Multiple linear regression model is used to fit the observed data, in order to characterize the relationship between the response variable y and i covariates of interest.
The following regression model is used:
[tex]y=Xβ+ ε[/tex]
where:y denotes the n × 1 response vector.
[tex]β[/tex]represents the [tex]p × 1[/tex]parameter vector consisting of [tex]p = k + 1[/tex] regression coefficients.
[tex]X[/tex]denotes the n × p model matrix.
[tex]ε[/tex] denotes the [tex]n × 1[/tex] vector of error terms.
The entries of the first column of X are all equal to 1.
The entries of other columns of X correspond to the i covariates of interest.
It is given that the model matrix X is an n × p full-column-rank matrix.
The least squares estimator of [tex]β[/tex] is given by:
[tex]β^ = (X'X)^-1X'y[/tex]
The error terms are assumed to be independent and identically distributed normal random variables.
The variance-covariance matrix of the least squares estimator is given by:
[tex]Var(β^) = σ^2(X'X)^-1[/tex]
It is given that all covariates have the same variance-covariance structure.
Hence,
[tex]σ^2 = σ0^2[/tex] for every i.
It is also given that a, σ0 for every i, where a represents a positive constant.
Hence,
[tex]σ^2 = σ0^2[/tex]
= [tex]a^2[/tex]
Show that Var and the equality would be attained if
[tex]X'X = a^2I[/tex],
where [tex]β^[/tex] represents the ith entry of the least square estimator [tex]β[/tex].
From the given data, the variance-covariance matrix of the least squares estimator is given by:
[tex]Var(β^) = σ^2(X'X)^-1[/tex]
[tex]= (a^2/n)(X'X)^-1[/tex]
It is given that all covariates have the same variance-covariance structure.
Hence,
[tex]σ^2 = σ0^2[/tex] for every i.
It is also given that a, σ0 for every i, where a represents a positive constant.
Hence,
[tex]σ^2 = σ0^2[/tex]
[tex]= a^23[/tex]
Hence,
[tex]Var(β^) = (a^2/n)(X'X)^-1[/tex]
Now, let the diagonal entries of (X'X) be d1, d2, ..., dp.
Hence,
(X'X) = [dij]
i=1,2,...,p;
X¹ = [0, 0, ..., 1, ..., 0]'
Let X¹ denote the ith column of X.
Hence, X¹ is given by:
[tex]X¹ = [0, 0, ..., 1, ..., 0]'[/tex]
where 1 is in the ith position.
Hence, the ith diagonal entry of X'X is given by:
[tex](X'X)ii = Σj(Xj¹)^2[/tex]
where the sum is over all i.
From the given data, the entries of the first column of X are all equal to 1.
Hence, [tex]X1¹ = [1, 1, ..., 1]'.[/tex]
Hence, [tex](X'X)ij = nai[/tex] and
[tex](X'X)ij = nai[/tex] for [tex]i ≠ j.[/tex]
Hence,[tex](X'X) = a^2I + n11'[/tex]
The inverse of (X'X) is given by:
[tex](X'X)^-1 = (1/n)(I - (1/n)a^-2(1 1'))[/tex]
Hence,
[tex]Var(β^) = (a^2/n)(X'X)^-1[/tex]
=[tex]a^2[(1/n)(I - (1/n)a^-2(1 1'))][/tex]
The variance of the ith entry of the least square estimator is given by:
[tex]Var(β^i) = ai^2[(1/n)(I - (1/n)a^-2(1 1'))]ii[/tex]
Hence,
[tex]Var(β^i)[/tex]= [tex]ai^2[(1/n)(1 - (1/n)a^-2)][/tex]
= [tex]a^2/n[/tex]
Therefore, the variance of the ith entry of the least square estimator is given by:
[tex]Var(β^i) = a^2/n[/tex]
The equality would be attained if
[tex]X'X = a^2I,[/tex]
where β^i represents the ith entry of the least square estimator β^. Therefore, the required result has been obtained.
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PLZZZZ SOMEONE ANSWER I WOULD APPRECIATE IT A LOT!!!!!
PLZZZZ.
