3}\)
Then
x² = ( \(\frac{16}{3}\) )² = \(\frac{16^2}{3^2}\) = \(\frac{256}{9}\)
Related Questions
I need help immediately!!!
Answers
The limit as x approaches one is infinity.
\(lim_{x\to1}\frac{x + x {}^{2} + {x}^{3} + ... + {x}^{100} - 1000}{1 - x} =\infty\)
What is the limit of a function?The limit of a function, f(x) as x approaches a given value b, is define as the value that the function f(x) attains as the variable x approaches the given value b.
From the given question, as x approaches 1,
substituting x into 1 - x,
the denominator of the function approaches zero, because 1 - 1 = 0 and thus the function becomes more and more arbitrarily large.
Thus, the limit of the function as x approaches 1 is infinity.
Therefore,
The limit (as x approaches 1)
\(lim_{x\to1}\frac{x + x {}^{2} + {x}^{3} + ... + {x}^{100} - 1000}{1 - x} = \infty \)
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Find a basis for the eigenspace corresponding to each listed eigenvalue of A below. A = [2 -1 1 ]
[0 -3 -4]
[0 8 9], lambda = 2, 5, A basis for the eigenspace corresponding to lambda = 2 is
Answers
The basis for the eigenspace corresponding to the eigenvalue λ = 2 is {[1, 1, 0]}.
To find a basis for the eigenspace corresponding to the eigenvalue λ = 2, we need to solve the equation (A - λI)X = 0, where A is the given matrix, λ is the eigenvalue, X is the eigenvector, and I is the identity matrix.
Given matrix A:
[2 -1 1]
[0 -3 -4]
[0 8 9]
Eigenvalue: λ = 2
We subtract λI from A to get (A - λI):
[2 - 1 1]
[0 -3 -4]
[0 8 9] - 2 * [1 0 0]
[0 1 0]
[0 0 1]
Simplifying, we have:
[2 - 1 1]
[0 -3 -4]
[0 8 9] - [2 0 0]
[0 2 0]
[0 0 2]
= [0 -1 1]
[0 -5 -4]
[0 8 7]
Now we need to solve the equation (A - λI)X = 0 to find the eigenvectors.
Substituting λ = 2 into (A - λI), we have:
[0 -1 1]
[0 -5 -4]
[0 8 7]X = 0
To solve this homogeneous system of equations, we can use row reduction. We start with the augmented matrix:
[0 -1 1 0]
[0 -5 -4 0]
[0 8 7 0]
Performing row operations, we can obtain the row-echelon form:
[0 -1 1 0]
[0 0 -1 0]
[0 0 0 0]
From this, we can write the system of equations:
-x + y = 0 ---> x = y
-z = 0 ---> z = 0
0 = 0 ---> no restriction on any variable
In vector form, the eigenvectors can be expressed as:
X = [y, y, 0] = y[1, 1, 0]
This indicates that for any scalar value y, the vector [y, y, 0] is an eigenvector corresponding to the eigenvalue λ = 2.
Therefore, a basis for the eigenspace corresponding to λ = 2 is { [1, 1, 0] }.
In summary, the basis for the eigenspace corresponding to the eigenvalue λ = 2 is {[1, 1, 0]}.
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An accountant is modeling the annual tax expenditures, e, in thousands of dollars t years after january 1st, 2000 for a small business using two different models. Both of the accountant's models have tax expenditures of $5000 on january 1st, 2000. What is the value of e?
Answers
Both models assume that the tax expenditures are $5000 on January 1st, 2000.
The accountant is using two different models to estimate the tax expenditures for the small business. Both models assume that the tax expenditures are $5000 on January 1st, 2000. The accountant wants to estimate the tax expenditures, e, in thousands of dollars t years after January 1st, 2000.
Let's call the two models Model 1 and Model 2. Model 1 assumes that the tax expenditures increase by a fixed amount every year. Let's call this fixed amount a. Therefore, the tax expenditures, e, under Model 1 can be represented by the equation:
e = 5000 + at
Model 2 assumes that the tax expenditures increase at a constant percentage rate every year. Let's call this percentage rate r. Therefore, the tax expenditures, e, under Model 2 can be represented by the equation:
e = 5000(1 + r)ˣ
Now, let's substitute t = 0 into both equations to find the value of e on January 1st, 2000.
