To show that if the column vectors of matrix A and B are linearly independent, then the column vectors of matrix C = A*B will also be linearly independent, we can use the following proof:
Assume that the column vectors of matrix A and B are linearly independent. Let's assume that the column vectors of matrix C = A*B are linearly dependent.
This means that there exists a non-trivial linear combination of the column vectors of C that results in the zero vector.
C = AB => c_1 = Ab_1, c_2 = Ab_2, ... c_n = Ab_n
Let's assume that a linear combination of c1, c2, ..., cn results in the zero vector.
a_1c_1 + a_2c_2 + ... + a_n*c_n = 0
Now, we can substitute the matrix multiplication for ci:
a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n = 0
Since matrix A is non-singular, we can left multiply by A^-1
A^-1*(a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n) = A^-1*0
This simplifies to
a_1b_1 + a_2b_2 + ... + a_n*b_n = 0
This is a non-trivial linear combination of the column vectors of matrix B that results in the zero vector, which contradicts the assumption that the column vectors of matrix B are linearly independent. Therefore, the column vectors of matrix C = A*B must also be linearly independent.
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Which statement best describes a graph of paired points that form a proportional relationship? Responses A straight line can be drawn through all the points, and the line passes through the point (3, 0). A straight line can be drawn through all the points, and the line passes through the point , begin ordered pair 3 comma 0 end ordered pair, . A straight line can be drawn through all the points, and the line passes through the point (0, 3). A straight line can be drawn through all the points, and the line passes through the point , begin ordered pair 0 comma 3 end ordered pair, . A straight line can be drawn through all the points, and the line passes through the point (0, 0). A straight line can be drawn through all the points, and the line passes through the point , begin ordered pair 0 comma 0 end ordered pair, . A straight line cannot be drawn through all the points, but the line passes through the point (0, 0).
The proportional relationship is given by the equations -
y = mx or y = kx or y = mx + b or y = kx + b.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given are the statements describing proportional relationship.
Direct variation of {y} with {x} means that as {x} increases, the value of {y} increases uniformly with it. Mathematically →{K} = y/x = constant.
Scale factor {K} is a dimensionless value that indicates the constant ratio value indicating direct variation.The proportional relationship is given by the equation -y = mx or y = kx or y = mx + b or y = kx + b
Therefore, the proportional relationship is given by the equations -
y = mx or y = kx or y = mx + b or y = kx + b.
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Consider the equation 2x+7=5x−23
. Arrange the variable terms on the right side.
Enter your response in the box.
7=
−23
Answer:4x
Step-by-step explanation:
Points D, B, and E are collinear. Find the value of x so that points A, B, and C are collinear
The value of x in the figure is calculated using the linear pair theorem to be 6 degrees
What is linear pair theorem?The linear pair postulate or linear pair theorem in mathematics states that the sum of the measurements of two angles that make up a linear pair is 180°.
Because o the linear pair postulate we have that
150 + 5x = 180
collecting like terms
5x = 180 - 150
5x = 30
Isolating x
5x / 5 = 30 / 5
x = 6
hence x is solved to be 6
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a student drove to the university from her home and noted that the odometer reading of her car increased by 8.00 miles. the trip took 20 min. (note: 1 mile
Vector of the average velocity = 3.33 km\hr towards the south.
Distance covered by the student=7miles 7*6 = 42km
Total displacement = 10km in the south.
Total time is taken = 20mins 20/60=0.33hr
Note:
Odometer reading is the distance covered while straight-line distance is the displacement which is the shortest route
Distance is a scaler where as displacement is a vector quantity.
Scaler has only magnitude while vector has magnitude and direction.
We have to find the average velocity.
Average velocity = Ratio of total displacement and total time.
⇒ Average velocity (v) = displacement/time.
⇒ Plugging the values.
⇒ (v) = 10/0.33
⇒ (v) =3.33 km\hr
So the vector of the average velocity = 3.33 km\hr towards the south.