Answer:
Angle B: 26 degrees
Angle C: 26 degrees
Angle BSD: 97 degrees
Angle CSE: 97 degrees
Angle E: 57 degrees
Arc BC: 114 degrees
Step-by-step explanation:
Good luck boi
3. You are helping build a community playground. Your group is building the frame
for a rectangular sandbox. The sides are already cut for you but you do not have
any tools to make sure the corners are 90 degrees. Explain how you can use 2 ropes
to make sure the frame is a rectangle.
don't use rope, us duck tape it always works
this is a haha funny joke by the way
8 ft
Find the area of the figure.
I’ll give the Brainliest to who answers these questions with reasonable explanations.
1. The manager of the soda shop decides that you’re a good customer, you may use a flavour as many times as you’d like. How many different possible mixtures can you make know (remember, you have 13 flavours available to you and you must choose 6)?
2. You realize that, despite what you expected, the soda actually tastes different depending on which flavour you choose first, second, etc. With this in mind, how many different sodas can you make?
Answer:
Answer
5.0/5
1
sana112704
Answer:
i think the answer is 10,296
Step-by-step explanation:
The first one is still 13C6
Review the definition of combina
Step-by-step explanation:
Answer:
what is the answer to part one ?
and the answer too part 2 ?
the answers down below wereńt clear enough .
Step-by-step explanation
You may need to use the appropriate appendix table to answer this question. A person must score in the upper 2% of the population on an admissions test to qualify for membership in society catering to highly intelligent individuals. If test scores are normally distributed with a mean of 140 and a standard deviation of 5, what is the minimum score a person must have to qualify for the society? (Round your answer to the nearest integer).
To calculate the minimum, score a person must have to qualify for the society catering to highly intelligent individuals, given that test scores are normally distributed with a mean of 140 and a standard deviation of 5, you may need to use the appropriate appendix table to answer this question.
The Z-score formula is useful for calculating the minimum score a person must have to qualify for the society catering to highly intelligent individuals. The formula for Z-score is given below: Z = (X - μ) / σwhere, X is the minimum scoreμ is the mean σ is the standard deviation Z is the z-score
From the given data, we can find the Z-score as follows: Z = (X - μ) / σZ = (X - 140) / 5
Given that the person must score in the upper 2% of the population, the area under the normal curve is 0.02.
Since the curve is symmetric, the area under the curve to the right of the mean will be 0.5 - 0.02/2 = 0.49. Since we need to find the minimum score, we can use the inverse normal distribution table or appendix table. From the inverse normal distribution table, the Z-score for 0.49 is 2.06. Now we can substitute the Z-score in the above formula and find the minimum score: X = Zσ + μX = (2.06) (5) + 140X = 151.3The minimum score a person must have to qualify for the society catering to highly intelligent individuals are 151 (rounded to the nearest integer).
Therefore, the correct answer is 151.
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Solve for x. Please I am a little lost on this one .
Answer:
10
Step-by-step explanation:
Hello,
I believe this is the explanation of this problem. If not, anyone else is free to correct me on this.
Look at the line segment passes through the center and is divided into 8 and 4. Since this line segment passes through the center of the circle, this is the diameter and is equal to 12 simply by adding 8 and 4..
Let's look at the next segment. It passes through the center as well, so it must have the same diameter. After all, the circle has the same diameter no matter what as long as the segment passes through the center.
You have the length of one part of that segment, which is 2. Subtract it from 12, the diameter you got from the other segment, and the answer is 10.
Hope this helps!
Please ASAP let me know the answer!
PLEASE HELP!!
Daniel deposits $1250 into each of two savings accounts shown in the image. Daniel does not make any additional deposits or withdrawals.
- Account A earns 5% annual simple interest.
- Account Bearns 5% interest compounded annually
What is the sum of the balances of Account A and Account B at the end of 4 years?
A. $1,500.00
B. $1,519.38
C. $3,019.38
D. $2,857.59
Answer:
234
Step-by-step explanation:
x=2+3
ANSWER PLEASEEEEEE !!!!!!!!!!!!!!
Kelly has a hula hoop with a radius of 2.4 feet. what is the diameter.