For Model 1:
e = 5000 + a(0) = 5000
For Model 2:
e = 5000(1 + r)⁰ = 5000
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Please help me with this question
Answers
if any doubt leave a comment
PLEASE HELP WILL GIVE BRANLIEST
Answers
In ΔJKL, k = 430 cm, l = 590 cm and ∠J=117°. Find the length of j, to the nearest centimeter.
Answers
Answer:
874
Step-by-step explanation:
The length of j from the given triangle JKL is 550 cm.
What is cosine rule?The law of cosine states that the square of any one side of a triangle is equal to the difference between the sum of squares of the other two sides and double the product of other sides and cosine angle included between them. The law of cosine states that: a² = b² + c² − 2bc·cosA.
Given that, in ΔJKL, k = 430 cm, l = 590 cm and ∠J=117°.
Using cosine rule
j²=k²+l²-2kl.cosJ
j²=430²+590²-2×430×590·cos117°
j²=184900+348100-507400×0.4539
j²=533000-230308.86
j²=302691.14
j=√302691.14
j=550 cm
Therefore, the length of j from the given triangle JKL is 550 cm.
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3)
The height of a triangle is 6 centimeters more than double its base in centimeters. What is the height of the triangle, in
centimeters, if the base of the triangle is 8 centimeters?
Answers
Answer:
22centimetres
Step-by-step explanation:
..............,
Answer:
22 centimeters.
Step-by-step explanation:
Since the base is 8 centimeters, you multiply it by 2.
8•2=16
Then you add 6.
16+6=22
. Alexandro, o professor de Física, avisou aos alunos que a média bimestral seria calculada conforme o seguinte critério: Adicionam-se as notas obtidas no projeto, na prova e no trabalho em grupo e divide-se o resultado obtido por três. Laura é uma das alunas do professor Alexandre. As notas de Laura: Projeto: 5,0 Prova: 6,5 trabalho em grupo: 9,5. A) Por que o resultado deve ser dividido por 3?
Answers
Answer:
É necessário dividir por 3 para calcular a média simples das 3 notas.
Média = 7.
Step-by-step explanation:
O resultado deve ser dividido por três porque três notas com pesos iguais (projeto, prova e trabalho em grupo) foram somadas, portanto para calcular a média bimestral de Laura, a soma das três notas deve ser dividida por 3. A média de Laura é:
\(M=\frac{5,0+6,5+9,5}{3}\\ M=7\)
PLZ answer fast Part 1: Celebration!
You and three friends will be going out to eat at your favorite restaurant!
Each of you will order three items of your choice. You will also need to leave a tip. If you split the bill evenly among all four of you, how much will each person need to contribute?
Locate a menu for your favorite restaurant and pick three items for yourself and each friend. Make sure all or some of the items differ for each person.
List the restaurant name, items, and prices in a table like the one below.
In the column labeled Subtotal, determine the cost of each person’s order.
The customary tip is 20% of the total bill. In the column labeled Tip, calculate the amount of tip each person should leave.
In the column labeled Total Bill, determine the total of each person’s order, including the tip.
Restaurant name:
Names Item 1 Price Item 2 Price Item 3 Price Subtotal Tip Total Bill
Your name
Friend #1
Friend #2
Friend #3
Part 2: Questions
Answer the following questions, showing all work:
What is the total cost of the entire bill (all four meals), including tip?
Write an expression using fractions to show how to determine the amount that each person will pay. Then calculate each person's contributions showing all steps in long division.
If each person paid an equal amount, who would save the most money? Explain your reasoning using at least two complete sentences.
Answers
Answer:
Ok for the 1st and 2nd paragraph, each person will need to pay 20 dollars each. For the 3rd paragraph, you could say your favorite restaurant is McDonalds. For you, you could have a strawberry milkshake, medium fries, and a Big Mac. For your second friend, they could have a coca cola, a salad, and large fries. For your third friend, they could have a lemonade, lettuce hamburger, and chocolate ice cream. For your fourth friend, they could have a Oreo Mc flurry, chicken nuggets, and small fries.