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7.) +9 – +16 = N
8.) +85 – +12 = N
9.) –6 –2 = N
10.) +13 – +12 = N SOMEONE PLEASE ANSWER
Answer:
here are the answers to the questions in order
Step-by-step explanation:
7. -7
8. 73
9. -8
10. 1
-6z - 18z + 2z + 6z = 16
nvm don't answer
The value of z in the given linear equation in one variable is -1.
What is a single-variable linear equation?By equating it to zero, a linear polynomial over a field, from which the coefficients are extracted, can be converted into a linear equation.
The values that, when employed as stand-ins for the unknowns in such an equation, lead to the equality being true are the solutions.
When there is just one variable, there is just one solution. The term "linear equation" is generally understood to relate to this particular circumstance, when the variable is accurately referred to as the "unknown."
It is possible to interpret each solution as the Cartesian coordinates of a specific point on the Euclidean plane when there are only two variables.
-6z -18z + 2z +6z = 16
-16z = 16
z = -1
Hence, the value of z is -1
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Let m = 3.
Enter >, <, or = to compare the expressions.
4m−145m−4
Answer:
=
Step-by-step explanation: Please mark brainliest
Substitute 3 in for m and solve
4m - 1 = 4(3) - 1 = 11
5m - 4 = 5(3) - 4 = 11
Answer:
Step-by-step explanation:
P&R3 model and solve
problems using linear
equations of the form:
ax = b *
x
a
= b, a ≠ 0*
ax + b = c *
x
a
+ b = c ,a ≠ 0*
a x( ) + b = c *
ax + b = cx + d *
* a bx ( ) + c = d ex ( ) + f
*where a, b, c, d, e, and f [C,
CN, PS, V]
1. Introduction to Solving Linear Equations (pg. 4-7)
• Model the solution of a given linear equation using concrete or
pictorial representations, and record the process.
Algebra stones and Algebra Tiles.
Solve x + 5 = 10 , x − 7 = 10 , 2x = 10 , x
3 = 10
2. Solving equations of the form ax+b=c and a/c +b=c (pg. 8-10)
• Solve a given linear equation symbolically.
• Solve a given problem using a linear equation and record the
process.
Solve. 4m+3=31 & 2
5
m − 5 = 3
3. Solving equations of the form a(x+b)=c and ax+b=cx+d(pg. 11-14)
• Determine, by substitution, whether a given rational number is a
solution to a given linear equation.
Solve. 4(m+3)=40 & 6m+3=2m+15
Is m=5 a solution to the equation 2(m + 2) = 14 ?
4. More practice with ax+b=cx+d (Pg. 15-18)
• Identify and correct an error in a given incorrect solution of a
linear equation
Solve. 2(m+1)+4m=4(m-2)+6.
5. Solve equation with fractions. (Pg. 19-23)
• Solve a given linear equation symbolically. Solve m
3 + 2m
5 − 1
2
= 2
6. Solve Linear equations without numbers. (Pg. 24-28)
• Solve a given linear equation symbolically.
Solve for m. )( =+ BnmA
P&R4 explain and illustrate
strategies to solve single
variable linear inequalities with
rational coefficients
within a problem-solving
context.
7. Introduction to linear inequalities (Pg. 29-33)
• Translate a given problem into a single variable linear inequality
using the symbols ≥, >, < , or ≤ .
• Determine if a given rational number is a possible solution of a
given linear inequality.
Write an expression to represent the following
statement: Melanie needs at least $280 for snow
boarding.
8. Inequalities that Include Addition and Subtraction (Pg. 34-37)
• Generalize and apply a rule for subtracting a positive or
negative number to determine the solution of a given inequality.
• Generalize and apply a rule for multiplying or dividing by a
positive or negative number to determine the solution of a given
inequality.
• Solve a given linear inequality algebraically and explain the
process orally or in written form.
• Verify the solution of a given linear inequality using
substitution for multiple elements in the solution.
Solve 2x + 5 < 25 and verify your solution.
True or False.
If −2x > −10 then x > 5 .
9. Solving Problems with Linear Inequalities (Pg. 38-41)
• Compare and explain the process for solving a given linear
equation to the process for solving a given linear inequality.
• Graph the solution of a given linear inequality on a number line.
• Compare and explain the solution of a given linear equation to
the solution of a given linear inequality.