Answer:
4.8
Step-by-step explanation:
Radius is 1/2 of the diameter so just do 2.4 and multiply it by 2
hope this helps :)
A spring attached to a wall was displaced horizontally to model simple periodic motionWhich function represents the oscillations of the spring it it had amplitude of 5, a frequency of 3/(2pi) and midine of 4? Do not show me a link I want the answer the question is in the picture
Answer:b
Step-by-step explanation:
The function f(t) = 5sin(3t) + 4 represents the oscillations of the spring it had an amplitude of 5, a frequency of 3/(2pi), and a midline of 4 option second is correct.
What is the frequency?It is defined as the number of waves that crosses a fixed point in one second known as frequency. The unit of frequency is per second.
We have amplitude A = 5
Frequency f = 3/(2π)
Midline m = 4
The equation for the simple periodic motion is given by:
f(t) = Asin(wt+∅) + m
Here phase angle ∅ = 0
We know:
w = 2πf
w = 2π(3/2π)
w = 3
Put all the values in the equation:
f(t) = 5sin(3t) + 4
Thus, the function f(t) = 5sin(3t) + 4 represents the oscillations of the spring it had an amplitude of 5, a frequency of 3/(2pi), and a midline of 4
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Can someone help me with this. Will Mark brainliest.
Answer:
[tex]\sqrt{53}[/tex]
Step-by-step explanation:
I need help with this question please help.
Its gt geometry
Answer: First one is 140, second one is 70
Step-by-step explanation: They’re corresponding angles
A technique of: 40 mAs with 60 kV an exposure of 100mR. If we change to 20 mAs value what should the new kV value be to maintain exposure?
To maintain the same exposure of 100mR with a new technique of 20 mAs, the new kV value should be increased to approximately 120 kV.
The exposure received by a patient during an X-ray examination is determined by the product of milliamperes-seconds (mAs) and kilovolts (kV).
In this case, the initial technique of 40 mAs with 60 kV resulted in an exposure of 100mR.
To calculate the new kV value, we can use the mAs reciprocity law, which states that if the mAs is halved, the kV should be doubled to maintain the same exposure. In other words, the product of mAs and kV should remain constant.
In the initial technique, the product of mAs (40) and kV (60) is 2400. When the mAs value is reduced to 20, we need to find the new kV value that, when multiplied by 20, gives the same product of 2400.
By rearranging the equation, we find that the new kV value should be approximately 120, obtained by dividing the constant (2400) by the new mAs value (20).
To maintain the same exposure of 100mR with a reduced mAs value of 20, the new kV value should be increased to approximately 120 kV. This adjustment ensures that the product of mAs and kV remains constant, as dictated by the mAs reciprocity law.
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Pls help with this math
1/4(16p+8) ÷ 2(p+2)
someone help
Answer: 1/2(16p+8)(p+2)
Explanation: There is a lot to do, but basically you must simplify the equation and then multiply by 4 on both sides
I will give 50 points for the answer
Answer:
nosee
Step-by-step explanation:
Zack placed $900 into a savings account. What is the total value of the account in 6 months at 3.5% rate?
Step-by-step explanation:
Interest=P×R×T/100
= 900×3.5×6/100
= 9×3.5×6. cancelling 900 by 100
= $189 Answer
What statement is used to prove that quadrilateral ABCD is a parallelogram? (5 points)
a) Opposite angles are congruent.
b) Opposite sides are congruent.
c) Diagonals bisect each other.
d) Consecutive angles are supplementary.
The statement that is used to prove that quadrilateral ABCD is a parallelogram is opposite sides are congruent. In a parallelogram, opposite sides are parallel, and their opposite angles are congruent. A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral with two pairs of opposite and parallel sides, which means that the opposite sides in the parallelogram are congruent.