Restaurant name: McDonalds
Price: 59.99/60.00
Cost of each persons order: you pick
Each person should leave 5 dollars as a tip.
The total including tip: 79.99/80.00
Each person will pay 15.00
I dont know the last one sorry
Need help only have 10 minutes and can’t get a tutor please help
Answers
STEP - BY - STEP EXPLANATION
What to find?
Measure of line DC using the properties of an isosceles triangle.
Given:
• Line DE = 30
,• Line CE= 32
To solve the given problem, we will follow the steps below:
Step 1
Recall the property of the isosceles triangle.
That is;
An isosceles triangle is a type of triangle with two sides angles equal and this follows that two of its sides are also equal.
Hence, the property we will be using to solve this is:
Two of the sides of the given triangles are equal.
Step 2
Identify the side that is equa to side DC
Since angle C = angle E, this follows that;
DC = DE = 30
Therefore, DC = 30
what is the goal of a hypothesis test? select one: a. to give a strong statement about a statistic b. to determine if the data can refute an assumption about a parameter or population c. to create a larger sample d. to provide a range of values that are reasonable for the parameter of interest
Answers
The goal of a hypothesis test is to determine if the data can refuse an assumption a parameter or a population.
Statistical hypothesis test:
A statistical hypothesis test is a technique for determining whether the available data are sufficient to support a specific hypothesis. We can make probabilistic claims about population parameters through hypothesis testing.
The purpose of a hypothesis test is to ascertain whether the data can refute a parameter or population assumption.
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Mrs karki purchased a sari for rs 8000 and sold it for Rs 11300 with 13% VAT , find her profit and loss percent.
Answers
The profit Mrs karki made from the Sari she sold at RS 11300 was 28.25%
What is an equation?An equation is an expression that shows the relationship between two numbers and variables.
An independent variable is a variable that does not depend on any other variable for its value whereas a dependent variable is a variable that depend on any other variable for its value.
VAT = 13% = 0.13 * 8000 = 1040
Selling price - VAT = 11300 - 1040 = 10260
Profit = [(10260 - 8000)/8000] * 100% = 28.25%
The profit Mrs karki made from the Sari she sold at RS 11300 was 28.25%
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In a city in Ohio, the sales tax rate is 7.25%. Complete the table.
Answers
The regression equation y=3.648⋅1.182x approximates the cost to go on a safari, y, given the number of years since it opened in 2005, x.
Which amount is the best estimate for the cost of a vehicle to drive through the safari in 2009?
$7.12
$16.43
$8.38
$17.25
Answers
Answer:
7.12 thank me later
Step-by-step explanation:
Find the slope and the y-intercept of the graph of y=4x−5
.
Answers
y
=
m
x
+
b
y
=
m
x
+
b
, where
m
m
is the slope and
b
b
is the y-intercept.
y
=
m
x
+
b
y
=
m
x
+
b
Find the values of
m
m
and
b
b
using the form
y
=
m
x
+
b
y
=
m
x
+
b
.
m
=
4
m
=
4
b
=
−
5
b
=
-
5
The slope of the line is the value of
m
m
, and the y-intercept is the value of
b
b
.
Slope:
4
4
y-intercept:
(
0
,
−
5
)
select all that apply. the probability of rolling an even number with a six-sided, equally weighted die is group of answer choices 1/3. 1/2. 1/6. 3/6.
Answers
Using the sample space, the probability of rolling an even number with a six-sided dice is 3/6 or 1/2.
In the given question we have to find the probability of rolling an even number with a six-sided dice.
The sample space of dice through the experiment is
{1,2,3,4,5,6}
The even numbers is {2,4,6}.
The probability of rolling an even number with a six-sided dice is
P(E)=Total number of even outcome/Total number of outcome
P(E)=3/6
P(E)=1/2
Hence, the probability of rolling an even number with a six-sided dice is 3/6 or 1/2.
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The half life for a first order reaction is 20 min. What is the
rate constant in units of s-1?
Select one:
Answers
The rate constant for the first-order reaction is approximately 0.035 s⁻¹. The correct answer is B
To find the rate constant in units of s⁻¹ for a first-order reaction, we can use the relationship between the half-life (t1/2) and the rate constant (k).