• Solve a given problem involving a single variable linear
inequality and graph the solution.
Vertical Wireless charges $50/ month plus $0.25 for
all minutes above 400 minutes per month. Frue Gal has
decided that she does not want to pay more than $70
per month. Write an inequality to represent how many
minutes she can use per month without going over her
$70 limit. Approximate your solution on the number
line
10. Chapter Review and Practice Test
• Help students develop sound study habits.
• Many students will graduate high school saying they do not
know how to study for math tests.
11. Go over Practice Test
12. Unit Evaluation
Determine the slope given graph
Answer:
slope = 4
Step-by-step explanation:
Add and Subtract Fractions with Unlike Denominators
Daniel decorated the backyard with 4 2/5 yards of pumpkin string lights and
6 3/4 yards of witch-hat string lights to spice up a Halloween party. How many yards
the two string lights did Daniel use?
Answer:11.15
Step-by-step explanation:
sry if its wrong :)
just add them
What number gives a result of −4
when 11
is subtracted from the quotient of the number and 5
?
The required number is 42 which gives a result of −4 when 11 is subtracted from the quotient of the number and 5.
What is the Quotient?A quotient is defined as the outcome of dividing an integer by any divisor that can be said to be a quotient. The dividend contains the divisor a specific number of times.
Let n be the number in question; then, we can write the word problem as an equation like this:
⇒ (n/6) - 11 = -4
Add 11 on both sides of the equation,
⇒ (n/6) - 11 + 11 = 11 - 4
⇒ (n/6) = 7
Multiply by 6 on both sides of the above equation, and we get
⇒ n = 42
Hence, the number is 42.
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1. 49, 34, and 48 students are selected from the Sophomore, Junior, and Senior classes with 496, 348, and 481 students respectively. Identify which type of sampling is used and why
2. The name of each contestant is written on a separate card, the cards are placed in a bag, and three names are picked from the bag. Identify which type of sampling is used and why
3. An education expert is researching teaching methods and wishes to interview teachers from a particular school district. She randomly selects ten schools from the district and interviews all of the teachers at the selected schools. Does this sampling plan result in a random sample? Simple random sample? Explain.
4. A polling company obtains an alphabetical list of names of voters in a precinct. They select every 20th person from the list until a sample of 100 is obtained. They then call these 100 people. Does this sampling plan result in a random sample? Simple random sample? Explain.
5. The personnel manager at a company wants to investigate job satisfaction among the female employees. One evening after a meeting she talks to all 30 female employees who attended the meeting. Does this sampling plan result in a random sample? Simple random sample? Explain.
Case Assignment Expectations:
Use information from the modular background readings as well as any good quality resource you can find. Please cite all sources and provide a reference list at the end of your paper.
The following items will be assessed in particular:
Your ability to identify situations when sampling is appropriate.
Your ability to draw the proper inferences about the population after the sample has been evaluated.
Your ability to describe the process of selecting and evaluating a samp
the ability to identify situations when sampling is appropriate and to draw the proper inferences about the population after the sample has been evaluated. The answer describes the process of selecting and evaluating a sample, including examples of stratified, random, and systematic sampling. It also discusses the limitations of not using a random sample in order to obtain unbiased results.
1. This is an example of stratified sampling, since the students are divided into three groups (Sophomores, Juniors, and Seniors) and a sample is taken from each group. This allows for a more representative sample of the overall population.
2. This is an example of random sampling, since each name has an equal chance of being selected from the bag. This ensures that the sample is unbiased and allows for a fair representation of the population.
3. This sampling plan results in a random sample, since the schools were randomly selected from the district. A random sample ensures that every member of the population has an equal chance of being selected.
4. This sampling plan results in a systematic sample, since the polling company selected every 20th person from the list. Systematic sampling allows for a more representative sample of the population, since it ensures that each person in the population is equally likely to be selected.
5. This sampling plan does not result in a random sample, since only the female employees who attended the meeting were selected. This means that those who did not attend the meeting had no chance of being selected in the sample, thus biasing the results.
This answer demonstrates the ability to identify situations when sampling is appropriate and to draw the proper inferences about the population after the sample has been evaluated. The answer describes the process of selecting and evaluating a sample, including examples of stratified, random, and systematic sampling. It also discusses the limitations of not using a random sample in order to obtain unbiased results.
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Use the grouping method to factor x^3 + x^2 + 2x + 2.
A. (x + 1)(x + 2)
B. x(x + 2)(x + 1)
C. (x + 1)(x2 + 2)
D. (x^2 + 1)(x + 2)
find the area under the normal distribution curve whose mean is and whose standard deviation is using the given value for as a boundary.
The area under the normal curve with a mean of 20 and a standard deviation of 5 is 0.8413 when x = 25.
The area under the normal distribution curve is calculated using the z-score, which is the number of standard deviations a value is from the mean. To calculate the area under the normal curve for a given mean and standard deviation, one must use the formula:
Area = ½[1 + erf(z)]
Where erf is the Gaussian error function, and z is the z-score.
To calculate the area under the curve given a boundary x, the z-score must be calculated first. This is done using the formula:
z = (x - mean) / standard deviation
Substituting this into the formula above gives:
Area = ½[1 + erf((x - mean) / standard deviation)]
For example, if the mean is 20 and the standard deviation is 5, and the boundary value is x = 25, the area under the normal curve is calculated as follows:
z = (25 - 20) / 5 = 1
Area = ½[1 + erf(1)] = 0.8413
Therefore, the area under the normal curve with a mean of 20 and a standard deviation of 5 is 0.8413 when x = 25.
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use the given data to find the equation of the regression line. examine the scatterplot and identify a characteristic of the data that is ignored by the regression line. x y
The equation of the regression line is Y=2.83+0.52x .
What do you mean by regression line?Linear regression is the statistical technique that is most frequently used to simulate a relationship between two sets of variables. As a result, a data-applicable linear regression equation is produced.
Plotted your points and drew a linear regression line.
What is the formula to calculate the line regression?y = a + bₐₓx
bₐₓ = n∑xy-∑x-∑y
n∑x²- ∑x²
x = 8.81
y = 7.45
∑x² = 947
∑xy=770.95
bₐₓ = 0.52
a = 2.83
y=1.3082X + 3.3436.
Its equation is
y = 2.83+0.52x .
Therefore, the regression line in a scatter plot ignores non-linear properties, so the equation of the reggresion line will be y = 2.83+0.52x .
Your question is incomplete but most probably your full question was
Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.
x y
9 7.38
9 6.75
12 13.08
9 7.42
11 7.56
13 8.53
7 5.93
4 5.33
12 8.1
6 6.46
5 5.44
Required:
Create a scatterplot of the data.
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you win the lottery and get 1000000. the bank you invest your winnings in gives you a 6% interest rate compounded quaterly. how much will you have in 3 years ?
The amount that you will have in 3 years, given the interest rate and the amount won, is 1, 195, 618. 17 currency units
How to find the amount earned ?To find the amount earned in three years as a result of your lottery being invested, use the future value formula which is:
= Amount won x ( 1 + periodic rate ) ^ number of periods
Periodic rate :
= 6 % / 4 quarters per year
= 1. 5 %
The number of periods is:
= 3 years x 4 quarters per year
= 12 quarters
The amount you will have in 3 years is:
= 1,000,000 x ( 1 + 1. 5 %) ¹²
= 1, 195, 618. 17 currency units
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Solve the problem by using addition, subtraction, multiplication, or division. First use front end rounding to estimate the answer. Then find the exact answer.
The foggiest place in a certain country is City A. It is foggy there an average of days each year. How many days is it not foggy each year? Find the estimated and the exact number of days.
The notion of an equation in algebra is a statistical statement that demonstrates the equality of two mathematical expressions.
For the stated question;
Use addition, subtraction, multiplication, or division to find the solution. To first estimate the response, use front end rounding. Find the precise response next. City A is located in the nation with the most haze. how many days a year are clear of fog.365 days are included in a year.
There are 256 days a year on average when it is foggy.
Determine the approximate number of clear days by using the formula:
We round the figures.