So, a statement that is used to prove that quadrilateral ABCD is a parallelogram is that opposite sides are congruent. It is one of the necessary and sufficient conditions to prove a quadrilateral as a parallelogram. Therefore, option (b) is the correct answer.Apart from this, the other statements that could have been options are:Option (a) - Opposite angles are congruent. It is a property of a parallelogram but is not sufficient to prove that a quadrilateral is a parallelogram. If only opposite angles are congruent, then it is not necessary that opposite sides are parallel.Option (c) - Diagonals bisect each other. This property is only applicable to a parallelogram, and not to any other quadrilateral. However, it is not sufficient to prove a quadrilateral as a parallelogram because this property is only one of the properties of a parallelogram.Option (d) - Consecutive angles are supplementary. This property is common to all quadrilaterals, not just parallelograms. Therefore, it is not sufficient to prove that a quadrilateral is a parallelogram.
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The statement that is used to prove that quadrilateral ABCD is a parallelogram is "Opposite sides are congruent".
A parallelogram is a quadrilateral with two pairs of parallel sides.
There are some properties of parallelograms which include:
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
Now let's look at the answer options available:
a) Opposite angles are congruent: This statement is used to prove that a quadrilateral is not necessarily a parallelogram, but it is a kite.
b) Opposite sides are congruent: This statement is used to prove that quadrilateral ABCD is a parallelogram.
c) Diagonals bisect each other: This statement is used to prove that quadrilateral is a parallelogram or rectangle. But it cannot prove that ABCD is a parallelogram.
d) Consecutive angles are supplementary: This statement is used to prove that quadrilateral is a parallelogram or rectangle. But it cannot prove that ABCD is a parallelogram.
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.2 How many millilitres of chutney are needed?
Answer:
Step-by-step explanation:
In this scenario, You've asked about the Chutney (Sauce/Raita).
Ok, So let's suppose you are making a "Chutney"
The first you need is the right amount of ingredients.Their quantity and they must be fresh "To make every effort of yours worth it!"Add it into a bowl or a dish that you have at the time (Or a big one to be on the safe side)At the end mix the ingredients well, So that they don't spoil taste of the one who is taking it.The amount of milliliters (Weight measurement) depends upon the amount of usage/intake that one is taking.
It could be a few hundred (gms. or some Kg's).
That totally depends up on your choice
Mr clay has 75 geese and 125 turkeys. what is the ratio of the number of geese to the total number of birds in the simplest form
Answer:
3/8
Step-by-step explanation:
125+75= 200
75:200= 3/8
3/8 is the ratio of the number of geese to the total number of birds in the simplest form.
Suppose we test H0: p=0.3 versus Ha: p≠0.3 and that a random sample of n=100 gives a sample proportion p ˆ = 0.2. Use the p-value to test H0 versus Ha by setting the level of significance α to 0.10, 0.05, 0.01 and 0.001. What do you conclude at each value of α
At α = 0.10, 0.05,0.01, and 0.001, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population proportion is not equal to 0.3.
What are the verdicts ?Given
H₀: p = 0.3 (null hypothesis)
Hₐ: p ≠ 0.3 (alternative hypothesis)
Sample size (n) = 100
Sample proportion (p)= 0.2
To calculate the p-value, we can follow these steps -
Calculate the test statistic z -
z = (pa - p₀) /√(p₀ * (1 - p₀) / n)
where pa is the sample proportion, p₀ is the hypothesized population proportion,and n is the sample size.
Calculate the p-value -
For a two-tailed test, the p-value is calculated as:
p-value =2 * P(Z ≤ -|z |), where Z is the standard normal distribution.
Now let's calculate the test statistic and p-value for each level of significance α
For α = 0.10:
p₀ = 0.3
The test statistic is
z =(0.2 - 0.3) / √(0.3 * (1 - 0.3) / 100)
z ≈ -4.714
The p-value for a two-tailed test is calculated like this
p-value = 2 * P(Z ≤ -|z|)
≈ 2 * P(Z ≤ -4.714)
Using a standard normal distribution table or calculator,the p-value is approximately < 0.001
At α = 0.10 - We reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population proportion is not equal to 0.3.
At α = 0.05 - We reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population proportion is not equal to 0.3.
At α = 0.01 - We reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population proportion is not equal to 0.3.
At α = 0.001 - We reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population proportion is not equal to 0.3.
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z=xe^y,x=u^2−v^2,y=u^2 v^2, find ∂z/∂u and ∂z/∂v. the variables are restricted to domains on which the functions are defined.