The half-life for a first-order reaction is given by the formula:
t1/2 = (ln(2)) / k
Given that the half-life is 20 minutes, we can substitute this value into the equation:
20 = (ln(2)) / k
To solve for the rate constant (k), we can rearrange the equation:
k = (ln(2)) / 20
Using the natural logarithm of 2 (ln(2)) as approximately 0.693, we can calculate the rate constant:
k ≈ 0.693 / 20
k ≈ 0.03465 s⁻¹
Therefore, the rate constant for the first-order reaction is approximately 0.0345 s⁻¹. The correct answer is B
Your question is incomplete but most probably your full question was attached below
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The weekly demand function for x units of a product sold by only one firm is p = 700 − 1/2 x dollars, and the average cost of production and sale is C = 400 + 2x dollars.(a)Find the quantity that will maximize profit.units(b) Find the selling price at this optimal quantity.$ per unit(c) What is the maximum profit?$
Answers
To find the quantity that will maximize profit, we need to first calculate the total revenue and total cost functions. Total revenue is given by the product of the price and quantity, which is p*x.
Therefore, the total revenue function is R = (700-1/2x)*x = 700x - 1/2x^2. Total cost is given by the sum of the average fixed cost and average variable cost, which is C = 400 + 2x. Therefore, total cost function is C = (400+2x)*x = 400x + 2x^2.
Next, we can find the profit function by subtracting total cost from total revenue:
P = R - C = 700x - 1/2x^2 - 400x - 2x^2 = -5/2x^2 + 300x.
To maximize profit, we need to take the first derivative of the profit function and set it equal to zero:
dP/dx = -5x + 300 = 0
x = 60
Therefore, the quantity that will maximize profit is 60 units.
To find the selling price at this optimal quantity, we can substitute x=60 into the demand function:
p = 700 - 1/2(60) = $670 per unit.
Therefore, the selling price at this optimal quantity is $670 per unit.
To find the maximum profit, we can substitute x=60 into the profit function:
P = -5/2(60)^2 + 300(60) = $12,000.
Therefore, the maximum profit is $12,000.
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Maggie's brother is 4 years younger than twice her age. The sum of their ages is 17.
How old is Maggie?
Answers
Answer:
Maggie is 7 years old
Step-by-step explanation:
m+2m-4=17
3m-4=17
3m=21
m=7
Answer:
Maggie is seven years old
Step-by-step explanation:
Maggie's Brothers Age = B
Maggie's Age = M
2M - 4 = B
B + M = 17
Plug in 2M - 4 for B
( 2M - 4 ) + M = 17
Combine Like Terms
3M - 4 = 17
Add 4 to either side
3M = 21
Divide either side by 3
M = 7
Check your answer
2M - 4 = B -> 2 x 7 - 4 = B -> 14 -4 = B -> B = 10
B + M = 17 => 7 + 10 = 17 => 17 = 17
Use the function to predict the minimum target heart rate for a person 20 years old. beats per minute
Answers
The minimum target heart rate for a person 20 years old is 120 beats per minute
How to predict the minimum target heart rate?The equation of the function is given as:
y = -0.6x + 132
Where x represents the age, and y represents the minimum target heart rate
From the question, we have:
x = 20
Substitute x = 20 in y = -0.6x + 132
This gives
y = -0.6 *20 + 132
Evaluate
y = 120
Hence, the minimum target heart rate for a person 20 years old is 120 beats per minute
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Answer:
Answer is 120 !!
Step-by-step explanation:
~Hope this helps! :)
If a card is drawn from a deck, find the probability of getting these results:
a. a 6 and a spade
b. a black king
c. a red card and a 7
d. a diamond or a heart
e. a black card
Answers
Answer:
Below in bold,
Step-by-step explanation:
a. There's is only one card that fits this description so
Probability = 1/52.
b. There are 2 black Kings so
Probability = 2/52 = 1/26.
c. There are 2 of these - 7 of diamonds and 7 of hearts
so
Probability = 2/52 = 1/26.
d. There 13 diamonds and 13 hearts so
Probability = 26/52 = 1/2.
e. There 26 black cards
Probability = 26/52 = 1/2.