To the nearest tens, 365 days
= 370 days
The nearest tenth of 256 days
= 260 days
370 - 260 days, then
= 110 days
The precise number of days without fog is =
= 365 − 256
= 109
Thus, the precise number of days without fog is found as 109 days.
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The complete question is -
Solve the problem by using addition, subtraction, multiplication, or division. First use front end rounding to estimate the answer. Then find the exact answer. The foggiest place in a certain country is City A. It is foggy there an average of 256days each year. How many days is it not foggy each year? Find the estimated and the exact number of days.
Now Jimmy wants to save money to buy concert tickets, so he draws up a new savings plan. Each week, he will raise $3 to the number of weeks he has been working and add that amount to his savings during the first 3 weeks of his new job. Use summation notation to write the sum and find the total amount he will save.
a) Using the summation notation to write the sum is 3∑n=1 3n.
b) The total amount that Jummy will save during the first 3 weeks of his new job is $18.
What is the summation notation?The summation notation is the representation of a series in a compact form, using the capital sigma symbol (Σ) to indicate the sum of a number of similar terms.
Using the summation notation, one can write a long sum in a single expression.
In this situation, the expression can be read as the sum of 3n as n goes from week 1 to week 3, representing the first three weeks Jimmy will spend on the job.
3∑n=1 3n
= 3 x 1 + 3 x 2 + 3 x 3
= 3 + 6 + 9
= 18
Thus, we have used the summation notation to find that Jimmy will save $18 during the first three weeks at his new job.
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Answer:
Step-by-step explanation:
A manufacturer of cell phone screens has found that 5 percent of all screens produced have defects. Let pd represent the population proportion of all cell phone screens with a screen defect, therefore pd=0.05. For the sampling distribution of the sample proportion of cell phone screens from this manufacturer with a screen defect for sample size 400, μpˆd=0.05. Which of the following is the best interpretation of μpˆd=0.05 ?For a randomly selected cell phone screen from this population, the mean number of screen defects for the selected screen will be equal to 0.05.A For a randomly selected cell phone screen from this population, the mean number of screen defects for the selected screen will be equal to 0.05.B For every sample of size 400 from this population, the proportion of cell phone screens with a screen defect will be 0.05.C For every sample of size 400 from this population, the proportion of cell phone screens with a screen defect will be 0.05.D For all samples of size 400 from this population, the mean number of screen defects for the samples is 0.05.F For all samples of size 400 from this population, the mean number of screen defects for the samples is 0.05.G For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05.H For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05.I For all samples of size 400 from this population, the standard deviation of all resulting sample proportions of cell phone screens with a screen defect is 0.05.
Using the Central Limit Theorem, it is found that the correct option is: C
For all samples of size 400 from this population, the mean of all resulting sample proportions of cell phone screens with a screen defect is 0.05.
By the Central Limit Theorem, for a proportion p in a sample of size n, the mean of the sampling distribution of sample proportions is u=5, while the standard error is [tex]SE=\sqrt{\frac{p*(1-p)}{n} }[/tex]
In this problem, the mean of the sampling distribution of sample proportions is 0.05, that is, the mean proportion of all samples, hence, the correct option is C.
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Which group cannot form 8 lines with an equal number of members in each line? A. 881 B. 440 C. 353 D. 224
Answer:
A. 881
C. 353
Step-by-step explanation:
In order to form 8 lines with each line having equal length, the total number of members has to be divisible by 8:
440/8 = 55, so 440 isn't the answer
224/8 = 28, so 224 isn't the answer
881/8 = 110.125. In addition, even numbers can't be divided evenly by odd numbers like 881, so A. 881 is the answer.
Same goes to 353 since 353 is also an odd number that can't evenly divide the even number 8. Hence, C. 353 is the answer as well.
1. Match each equation with an equivalent equation. Some of the answer choices are
not used.
A. 3x + 6 = 4x+7
B. 3(x + 6) = 4x+7
C. 4x + 3x = 7-6
1.9x = 4x +7
2. 3x + 18 = 4x+7
3.3x = 4x + 7
4. 3x1 = 4x
5.7x = 1
Answer: Match each equation with an equivalent equation. Some of the answer choices are
not used.