[tex]z=xe^y,\ x=u^2-v^2,\ y=u^2 v^2[/tex]. Using partial differentiation, ∂z/∂u = [tex]2ue^y + 2uv^2xe^y[/tex] and ∂z/∂v = [tex]-2ve^y + 2u^2\ vxe^y[/tex].
How to apply partial differentiation?
To find the partial derivatives ∂z/∂u and ∂z/∂v, we will differentiate the function [tex]z = xe^y[/tex] with respect to u and v while keeping other variables constant.
Given:
[tex]z = xe^y,\\x = u^2 - v^2,\\y = u^2v^2.[/tex]
First, let's express z in terms of u and v by substituting the expressions for x and y into z:
[tex]z = (u^2 - v^2)e^{u^2v^2}[/tex]
Now, we can find the partial derivatives.
Partial derivative ∂z/∂u:
To find ∂z/∂u, we differentiate z with respect to u while treating v as a constant. Applying the chain rule, we have:
∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u).
∂z/∂x = [tex]e^y[/tex],
∂x/∂u = 2u,
∂z/∂y =[tex]x * e^y[/tex],
∂y/∂u = [tex]2uv^2[/tex].
Substituting these values, we get:
∂z/∂u = [tex](e^y)(2u) + (x * e^y)(2uv^2)[/tex].
∂z/∂u = [tex]2ue^y + 2uv^2xe^y[/tex].
Partial derivative ∂z/∂v:
To find ∂z/∂v, we differentiate z with respect to v while treating u as a constant. Applying the chain rule, we have:
∂z/∂v = (∂z/∂x)(∂x/∂v) + (∂z/∂y)(∂y/∂v).
∂z/∂x = [tex]e^y[/tex],
∂x/∂v = -2v,
∂z/∂y = [tex]x * e^y[/tex],
∂y/∂v = [tex]2u^2v[/tex].
Substituting these values, we get:
∂z/∂v = [tex](e^y)(-2v) + (x * e^y)(2u^2v)[/tex].
∂z/∂v = [tex]-2ve^y + 2u^2vxe^y[/tex].
Therefore, the partial derivatives are:
∂z/∂u = [tex]2ue^y + 2uv^2xe^y[/tex],
∂z/∂v =[tex]-2ve^y + 2u^2\ vxe^y.[/tex]
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A semi circle of diameter 6cm is cut from rectangle with sides 6cm and 8cm.
Calculate the perimeter of the shaded shape correct to 1 decimal place.
(My words) Add working steps if possible :)
(-4,9);m=-1/2
Write the equation in point slope form
Peggy had a cutting board measuring 2 m 4 cm. She cut off 78 cm. How long is the board now, in meters?
Olivia is sending a recipe to her mother in Mexico. Among other things, the recipe calls for 4 ounces of rice and a baking temperature of 350 degrees fahrenheit. Convert these measurements to metric, rounding to the nearest gram and nearest degree.
Imported wool fabric is $12.99 per meter. What is the cost, to the nearest cent of a piece that measures 3 m to 70 cm?
While on vacation in Canada, Jalo became ill and went to a health clinic. They said he weighed 80.9 kilograms and was 1.83 meters tall. Find his weight in pounds and height in feet. Round to nearest tenth.
Each serving of punch at the wedding reception will be 180 mL. How many liters of punch are needed for 75 people?
The answers are =
1) The board is now 1.26 meters long.
2) Olivia's recipe calls for approximately 113 grams of rice and a baking temperature of approximately 176.67 degrees Celsius.
3) The cost of a piece measuring 3.70 meters is 3.70 × $12.99 = $48.03. Rounded to the nearest cent, the cost is $48.03.
4) Jalo's weight in pounds is approximately 80.9 × 2.20462 ≈ 178.6 pounds.
5) Approximately 13.5 liters of punch are needed for 75 people.
Peggy's cutting board initially measured 2 meters 4 centimeters, which is equivalent to 204 centimeters.
After cutting off 78 centimeters, the board's new length is 204 - 78 = 126 centimeters.
To convert this to meters, divide by 100: 126 cm ÷ 100 = 1.26 meters. Therefore, the board is now 1.26 meters long.
2) To convert 4 ounces of rice to grams, we need to know the conversion factor.
1 ounce is approximately equal to 28.35 grams.
Therefore, 4 ounces of rice is approximately equal to 4 × 28.35 = 113.4 grams.
To convert 350 degrees Fahrenheit to Celsius, we use the formula: Celsius = (Fahrenheit - 32) × 5/9.
Celsius = (350 - 32) × 5/9 = 318 × 5/9 ≈ 176.67 degrees Celsius.
Therefore, Olivia's recipe calls for approximately 113 grams of rice and a baking temperature of approximately 176.67 degrees Celsius.
3) The length of the fabric is 3 meters 70 centimeters, which is equivalent to 3.70 meters.
The cost of the fabric per meter is $12.99.
Therefore, the cost of a piece measuring 3.70 meters is 3.70 × $12.99 = $48.03. Rounded to the nearest cent, the cost is $48.03.
4) Jalo's weight is given as 80.9 kilograms.
To convert this to pounds, we need to know the conversion factor. 1 kilogram is approximately equal to 2.20462 pounds.
Therefore, Jalo's weight in pounds is approximately 80.9 × 2.20462 ≈ 178.6 pounds.
Jalo's height is given as 1.83 meters.
To convert this to feet, we need to know the conversion factor. 1 meter is approximately equal to 3.28084 feet.
Therefore, Jalo's height in feet is approximately 1.83 × 3.28084 ≈ 6.003 feet, which can be rounded to 6 feet.
Therefore, Jalo weighs approximately 178.6 pounds and is approximately 6 feet tall.
5) Each serving of punch is 180 mL.
To find the total amount of punch needed for 75 people, we multiply the serving size by the number of people: 180 mL × 75 = 13,500 mL.
Since 1 liter is equal to 1,000 mL, we can convert the amount of punch to liters by dividing by 1,000: 13,500 mL ÷ 1,000 = 13.5 liters.
Therefore, approximately 13.5 liters of punch are needed for 75 people.
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Jake wants to buy two pairs of a particular style of sunglasses. The sunglasses are sold at two different stores. Sunglass House sells the sunglasses go 20% off the $40 retail price. The Eyewear Shoppe sells the sunglasses for $60 but is having a “Buy 1 Pair, Get 1 Pair Free” promotion. Which store has the better buy?
Answer:
Based on the calculations, the store with the better buy is The Eyewear Shoppe
Step-by-step explanation:
Jake wants to buy two pairs of a particular style of sunglasses. The sunglasses are sold at two different stores.
For The Eyewear Shoppe:
The Eyewear Shoppe sells the sunglasses for $60 but is having a “Buy 1 Pair, Get 1 Pair Free” promotion.
Based on this promotion, you can get 2 sunglasses for $60
For Sunglass House
Sunglass House sells the sunglasses go 20% off the $40 retail price.
A Sunglass = $40 - ( 20%× $40)
= $40 - $8
= $32
Two sunglasses will cost = $32 × 2
= $64
Therefore, based on the above calculations, the store with the better buy is The Eyewear Shoppe
If Jake buys 2 sunglasses from Sunglass House the total price would be $64. The Eyewear Shoppe would get you two sunglasses for $60. This means that the The Eyewear Shoppe is a better buy.
need help w this! will give brainliest and thanks to best answer :o) tysm<3
Answer:
A. They will pay more with the new price plan.
B. The new price plan would be cheaper.
Step-by-step explanation:
A. They will pay more with the new price plan.
For the current price plan, you would add the $3 rent to the two games (which are $4 each). This basically means:
$3 + $4 + $4 = $11
For the new price plan, you would add the $11 rent to the two games (which are $2 each). This basically means:
$11 + $2 + $2 = $15
Therefore, you pay more for the new price plan.
B. Using similar logic as part A, the current price plan 7 games would cost:
$3 + [7 x ($4)] = $31 (multiply by 7 since they play 7 games)
For the newprice plan, 7 games would cost:
$11 + [7 x ($2)] = $25 (multiply by 7 since they play 7 games)
Therefore, the new price plan would be cheaper.