Please help
Given the angle measure
Answers
The answer choices are:A. Definition of complementary anglesB. Definition of a linear pairC. Simplification of equation from previous step D. Substitution
Answers
For the fourth step shown in the "Reasons"
\(\angle ANV+\angle ANR=180\)Note that both angles lie on the same straight line which is line RV.
Angles on a straight line sum up to 180 degrees.
Therefore,
\(\begin{gathered} \angle ANV+\angle ANR=180\text{ } \\ \text{Both angles are a linear pair} \end{gathered}\)The answer is option B; Definition of a linear pair
The radius r and height h of a circular cone change at a rate of 2 cm/s. How fast is the volume of the cone increasing when r = 10 and h = 20?
Answers
At r = 10 cm and h = 20 cm, the cone's volume is growing at a rate of 1047 cm³/s.
Describe the term rate of change?Volume changes at a rate of dV/dt. The rate at which the side length is varying is what is being questioned ds/dt would then apply. The result is dV/dt = d/dt[s^3]. if we consider the derivative from both side of V = s3.The following formula determines the cone's volume:
V = 1/ 3 πr²h
In which r and h are specified in cm, whereas V is given in cm³.
Assume that a circular cone's height h and radius r are both rising at a speed of 2 cm/s.
This indicates that; the rate of change of volume
dh/dt = dr/dt = 2 cm/s.
Increasing the cone's volume at r = 10 cm and h = 20 cm.
dV/dt for r = 10 cm and h = 20 cm.
V = 1/ 3 πr²h
Implicit distinction in use:
V, r, and h are the three variables we have. So
dV/dt = (1/3) [2r.(dr/dt).hπ + r²π(dh/dt)]
dV/dt = (1/3) [2(10).(2).(20)π + (10)²π(2)]
dV/dt = 1047 cm³/s
At r = 10 cm and h = 20 cm, the cone's volume is growing at a rate of 1047 cm³/s.
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in a certain industrial facility, accidents occur infrequently. it is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other. what is the probability that in any given period of 400 days there will be an accident on one day? use binomial approximation to poisson distribution.
Answers
The probability that in any given period of 400 days there will be an accident on one day is 0.2707
Given that;
In a certain industrial facility, accidents occur infrequently. it is known that the probability of an accident on any given day is 0.005 and accidents are independent of each other.
n = 400, p = 0.005
Poisson distribution is given by;
e P(x) = (λˣe^λ ) / x!
Here, Mean(λ) = np
= 400*0.005
=2
The probability that there will be an accident on one day;
(P(x = 1)) = (λxe-λ)/x!
= 2*e-2
= 0.2707
The probability that in any given period of 400 days there will be an accident on one day is 0.2707
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Search your home for a rectangular prism. Some examples are a cereal box, a CD case, or a coffee table. Measure your prism using appropriate units, such as inches, centimeters, or feet.
Complete the following. Show all work for calculations.
List the dimensions of your box. Be sure to include the units (in, cm, ft, etc.).
Describe the shape of the cross section when the box is cut parallel to the base.
What is the surface area of the box?
What is the surface area of the box if it is scaled up by a factor of 10?
What is the volume of the box?
What is the volume of the box if it is scaled down by a factor of
1
10
?
ill give 40 points and brainlysest please i really need this please
Answers
Answer:
541/2 square in I hope this helps
Step-by-step explanation:
i hope this helps if it does can i get brainlists
Answer:
Rectangular prism
I chose a box of cereal.
Part 1)List the dimensions of your box. Be sure to include the units (in, cm, ft, etc.).
Length: 20 cm
Width: 8 cm
Height: 32 cm
Part 2).
Describe the shape of the cross-section when the box is cut parallel to the base.
The shape of the cross-section would be a rectangle in dimensions.