A. 3x + 6 = 4x+7
B. 3(x + 6) = 4x+7
C. 4x + 3x = 7-6
1.9x = 4x +7
2. 3x + 18 = 4x+7
3.3x = 4x + 7
4. 3x1 = 4x
5.7x = 1
A. 3x + 6 = 4x+7 = 3
B. 3(x + 6) = 4x+7 = 2
C. 4x + 3x = 7-6 = 5
1.9x = 4x +7 = 3
3x + 18 = 4x+7 = 2
3.3x = 4x + 7 = 1
3x1 = 4x = not used
5.7x = 1 = not used
So,
3x + 6 = 4x+7 matches with 3
3(x + 6) = 4x+7 matches with 2
4x + 3x = 7-6 matches with 5
1.9x = 4x +7 matches with 3
3x + 18 = 4x+7 matches with 2
3x = 4x + 7 matches with 1
3x1 = 4x is not used
7x = 1 is not used
Step-by-step explanation:
What is 11/12 = 3/4h?
h=
The value of h in the algebraic equation in fraction 11/12 = 3/4h is equal to 9/11
How to solve for the value of h in the algebraic equationWe shall first find the lowest common multiple LCM of the denominators to help us clear the fractions, before then simplifying further for the value of h as follows:
LCM of the denominators 12 and 4h is 12h
multiply both sides by the LCM
11/12 × 12h = 3/4h × 12h
11 × h = 3 × 3
11h = 9
divide both sides by the coefficient of h
11h/11 = 9/11
h = 9/11
Therefore, the value of h in the algebraic equation in fraction 11/12 = 3/4h is equal to 9/11.
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The integral with respect to time of a force applied to an object is a measure called impulse, and the impulse applied to an object during a time interval determines its change in momentum during the time interval. The safety of a t-shirt launcher, used to help get crowds cheering at baseball games, is being evaluated. As a first step in the evaluation, engineers consider the design momentum of the launched t-shirts. The springs in the launcher are designed to apply a variable force to a t-shirt over a time interval of tL=0.5s. The force as a function of time is given by F(t)=at2+b, where a=−28N/s2 and b=7.0N. The momentum of the t-shirt will be its initial momentum (p0=0) plus its change in momentum due to the applied impulse: pf=p0+∫t=tLt=0F(t)dt. By applying the given time dependent function for F(t) and performing the integration, which of the following is the correct expression for pf?
The impulse equation is given by pf = p0 + ∫tL₀F(t)dt. With the given information, the integral is ∫tL₀(-28t² + 7.0)dt.
This integral can be solved using the power rule for integration, which states that the integral of xⁿ is xⁿ⁺¹/(n+1). Thus, the integral of -28t² + 7.0 is -28/3t³ + 7t. This can be evaluated from t=0 to t=0.5 to get the change in momentum due to the applied impulse. Substituting the values into the impulse equation, pf = 0 + (-28/3(0.5)³ + 7(0.5)) = -2.167. Thus, the correct expression for pf is pf = -2.167.
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Identify the kind of sample that is described.
A football coach randomly selected the varsity or junior varsity team and then asks all players from that team their opinion on a new logo.
A. Stratified
B. Simple Random
C. Voluntary Response
D. Cluster
E. Systematic
The given sample of the randomly selected players from the teams as described is a stratified sample. Hence, the right answer is option A.
In a stratified sample, the population is divided into groups (or strata) based on some characteristic, and a random sample is selected from each group. In this case, the population is the football players, and the groups are the varsity and junior varsity teams. The coach randomly selects one of these teams and then asks all players from that team their opinion on a new logo.
This type of sampling is used when the population is divided into subgroups that are different in some way, and the researcher wants to ensure that these subgroups are represented in the sample. In this case, the coach wants to ensure that both varsity and junior varsity players are represented in the sample, as they may have different opinions on the new logo.
It's not a simple random sample because he selected the team first and then the players. Also, it's not a voluntary response sample because the coach is selecting the players.
It's not a cluster sample because the coach is not selecting the players based on location or geographic proximity, and it's not a systematic sample because he is not selecting the players based on the order in which they appear in a list or a database.
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use dynkin's theorem to prove that the infinite product measure is uniquely defined on the product measure space
The infinite product measure [tex]$\mu$\\[/tex] is uniquely defined on the product measure space[tex]$(\Omega, \mathcal{A}, \mu)$[/tex] by Dynkin's theorem, since [tex]$\mu$\\[/tex] is finitely additive, countably subadditive, and [tex]$\sigma$[/tex]-finite.
Let [tex]$\mathcal{A}_1, \mathcal{A}_2, \ldots$[/tex] be a sequence of sigma algebras defined on [tex]$\Omega_1, \Omega_2, \ldots$[/tex] respectively. Let [tex]$(\Omega, \mathcal{A}, \mu)$[/tex] be the product measure space, where [tex]\Omega = \prod_{i=1}^{\infty} \Omega_i$ and $\mathcal{A} = \bigotimes_{i=1}^{\infty} \mathcal{A}_i$[/tex].
We want to prove that the infinite product measure [tex]$\mu$[/tex] is uniquely defined. To do this, we will use Dynkin's theorem.
According to Dynkin's theorem, a measure [tex]$\mu$[/tex] is uniquely determined on a product measure space [tex]$(\Omega, \mathcal{A}, \mu)$[/tex] if the following conditions are satisfied:
1. [tex]$\mu$[/tex] is finitely additive on [tex]$\mathcal{A}$[/tex]
2. [tex]$\mu$[/tex] is countably subadditive on [tex]$\mathcal{A}$[/tex]
3. [tex]$\mu$[/tex] is [tex]$\sigma$[/tex]-finite on [tex]$\mathcal{A}$[/tex]
It is easy to see that the infinite product measure [tex]$\mu$[/tex]satisfies these conditions, since it is defined as a product of finitely additive, countably subadditive, and [tex]$\sigma$[/tex]-finite measures. Thus, by Dynkin's theorem,[tex]$\mu$[/tex] is uniquely determined on the product measure space [tex](\Omega, \mathcal{A}, \mu)$.[/tex]
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If Mario eats two cups of pretzels every night while he watches television how many cup dose he eat in one week
the possible answers in the question mark are half or twice or eight times for both of the question marks. ASAP I NEED HELP
The size of each group will be half as big.
There will be twice as many groups.
How to divide fractions?We want to find out what is going to happen if we divide by 8/4 instead of 4/5.
The total of the tape diagram is 3¹/₅ = 16/5 and as such if we are going to divide by 4/5 we will have;
16/5 ÷ 4/5 = 4
However if we are dividing by 8/5, it means that the solution is;
16/5 ÷ 8/5 = 2
Thus, the solution will be half as big.
There will be twice as many groups.
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Is this a polynomial?
suppose that for a group of consumers the probability of eating pretzels is .75 and the probability of drinking coke in .65. further suppose that the probability of eating pretzels and drinking coke is about.55. determine if these two events are independent
Marty buys 12 ounces of granola what fraction of a pound of granola did Marty buy? Please help
The fraction of a pound of granola Marty buys is 3/4.
What is a fraction?A fraction is a part of a whole number, value, and variable.
There are proper and improper fractions, including complex fractions.
Proper fractions are those with the numerator less than the denominator. An improper fraction produces a whole number with a remainder when the denominator divides the numerator.
On the other hand, complex fractions are rational expressions that have some fractions in their numerators, denominators, or both.
1 pound = 16 ounces
The quantity of granola Marty buys = 12 ounces
The fraction of a pound of granola Marty buys = 3/4 (12/16)
Thus, we can conclude that Marty buys 3/4 or 75% of a pound of granola.
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which one shows a line with a slop of 2
The graph with the slope of 2 is the second graph.
How to find the slope of a line?The slope of a line is a measure of its steepness. The slope of a line is the change in dependent variable with respect to the change in the independent variable.
Therefore, let's find the slope of the graph to know the one with the slope of 2.
Hence,
slope = y₂ - y₁ / x₂ - x₁
Using the second graph,
(-1, -1)(0, 1)
slope = 1 + 1 / 0 + 1
slope = 2 / 1
slope = 2
Therefore, the graph with slope of 2 is the second graph.
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