an experiment consists of choosing objects without regards to order. determine the size of the sample space when you choose the following:

Answer:

C. 454 students

Step-by-step explanation:

Students who play video games = 259°

Students who don't play video games = 360° - 259°

= 101°

Students who don't play video games = 101°/360° * 1,619

= 0.2805555555555 × 1,619

= 454.219 students

Approximately 454 students

C. 454 students

Due to the negative reciprocal of -1/2, the straight line will have a slope of 2.

The slope of a line in mathematics serves as a gauge for how steep it is. It represents the rate of rise or fall of the line as you move along it from left to right. The character "m" is frequently used to indicate slope. The coordinates of two locations on a line can be used to determine the slope of the line. Let the coordinates of two locations on the line be (x1, y1) and (x2, y2). The line's inclination is then determined by:

(y2 - y1) / (x2 - x1) (x2 - x1) = m

Depending on how the line is oriented, the inclination can be either positive, negative, zero, or undefinable.

given

On each set of points, the slope formula would be applied.

(y2 -y1)/(x2 - x1) (x2 - x1) = m

Due to the fact that -1/2's negative reciprocal is equal to 2, the slope of the vertical line will be 2.

Due to the negative reciprocal of -1/2, the straight line will have a slope of 2.

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Solution

Given that

sample size n = 11

The degree of freedom df = n - 1 = 11 - 1 = 10

At 99% confidence level,

Checking the table,

Therefore, the Critical value t value = 3.169

In a bag containing 5 red, 3 blue, and 12 yellow marbles, we will predict the number of times Hazel will select a blue marble out of 500 trials.

Step 1: Calculate the total number of marbles in the bag:
Total marbles = 5 red + 3 blue + 12 yellow = 20 marbles

Step 2: Determine the probability of selecting a blue marble:
Probability of selecting a blue marble = (number of blue marbles) / (total marbles) = 3 blue / 20 marbles = 3/20

Step 3: Predict the number of times Hazel will select a blue marble in 500 trials:
Predicted blue marbles selected = (probability of selecting a blue marble) x (total trials) = (3/20) x 500

Step 4: Perform the calculation:
(3/20) x 500 = 75

In conclusion, we predict that Hazel will select a blue marble 75 times out of 500 trials, given that the bag contains 5 red, 3 blue, and 12 yellow marbles.

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Answer:

Total chocolates Jo bought for her birthday  = 4.25 pounds

Step-by-step explanation:

Given - Fourteen pounds less than eight times the pounds of chocolate Jo got for her birthday is three more than four times the pounds of chocolate Jo got.

To find - How much chocolate did Jo get for her birthday?

Proof -

Let Jo bought total chocolates = x pounds

Then,

8x - 14 = 4x + 3

⇒8x - 4x = 3 + 14

⇒4x = 17

⇒x = \(\frac{17}{4}\) = 4.25

∴ we get

Total chocolates Jo bought for her birthday  = 4.25 pounds

Justification -

8(4.25) - 14 = 4(4.25) + 3

⇒34 - 14 = 17 + 3

⇒20 = 20

Satisfied.

To derive the Lagrangian dual for the given convex optimization problem, we introduce the Lagrange multiplier λ and construct the Lagrangian function as follows:

L(w, λ) = WERD + λ(1 - W^T x)

Here, w represents the optimization variable, λ is the Lagrange multiplier, and x is a fixed parameter. WERD represents the objective function of the original problem.

To find the Lagrangian dual, we need to minimize the Lagrangian function L(w, λ) with respect to w, while keeping λ fixed. This can be expressed as follows:

min_w L(w, λ) = min_w (WERD + λ(1 - W^T x))

To minimize the Lagrangian with respect to w, we take the derivative of L with respect to w and set it to zero:

∇w L(w, λ) = 0

Differentiating L with respect to w, we get:

∇w L(w, λ) = 2w - λx = 0

Simplifying the equation, we have:

w = (1/2λ)x

Substituting this value of w back into the Lagrangian function, we get:

L(w, λ) = (1/4λ^2)x^T x E R D + λ(1 - (1/4λ^2)x^T x)

Simplifying further, we obtain:

L(w, λ) = (1/4λ^2)x^T x E R D + λ - (1/4λ)x^T x

Finally, the Lagrangian dual function g(λ) is obtained by maximizing the Lagrangian over w:

g(λ) = max_w L(w, λ) = max_w [(1/4λ^2)x^T x E R D + λ - (1/4λ)x^T x]

Hence, the Lagrangian dual of the given convex optimization problem is g(λ) = (1/4λ^2)x^T x E R D + λ - (1/4λ)x^T x.

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The variance is the correct option of property of Poisson distribution.

The Poisson distribution has the property that its mean is equal to its variance. This means that the average number of occurrences, represented by λ, is equal to the spread of the distribution.

The Poisson probability is calculated as P (x, λ) = (e^-λ * λ^x) / x! where e is approximately equal to 2.71828. The mean, represented as E(X), is equal to the variance, V(X). In a Poisson distribution, λ is the expected value of the distribution.

When certain conditions are met, there is a table available to make calculating probabilities easier using the Poisson distribution.

In Poisson distribution, the mean is known as E(X) = λ.

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Answer:

To find the derivative of g(x) = e^x / (e^x - x), we can first simplify the function using the quotient rule, and then apply the chain rule.

Using the quotient rule, we get:

g'(x) = [ (e^x)(e^x - x)' - (e^x - x)(e^x)' ] / (e^x - x)^2

g'(x) = [ (e^x)(-1) - (e^x - x)(e^x) ] / (e^x - x)^2 (using (e^x)' = e^x)

g'(x) = [ -e^x + xe^x ] / (e^x - x)^2

Now, applying the chain rule, we get:

g'(x) = [ (-e^x + xe^x)(e^x - x)' ] / (e^x - x)^2

g'(x) = [ (-e^x + xe^x)(e^x - 1) ] / (e^x - x)^2

Therefore:

g'(x) = [ (-e^x + xe^x)(e^x - 1) ] / (e^x - x)^2.

In order to divide these polynomials, let's use the following property:

So we have:

Answer:

x=-2/3

y=4/5

Step-by-step explanation:

solve for the first variable in one of the equations and then substitute the result into the other equation

Step-by-step explanation:

Let the number be \(\overline{abc}=100a+10b+c\).

\(100a+10b+c=34a+34b+34c \\ \\ 66a-24b-33c=0 \\ \\ 22a-8b-11c=0\)

Taking the equation mod \(11\) yields \(-8b \equiv \pmod{11}\), meaning \(b=0\).

So, \(22a-11c=0 \implies c=2a\).

So, the only possibilites are \((a,c)=(1,2), (2,4),(3,6), (4,8)\).

Therefore, there are \(4\) such numbers.

To show that the product of the sample observations is a sufficient statistic for θ > 0 in the case of a random sample taken from a gamma distribution with parameters α = θ and β = 6, we can use the factorization theorem for sufficient statistics.

Let's denote the random sample as X₁, X₂, ..., Xₙ, where each Xi is an independent and identically distributed random variable following a gamma distribution with parameters α = θ and β = 6.

The probability density function (pdf) of a gamma distribution with parameters α and β is given by:

f(x; α, β) = (1 / (β^α * Γ(α))) * (x^(α - 1)) * exp(-x / β)

where Γ(α) is the gamma function.

The joint pdf of the random sample can be expressed as:

f(x₁, x₂, ..., xₙ; α, β) = (1 / (β^(nα) * Γ(α)^n)) * (x₁ * x₂ * ... * xₙ)^(α - 1) * exp(-(x₁ + x₂ + ... + xₙ) / β)

By the factorization theorem, the product of the sample observations, denoted as T = x₁ * x₂ * ... * xₙ, is a sufficient statistic for θ if we can express the joint pdf as the product of two functions, one depending on the sample observations T and the other on the parameter θ.

Let's rewrite the joint pdf in terms of T:

f(x₁, x₂, ..., xₙ; α, β) = (1 / (β^(nα) * Γ(α)^n)) * T^(α - 1) * exp(-(x₁ + x₂ + ... + xₙ) / β)

Now, we can separate the terms depending on T and θ:

f(x₁, x₂, ..., xₙ; α, β) = (1 / (β^(nα) * Γ(α)^n)) * T^(α - 1) * exp(-(x₁ + x₂ + ... + xₙ) / β) = g(T; α) * h(x₁, x₂, ..., xₙ; β)

Here, we can observe that g(T; α) = (1 / (β^(nα) * Γ(α)^n)) * T^(α - 1) depends only on T and α, and h(x₁, x₂, ..., xₙ; β) = exp(-(x₁ + x₂ + ... + xₙ) / β) depends only on the sample observations and β.

Therefore, we have successfully factorized the joint pdf into two functions, one depending on T and α, and the other depending on the sample observations and β. This confirms that the product of the sample observations T = x₁ * x₂ * ... * xₙ is a sufficient statistic for the parameter θ when the random sample is taken from a gamma distribution with parameters α = θ and β = 6.

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Answer:

\(C(x) = 540x + 72120\)

Step-by-step explanation:

Given

Cost = $78,600 when Printing Press = 12

Cost = $82,380 when Printing Press = 19

Required

Determine the linear equation

From the given parameters; Cost is a function of Printing Press

Represent Cost by C and Printing press by x;

This implies that C = f(x)

The given parameters can then be modeled as; \((x, C)\)

\((x_1, C_1) = (12,78600)\)

\((x_2, C_2) = (19,82380)\)

The first step is to calculate the slope, m;

\(m = \frac{C_1 - C_2}{x_1 - x_2}\)

\(m = \frac{78600 - 82380}{12 - 19}\)

\(m = \frac{-3780}{-7}\)

\(m = 540\)

The linear equation can then be calculated using slope formula

\(m = \frac{C- C_2}{x - x_2}\)

Substitute 540 for m and \((x_2, C_2) = (19,82380)\)

\(540 = \frac{C- 82380}{x - 19}\)

Multiply both sides by x - 19

\(540 * (x-19)= \frac{C- 82380}{x - 19} * (x-19)\)

\(540 * (x-19)= C- 82380\)

Open bracket

\(540x - 10260 = C - 82380\)

Add 82380 to both sides

\(540x - 10260 + 82380= C - 82380 + 82380\)

\(540x - 10260 + 82380= C\)

\(540x + 72120= C\)

\(C = 540x + 72120\)

Hence;

\(C(x) = 540x + 72120\)

Answer:

y=3

Step-by-step explanation:

y=3  ZERO slope

Since each point has 3 (same number ) as y coordinate,this is a zero slope

You will see an horizontal line on the coordinate plan

The median is the middle value, which in this case is $162,000. When determining the measure of central tendency for a given set of data, several measures can be considered, including the mean, median, and mode.

In this case, it would be advisable to use the median as the measure of central tendency. The median represents the middle value when the data is arranged in ascending or descending order. It is less influenced by extreme values or outliers, making it a suitable choice for situations where the data set may contain extreme values, such as the significantly higher value of $1,232,000 in this case.

By arranging the data in ascending order, we have:

$152,000, $154,000, $162,000, $182,000, $1,232,000

The median is the middle value, which in this case is $162,000.

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Answer:

350

Step-by-step explanation:

The revenue from hardcover books was $24,860 and the revenue from paperback books was $99,440.

Let's assume the revenue from hardcover books as "x" dollars.

Then, the revenue from paperback books will be 4 times the revenue from hardcover books, i.e., 4x dollars.

The total revenue is given as $124,300, so we can set up the following equation:

x + 4x = 124300

Simplifying the above equation, we get:

5x = 124300

x = 24860

Therefore, the revenue from hardcover books was $24,860 and the revenue from paperback books was 4 times that amount, i.e., $99,440.

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Answer:

50x

Step-by-step explanation:

5x(7x + 3)

(7x + 3)= 10x

5x x 10x=50x

Let's call the greater number "x" and the lesser number "y". According to the problem, we know that:

x - y = 48  (since the difference between the two numbers is 48)

y = (1/3)x   (since the lesser number is one third of the greater number)

Now we can substitute the second equation into the first equation:

x - (1/3)x = 48

Simplifying this equation, we get:

(2/3)x = 48

Multiplying both sides by 3/2, we get:

x = 72

Now that we know x, we can use the second equation to find y:

y = (1/3)x = (1/3)(72) = 24

So the two positive numbers are 72 and 24.

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9514 1404 393

Explanation:

The closure property for any operation and set says that performing the operation on members of the set will result in a member of the set.

The closure property for addition of polynomials says that the addition of any polynomials will result in a polynomial.

Examples:

1)  1 and x are polynomials, as is their sum: 1+x.

2) x^3 -5 and x+5 are polynomials, as is their sum: (x^3 -5) +(x+5) = x^3 -x.

Answer:

3√13

Step-by-step explanation:

4 : y = y = 9

y^2 = 36

y = 6

z = \(\sqrt{9^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117}\)

117 = 3^2 * 13

\(\sqrt{117} = \sqrt{3^2 * 13} = 3\sqrt{13\)

To find the particular solution that satisfies the given differential equation and initial condition, we need to integrate the derivative of f(x) with respect to x to obtain the function f(x).

Given: f'(x) = 7x and f(0) = 6

Integrating f'(x) with respect to x:

∫7x dx = (7/2)x^2 + C

Here, C represents the constant of integration.

To find the value of C, we can use the initial condition f(0) = 6:

f(0) = (7/2)(0)^2 + C

6 = C

Therefore, C = 6.

Substituting the value of C back into the integrated function, we have:

f(x) = (7/2)x^2 + 6

Thus, the particular solution that satisfies the given differential equation f'(x) = 7x and initial condition f(0) = 6 is f(x) = (7/2)x^2 + 6.

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Answer:

y = 10x - 46

Step-by-step explanation:

slope intercept form: y = mx + b

m is slope and b is y-intercept

given: y - 4 = 10(x-5)

1. distibute 10

y - 4 = 10x - 50

2. we need to get y by itself so bring -4 to other side

y = 10x - 50 + 4

y = 10x - 46

slope-intercept form is \(y=mx+b\)

\(y=10(x-5)+4\)          ↣ We move -4 to another side and become positive number.

\(y=(10x-50)+4\)             ↣ Distribute 10 inside the brackets.

\(y=10x-50+4\\y=10x-46\)            ↣ Then we get the slope-intercept form.

Thus, the answer is \(y=10x-46\)

Shandra is expected to select a pineapple chew about 56 times in the next 400 selections.

Based on the given results, Shandra selected a pineapple chew 8 times out of 57 selections. To estimate the expected number of times she would select a pineapple chew in the next 400 selections, we can use the proportion of pineapple chews selected in the first 57 selections:

The proportion of pineapple chews selected = 8/57

To estimate the expected number of pineapple chews in the next 400 selections, we can multiply the proportion by the total number of selections:

Expected number of pineapple chews in the next 400 selections = (8/57) x 400

expected number of pineapple chews in the next 400 selections = 56.14

Rounding to the nearest whole number, we would expect Shandra to select a pineapple chew about 56 times in the next 400 selections.

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The value of differentiation of h(1) is 33.

We can use the product rule of differentiation to find h'(x)

h(x) = f(x)g(x)

h'(x) = f'(x)g(x) + f(x)g'(x)

At x = 1, we know g(1) = -5 and g'(1) = -4. To find f'(x), we first need to find f(x)

f(x) = -4x² - 5x + 1

Then, we can find f'(x) by taking the derivative of f(x)

f'(x) = -8x - 5

Now we can substitute in the values at x = 1

f(1) = -4(1)² - 5(1) + 1 = -8

f'(1) = -8(1) - 5 = -13

h'(1) = f'(1)g(1) + f(1)g'(1)

h'(1) = (-13)(-5) + (-8)(-4)

h'(1) = 65 - 32

h'(1) = 33

Therefore, h'(1) = 33.

The value of h'(1) is 33.

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Factorial design are useful in testing theories because they enable researchers to investigate a theory's construct validity. so the correct answer  is option  (a).

A key technique for identifying the impact of numerous variables on a response is the factorial design. Traditionally, experiments are planned to ascertain how one variable affects just one response.

The corpus of knowledge supporting the analysis and application of test results. The definition and measurement of reliability are of primary relevance. Classical test theory, generalizability theory, and item response theory are examples of theoretical frameworks.

a) They enable academics to investigate a theory's construct validity.

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The first step to isolate the variable term on one side of the equation is to move all constant terms to the other side by adding or subtracting the appropriate terms.

To isolate the variable term on one side of the equation, the first step is to gather all terms containing the variable on one side and move all constant terms to the other side.

In the given equation:

2/3x = -1/2x + 5

We have variable terms on both sides: 2/3x and -1/2x. To isolate the variable term, we can start by moving the -1/2x term to the left side by adding 1/2x to both sides of the equation.

Adding 1/2x to both sides:

(2/3x) + (1/2x) = (-1/2x) + (1/2x) + 5

Simplifying the left side:

(2/3x + 1/2x) = 5

To combine the fractions, we need a common denominator. The common denominator of 3 and 2 is 6, so we can rewrite the left side:

(4/6x + 3/6x) = 5

Combining like terms on the left side:

(7/6x) = 5

Now, the variable term 7/6x is isolated on one side of the equation. To completely isolate the variable, we can multiply both sides of the equation by the reciprocal of the coefficient of x, which in this case is 6/7.

Multiplying both sides by 6/7:

(6/7) * (7/6x) = (5) * (6/7)

Simplifying:

1x = 30/7

The variable x is now isolated on the left side, and the equation simplifies to:

x = 30/7

Moving all constant terms to the opposite side of the equation by appropriately adding or deleting terms is the first step towards isolating the variable term on one side of the equation.

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The finest-grain property simply means that all possible distinguishable outcomes are identified separately in probability statistics.

The requirement that outcomes be mutually exclusive says that if one outcome occurs, then no other outcome also occurs.

An outcome is often called a finest grain result of the model in the sense that an outcome ω contains no subsets other than the empty set ϕ and the singleton subset {ω}. A sample space is a set of all the possible outcomes or sample points, denoted as Ω. An outcome ω is often called a finest grain result when even {ω} contains no proper subsets. The outcomes in the sample space are mutually exclusive and collectively constitute the entire sample space.

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