20 cm x 8 cm
Part 3)
What is the surface area of the box?
surface area=2*area of the base + perimeter of the base*height
area of the base=20*8=160 cm²
perimeter of the base=2*[20+8]-----> 56 cm
height=32 cm
surface area=2*160+56*32------> 2,112 cm²
the answer part 3) is
2,112 cm²
Part 4)
What is the surface area of the box if it is scaled up by a factor of 10?
we know that
surface area of the larger box =[scale factor]²*surface area original box
scale factor=10
surface area original box=2,112 cm²
so
surface area of the larger box=10²*2,112-----> 211,200 cm²
the answer part 4) is
211,200 cm²
Part 5)
What is the volume of the box?
volume of the box = L*W*H 20*8*32-5,120 cm³
the answer Part 5) is
5,120 cm³
Part 6)
What is the volume of the box if it is scaled down by a factor of 1/10?
we know that
the volume of the smaller box =[scale factor]³volume original box
scale factor=1/10
volume original box=5,120 cm³
so
volume of the smaller box =[1/10]³*5,120 5.12 cm³
the answer part 6) is
5.12 cm³
The length is 6 in., the width is 2 in., and the height is 16 in.
f(x)=2x^2+x-4
Find f(−10)
Answers
Answer:
186
Step-by-step explanation:
f(x)=2x^2+x-4
f(-10)=2(-10)(-10)+(-10)-4=2(100)-10-4=200-10-4=190-4=186
Answer:
186
Step-by-step explanation:
Which statement about 7/3 is true
A. 7/3 = 7 x 3/1
B. 7/3 = 7 x 3/3
C. 7/3 = 7 c 1/7
D. 7/3 = 7 x 1/3
Answers
Answer:
D, the rest are not true about the statement 7/3
Answer:
D. 7/3 = 7 x 1/3
Step-by-step explanation:
\(7 * \frac{1}{3} = \frac{7}{1} * \frac{1}{3} = \frac{7}{3}\)
Suppose {ϵ i
} is a sequence of n independent normal random variables with mean 0 and variance σ 2
. Consider the simple linear regression model: Y i
=β 0
+β 1
X i
+ϵ i
, i=1,…,n. Recall from class that β
^
0
= y
ˉ
− β
^
1
x
ˉ
and β
^
1
= S xx
S xy
, where S xy
=∑ i=1
n
(x i
− x
ˉ
)(y i
− y
ˉ
) and S xx
=∑ i=1
n
(x i
The residual sum of squares (or sum of squared error) is defined by SSE=∑ i=1
n
e i
2
where e i
=y i
− y
^
i
and y
^
i
is the i-th fitted value, i.e., y
^
i
= β
^
0
+ β
^
1
x i
. The residual sum of squares (or sum of squared error) is defined by SSE=∑ i=1
n
e i
2
where e i
=y i
− y
^
i
and y
^
i
is the i-th fitted value, i.e., y
^
i
= β
^
0
+ β
^
1
x i
. Show that: (a) ∑ i=1
n
e i
=0 (b) ∑ i=1
n
x i
e i
=0 (c) E[∑ i=1
n
(ϵ i
− ϵ
ˉ
i
) 2
]=(n−1)σ 2
.
Answers
To prove the given statements, we'll use the definitions and properties of the linear regression model and the properties of the normal distribution. Let's go step by step:
(a) We want to show that ∑ni=1ei = 0.
Using the definition of the residuals (ei = yi - ŷi), we have:
∑ni=1ei = ∑ni=1(yi - ŷi).
Substituting ŷi = β^0 + β^1xi = y¯ - β^1x¯ + β^1xi, we get:
∑ni=1ei = ∑ni=1(yi - y¯ + β^1x¯ - β^1xi).
Rearranging the terms and regrouping, we have:
∑ni=1ei = (∑ni=1yi) - n⋅y¯ + (∑ni=1β^1x¯) - (∑ni=1β^1xi).
Since y¯, x¯, and β^1 are constants, we can take them out of the summation:
∑ni=1ei = (∑ni=1yi) - n⋅y¯ + (∑ni=1β^1x¯) - β^1(∑ni=1xi).
Using the property that ∑ni=1yi = n⋅y¯ and ∑ni=1xi = n⋅x¯, we simplify further:
∑ni=1ei = n⋅y¯ - n⋅y¯ + n⋅β^1x¯ - n⋅β^1x¯ = 0.
Therefore, we have shown that ∑ni=1ei = 0.
(b) We want to show that ∑ni=1xiei = 0.
Using the definition of the residuals (ei = yi - ŷi) and the definition of ŷi, we can write:
∑ni=1xiei = ∑ni=1xi(yi - ŷi) = ∑ni=1xi(yi - β^0 - β^1xi).