Hope this helps :)
Consider the following data.
−11, −11, −9, −11, 0, 0, 0
Step 1 of 3 :
Determine the mean of the given data. and median and no mode unimodal, bimodal or multimodal
The mean of the data set is -6.
The median of the data set is - 9.
The mode of the data set, is - 11.
What is the mean, median and mode of the data?The mean of the data set is calculated by applying the following formula as follows;
mean = total number of data sample /number of data sample
mean = ( -11 - 11 - 9 - 11 + 0 + 0 + 0 ) / 7
mean = ( -42 ) /7
mean = -6
The median of the data set is calculated as follows;
-11, - 11, - 11, - 9, 0, 0, 0
The median number = - 9
The mode of the data set, is the most occurring number or the number with the highest occurrence.
mode = - 11
The data sample is unimodal because it has only one mode.
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the bearing of c from a is 210°
(i) find the bearing of B from A
(ii) find the bearing of A from B
if you could also solve the question above I'd be very grateful
I need it in at least 10 minutes so please answer it!! ♡♡
Answer:
(a) His speed when he runs from C to A is [tex]4.\overline {285714}[/tex] m/s
(i) The bearing of B from A is approximately 127.18°
(ii) The bearing of A from B is approximately 307.18°
Step-by-step explanation:
(a) The given parameters are;
The distance from A to B = 120 m
The speed with which Olay runs from A to B, v₁ = 4 m/s
The distance from B to C = 180 m
The speed with which Olay runs from B to C, v₂ = 3 m/s
The distance from C to A = 150 m
His average speed for the whole journey = 3.6 m/s
We find
The total distance of running from A back to A, d = 120 m + 180 m + 150 m = 450 m
The time it takes to run from A to B, t₁ = 120 m/(4m/s) = 30 seconds
The time it takes to run from B to C, t₂ = 180 m/(3m/s) = 60 seconds
Let t₃ represent the time it takes Olay to run from C to A
We have;
The total time it takes to run from A back to A = t₁ + t₂ + t₃
Therefore;
[tex]Average \ velocity = \dfrac{Total \ distance }{Total \ time} = \dfrac{d}{t_1 + t_2 + t_3}[/tex]
Substituting the known values for the average velocity, 'd', 't₁' and 't₂' gives;
[tex]Average \ velocity = 3.6 \, m/s = \dfrac{450 \, m}{30 \, s + 60 \, s + t_3}[/tex]
3.6 m/s × (30 s + 60 s + t₃) = 450 m
3.6 m/s × 30 s + 3.6 m/s × 60 s + 3.6 m/s × t₃ = 450 m
108 m + 216 m + 3.6 m/s × t₃ = 450 m
∴ 3.6 m/s × t₃ = 450 m - (108 m + 216 m) = 126 m
t₃ = 126 m/(3.6 m/s) = 35 s
The speed with which Olay runs from C to A, v₃ = Distance from C to A/t₃
The speed with which he runs from C to A = 150 m/(35 s) = 30/7 m/s =[tex]4.\overline {285714}[/tex] m/s
(i) The given bearing of C from A = 210°
By cosine rule, we have;
a² = b² + c² - 2·b·c·cos(A)
∴ cos(A) = (b² + c² - a²)/(2·b·c)
Where;
a = The distance from B to C = 180 m
b = The distance from C to A = 150 m
c = The distance from A to B = 120 m
We find;
cos(A) = (150² + 120² - 180²)/(2 × 150 × 120) = 0.125
A = arccos(0.125) ≈ 82.82°
The bearing of B from A ≈ 210° - 82.82° ≈ 127.18°
The bearing of B from A ≈ 127.18°
(ii) The angle, θ, formed by the path of the bearing of A from B is an alternate to the supplementary angle of the bearing of B from A
Therefore, we have;
θ ≈ 180°- 127.18° ≈ 52.82°
The bearing of A from B = The sum of angle at a point less θ
∴ The bearing of A from B = 360° - 52.82° ≈ 307.18°
The bearing of A from B ≈ 307.18°.