Expanding the expression and regrouping terms, we get:
∑ni=1xiei = (∑ni=1xi⋅yi) - (∑ni=1xi⋅β^0) - (∑ni=1β^1xi^2).
Again, using the properties that ∑ni=1xi⋅yi = ∑ni=1xi⋅ŷi and ∑ni=1xi = n⋅x¯, we simplify further:
∑ni=1xiei = ∑ni=1xi⋅ŷi - n⋅β^0⋅x¯ - (∑ni=1β^1xi^2).
Using the definition of ŷi = β^0 + β^1xi, we can rewrite ∑ni=1xi⋅ŷi as:
∑ni=1xi⋅ŷi = ∑ni=1xi⋅(β^0 + β^1xi) = ∑ni=1xi⋅β^0 + β^1(∑ni=1xi^2).
Substituting this back into the equation, we have:
∑ni=1xiei = (∑ni=1xi⋅β^0 + β^1(∑ni=1xi^2)) - n⋅β^0⋅x¯ - (∑ni=1β^1xi^2).
Simplifying the terms, we get:
∑ni=1xiei = (∑ni=1xi⋅β^0 - n⋅β^0⋅x¯) + (β^1(∑ni=1xi^2) - ∑ni=1β^1xi^2).
Again, using the property that ∑ni=1xi = n⋅x¯, we have:
∑ni=1xiei = (∑ni=1xi⋅β^0 - n⋅β^0⋅x¯) + (β^1(∑ni=1xi^2) - ∑ni=1β^1xi^2) = 0.
Therefore, we have shown that ∑ni=1xiei = 0.
(c) We want to show that E[∑ni=1(ϵi - ϵ¯i)^2] = (n - 1)σ^2.
First, let's define ϵ¯i as the mean of the residuals ϵi. Since ϵi ~ N(0, σ^2), the mean ϵ¯i = E[ϵi] = 0.
Expanding the squared term and using the linearity of expectation, we have:
E[∑ni=1(ϵi - ϵ¯i)^2] = E[∑ni=1(ϵi^2 - 2ϵiϵ¯i + ϵ¯i^2)].
Taking the expectation inside the summation, we get:
E[∑ni=1(ϵi - ϵ¯i)^2] = ∑ni=1(E[ϵi^2] - 2E[ϵiϵ¯i] + E[ϵ¯i^2]).
Since E[ϵi] = E[ϵ¯i] = 0, we can simplify further:
E[∑ni=1(ϵi - ϵ¯i)^2] = ∑ni=1(E[ϵi^2] - 2⋅0⋅E[ϵi] + E[ϵ¯i^2]).
Using the property that E[ϵi^2] = Var[ϵi] + (E[ϵi])^2 and E[ϵ¯i^2] = Var[ϵ¯i] + (E[ϵ¯i])^2, we have:
E[∑ni=1(ϵi - ϵ¯i)^2] = ∑ni=1(Var[ϵi] + (E[ϵi])^2 - 2⋅0⋅E[ϵi] + Var[ϵ¯i] + (E[ϵ¯i])^2).
Simplifying, we get:
E[∑ni=1(ϵi - ϵ¯i)^2] = ∑ni=1(Var[ϵi] + Var[ϵ¯i]).
Since ϵi ~ N(0, σ^2) and ϵ¯i = 0, we have Var[ϵi] = σ^2 and Var[ϵ¯i] = 0.
Substituting these values back into the equation, we get:
E[∑ni=1(ϵi - ϵ¯i)^2] = ∑ni=1(Var[ϵi] + Var[ϵ¯i]) = ∑ni=1(σ^2 + 0) = nσ^2.
Finally, since the sum of squared errors (SSE) is given by ∑ni=1ei^2, we have:
E[∑ni=1(ϵi - ϵ¯i)^2] = E[∑ni=1ei^2] = SSE.
Therefore, we conclude that E[SSE] = nσ^2.
However, it's important to note that in the derived expression, we made an assumption that the residuals ϵi are normally distributed with mean 0 and variance σ^2.
Learn more about normal distribution here:
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Find the sum (7b -4) + (2b+ a + 1)
Answers
Answer: (9b + a - 3)
Step-by-step explanation: Good luck! :D
Answer:
a=9b-3
Step-by-step explanation: