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The pdf of x is f(x) = (x^4 * e^(-x/3)) / 1620.

the variance of x is var[x] = 45.

(a) Since x is an Erlang (n, λ) random variable with expected value e[x] = 15 and λ = 1/3, we have:

e[x] = n/λ = n/(1/3) = 3n

Therefore, we have:

3n = 15

n = 5

So the value of the parameter n is 5.

(b) The probability density function (pdf) of an Erlang (n, λ) random variable is given by:

f(x) = (λ^n * x^(n-1) * e^(-λx)) / (n-1)!

Substituting λ = 1/3 and n = 5, we have:

f(x) = (1/3)^5 * x^4 * e^(-x/3) / 4!

        = (x^4 * e^(-x/3)) / 1620

Therefore, the pdf of x is f(x) = (x^4 * e^(-x/3)) / 1620.

(c) The variance of an Erlang (n, λ) random variable is given by:

var[x] = n/λ^2 = n/(1/λ)^2

Substituting λ = 1/3 and n = 5, we have:

var[x] = 5/(1/(1/3))^2

        = 45

Therefore, the variance of x is var[x] = 45.

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An integral expression that would compute the volume of the solid is [tex]V = \int\limits^a_b {1/2 \pi [R(x)]^2} \, dx[/tex]

An integral expression is a mathematical statement that represents the area under a curve or the volume of a solid in three-dimensional space. It is written using integral notation, which involves an integral sign, a function to be integrated, and limits of integration.

If each cross section perpendicular to the x-axis is a semicircle, then the radius of each cross section depends on the x-coordinate of the center of the cross section. Let R(x) be the radius of the cross section at x.

To find the volume of the solid, we can integrate the area of the cross section over the interval of x that defines the base R. The area of each cross section is given by the formula for the area of a semicircle:

[tex]A(x) = (1/2)[/tex][tex]\pi[R(x)]^2[/tex]

The volume of the solid can be found by integrating A(x) over the base R:

[tex]V = \int\limits^a_b {1/2 \pi [R(x)]^2} \, dx[/tex]

where a and b are the limits of integration for x that define the base R.

Note that we are integrating with respect to x, so we need to express the radius R(x) in terms of x.

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The other root of the quadratic equation include the following (-4, 0).

In Mathematics and Geometry, the vertex form of a quadratic equation is given by this formula:

y = a(x - h)² + k

Where:

For the given quadratic function, we have;

y = a(x - h)² + k

0 = a(8 - 2)² - 5

0 = 36a - 5

5 = 36a

a = 5/36

Therefore, the required quadratic function in vertex form is given by;

y = 5/36(x - 2)² - 5

0 = 5/36(x - 2)² - 5

5 = 5/36(x - 2)²

36 = (x - 2)²

±6 = x - 2

x = -6 + 2

x = -4.

Other root = (-4, 0).

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Answer: Website A is cheaper, by an amount of, £0.29.

Step-by-step explanation: Here, the problem is simply about, initially adding, and then finding difference between the added results.

That is,

For Website A,

Net Cost = £49.95 + £4.39

= £54.34

Similarly,

For Website B,

Net Cost = £47.68 + £6.95

= £54.63

Therefore, we can clearly see,

Website A is cheaper by,

£(54.63 - 54.34) = £0.29

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Answer:

X = 24.4

Step-by-step explanation:

for the triangle we use sin b/c it contain both hyp and opposite so

sin(35°) = 14/x

sin(35) × X = 14

X = 14 / (sin(35)

X = 24.4 ... it is the answer of hypotenus of the

triangle

Answer:

Step-by-step explanation:

The required values are -2+√15, -2-√15.

What is a quadratic equation?

Any equation in algebra that can be written in the standard form where x stands for an unknown value, where a, b, and c stand for known values, and where a 0 is true is known as a quadratic equation.

Here, we have

Given:  x² + 4x - 11 = 0

we have to find the values of a and b to complete the solutions.

The given equation is x² + 4x - 11 = 0

The general form of a quadratic equation is ax² + bx + c = 0

Comparing with the given equation we have

a = 1

b = 4

c = -11

Rearranging the equation:

x² + 4x = 11

Finding (b/2)²

(4/2)² = 4

Adding to both sides of the equation

x² + 4x + 4 = 11 + 4

(x+2)² = 15

x + 2 = ±√15

x = -2  ±√15

Hence, the required values are -2+√15, -2-√15.

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The length of the third side of this isosceles triangle is 2 units.

We have,

If two sides of an isosceles triangle are equal, then the third side must also be equal in length.

So,

If two sides of the triangle are 2 and 2, the length of the third side must also be 2.

Thus,

The length of the third side of this isosceles triangle is 2 units.

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Answer: 5

5/8/1/8, you can do 5x8 and also do 8x1 because you can not divide fractions after that you get 40/8 then you divide 40/8 is 5 so the answer is 5

z-score for Stimulus 2 (-2.33) is more negative than the z-score for Stimulus 1 (-1.29), you are reacting relatively more quickly to Stimulus 2 compared to other individuals.

We need to calculate the z-scores for your reaction times for each stimulus.

For Stimulus 1:

z-score = (your reaction time - mean reaction time for Stimulus 1) / standard deviation for Stimulus 1

z-score = (4.2 - 6.0) / 1.4

z-score = -1.29

For Stimulus 2:

z-score = (your reaction time - mean reaction time for Stimulus 2) / standard deviation for Stimulus 2

z-score = (1.8 - 3.2) / 0.6

z-score = -2.33

The more negative the z-score, the farther away your reaction time is from the mean.

Therefore, since the z-score for Stimulus 2 (-2.33) is more negative than the z-score for Stimulus 1 (-1.29), you are reacting relatively more quickly to Stimulus 2 compared to other individuals.

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We have demonstrated that x² ≥ x for all integers x. Therefore, the statement x² ≥ x for all x ∈ Z is true.

An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made.

To prove that x² ≥ x for all x ∈ Z, we need to show that the inequality holds true for any arbitrary integer value of x.

We can prove this by considering two cases:

Case 1: x ≥ 0

If x ≥ 0, then x² ≥ 0 and x ≥ 0. Therefore, x² ≥ x.

Case 2: x < 0

If x < 0, then x² ≥ 0 and x < 0. Therefore, x² > x.

In either case, we have shown that x² ≥ x for all integers x. Therefore, the statement x² ≥ x for all x ∈ Z is true.

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The measure of the central angle is 86 degrees.

The length of the arc is 9 cm and the radius is 6 centimetres. Therefore, let's find the central angle as follows:

Hence,

length of an arc = ∅ / 360 × 2πr

where

Therefore,

length of arc = 9 cm

radius = 6 cm

Therefore,

9 = ∅ / 360 × 2 × 3.14 × 6

9 = 37.68∅ / 360

cross multiply

3240 = 37.68∅

divide both sides by 37.68

∅ = 3240 / 37.68

∅ = 85.9872611465

∅ = 86 degrees.

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Equation of the tangent line to the curve y=8x is y = 8x + 48.

Derivative of the curve, we get:

dy/dx = 8

This means that the slope of the tangent line to the curve at any point is 8.

So, at the point (2,64), the slope of the tangent line is 8.

By point-slope form of a line, we will find the equation of the tangent line:

y - y1 = m(x - x1)

where m is the slope and (x1,y1) is the given point.

Plugging in the values, we get:

y - 64 = 8(x - 2)

Simplifying, we get:

y = 8x + 48

Equation of the tangent line to the curve y=8x at the point (2,64) is y = 8x + 48.

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The dial's radius is approximately 13.99 millimeters.

The circumference of a circle is given by the formula:

C = 2πr

where C is the circumference, π is the constant pi (approximately equal to 3.14159), and r is the radius of the circle.

The circumference C in this instance is 87.92 millimeters. We can adjust the equation to address for the sweep:

r = C / 2π

Substituting the given value for C, we get:

r = 87.92 mm / (2π)

r ≈ 13.99 mm

As a result, the dial has a radius of about 13.99 millimeters.

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Referring to Exercise 3.39,

(a) f(y|2) for all values of y is f(2|2) = P(Y=2|X=2) = P(X=2, Y=2) / P(X=2) = (1/14) / (3/14) = 1/3

(b) P(Y = 0 | X = 2) = 1

To find f(y|2), we need to first calculate the conditional probability of Y=y given that X=2, which we can do using the joint probability distribution we found in part (a) of Exercise 3.39:
P(Y=y|X=2) = P(X=2, Y=y) / P(X=2)
We know that P(X=2) is equal to the probability of selecting 2 oranges out of 4 fruits, which can be calculated using the hypergeometric distribution:
P(X=2) = (3 choose 2) * (2 choose 0) / (8 choose 4) = 3/14
To find P(X=2, Y=y), we need to consider all the possible combinations of selecting 2 oranges and y apples out of 4 fruits:
P(X=2, Y=0) = (3 choose 2) * (2 choose 0) / (8 choose 4) = 3/14
P(X=2, Y=1) = (3 choose 2) * (2 choose 1) / (8 choose 4) = 3/14
P(X=2, Y=2) = (3 choose 2) * (2 choose 2) / (8 choose 4) = 1/14
Therefore, f(y|2) is:
f(0|2) = P(Y=0|X=2) = P(X=2, Y=0) / P(X=2) = (3/14) / (3/14) = 1
f(1|2) = P(Y=1|X=2) = P(X=2, Y=1) / P(X=2) = (3/14) / (3/14) = 1
f(2|2) = P(Y=2|X=2) = P(X=2, Y=2) / P(X=2) = (1/14) / (3/14) = 1/3
To find P(Y=0|X=2), we can use the conditional probability formula again:
P(Y=0|X=2) = P(X=2, Y=0) / P(X=2) = 3/14 / 3/14 = 1
Therefore, P(Y=0|X=2) = 1.

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No, it is not possible that CA = I4 for some 4 × 2 matrix C, where A is a 4 × 2 matrix and I4 is the 4 × 4 identity matrix.



1. Recall that the identity matrix I4 is a 4 × 4 matrix with ones on the diagonal and zeros elsewhere.

2. In matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.

3. If C is a 4 × 2 matrix and A is a 4 × 2 matrix, then matrix multiplication CA results in a 4 × 2 matrix, as the number of rows in C (4) and the number of columns in A (2) determine the dimensions of the resulting matrix.

4. Since CA produces a 4 × 2 matrix, it cannot be equal to the 4 × 4 identity matrix I4, as the dimensions are not the same.

Therefore, it is not possible for CA = I4 for some 4 × 2 matrix C.

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Answer:

x² + y² = 80

Step-by-step explanation:

We are given that a circle has the center at the origin (the point (0,0)) and passes through the point (-8,4).

We want to write the equation of this circle in the standard equation. The standard equation is (x-h)² + (y-k)² = r² where (h,k) is the center and r is the radius.

As we are given the center, we can plug its values into the equation.

Substitute 0 as h and 0 as k.

(x-0)² + (y-0)² = r²

This becomes:

x² + y² = r²

Now, we need to find r².

As the circle passes through (-8,4), we can use its values to help solve for r².

Substitute -8 as x and 4 as y.

(-8)² + (4)² = r²

64 + 16 = r²

80 = r²

Substitute 80 as r².

x² + y² = 80

The value of the series is: ∑ 1/n(n+2) = lim N→∞ S(N) = 1/2.

The series ∑ 1/n(n+2)n=1 converges. To determine its value, we can use the partial fraction decomposition:

1/n(n+2) = 1/2 * (1/n - 1/(n+2))

Using this decomposition, we can rewrite the series as:

∑ 1/n(n+2) = 1/2 * (∑ 1/n - ∑ 1/(n+2))

The first series ∑ 1/n is the harmonic series, which diverges. However, the second series ∑ 1/(n+2) is a shifted version of the harmonic series, and it also diverges. But since we are subtracting a divergent series from another divergent series, we can use the limit comparison test to determine whether the original series converges or diverges. Specifically, we can compare it to the series ∑ 1/n, which we know diverges. This gives:

lim n→∞ 1/n(n+2) / 1/n = lim n→∞ (n+2)/n^2 = 0

Since the limit is less than 1, we can conclude that the series ∑ 1/n(n+2) converges. To find its value, we can evaluate the partial sums:

S(N) = 1/2 * (∑_{n=1}^N 1/n - ∑_{n=1}^N 1/(n+2))
    = 1/2 * (1/1 - 1/3 + 1/2 - 1/4 + ... + 1/(N-1) - 1/(N+1))

As N approaches infinity, the terms in the parentheses cancel out except for the first and last terms:

S(N) → 1/2 * (1 - 1/(N+1))

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The partial derivatives of the function f(x,y) = xy*e^(-9y) with respect to x and y are: ∂f/∂x = ye^(-9y), and ∂f/∂y = x(-9y*e^(-9y)) + e^(-9y).

The first partial derivative concerning x is obtained by treating y as a constant and differentiating concerning x. The result is ye^(-9y), which means that the rate of change of f concerning x is equal to ye^(-9y).

The second partial derivative concerning y is obtained by treating x as a constant and differentiating concerning y. The result is x(-9ye^(-9y)) + e^(-9y), which means that the rate of change of f concerning y is equal to x times -9ye^(-9y) plus e^(-9y).

To better understand these partial derivatives, we can analyze the behavior of the function f(x,y) = xy*e^(-9y). As we can see, the function is the product of three terms: x, y, and e^(-9y). The term e^(-9y) represents a decreasing exponential function that approaches zero as y increases. Therefore, the value of f(x,y) decreases as y increases. The terms x and y represent a linear function that increases as x and y increase. Therefore, the value of f(x,y) increases as x and y increase.

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Answer: I believe -2 is the answer

Step-by-step explanation: To solve for the function over an interval, you need to know the equation of the function. If you have the equation, you can plug in the values of the interval into the equation to find the corresponding y-values. For example, if the function is y = 2x + 1 and the interval is [0,3], you can plug in x = 0 and x = 3 to find the corresponding y-values and get the ordered pairs (0,1) and (3,7).

As n increases, 3^n becomes larger, making the fraction 30/3^n approach zero. Therefore, the sequence {A_n} is convergent, and the limit of convergence is 0. (a) Σ(A_n) is convergent and (b) {A_n} is convergent with the limit of convergence equal to 0.

(a) To determine whether sigma _n = 1^infinity (A_n) is convergent, we need to take the sum of the sequence A_n from n=1 to infinity:
sigma _n = 1^infinity (A_n) = A_1 + A_2 + A_3 + ...
Substituting A_n = 30/3^n, we get:
sigma _n = 1^infinity (A_n) = 30/3^1 + 30/3^2 + 30/3^3 + ...
To simplify this, we can factor out a common factor of 30/3 from each term:
sigma _n = 1^infinity (A_n) = 30/3 * (1/3^0 + 1/3^1 + 1/3^2 + ...)
Now, we recognize that the expression in parentheses is a geometric series with first term a=1 and common ratio r=1/3. The sum of an infinite geometric series with first term a and common ratio r is:
sum = a / (1 - r)
Applying this formula to our series, we get:
sigma _n = 1^infinity (A_n) = 30/3 * (1/ (1 - 1/3)) = 30/2 = 15
Therefore, sigma _n = 1^infinity (A_n) is convergent, with a limit of 15.
(b) To determine whether {An} is convergent, we need to take the limit of the sequence A_n as n approaches infinity:
lim n->infinity (A_n) = lim n->infinity (30/3^n) = 0
Therefore, {An} is convergent, with a limit of 0.
(a) To determine if the series Σ(A_n) from n=1 to infinity is convergent, we can use the ratio test. The ratio test states that if the limit as n approaches infinity of the absolute value of the ratio A_(n+1)/A_n is less than 1, the series converges.
For A_n = 30/3^n, we have:
A_(n+1) = 30/3^(n+1)
Now let's find the limit as n approaches infinity of |A_(n+1)/A_n|:
lim(n→∞) |(30/3^(n+1))/(30/3^n)| = lim(n→∞) |(3^n)/(3^(n+1))| = lim(n→∞) |1/3|
Since the limit is 1/3, which is less than 1, the series Σ(A_n) converges.
(b) To determine if the sequence {A_n} is convergent, we need to find the limit as n approaches infinity:
lim(n→∞) (30/3^n)
As n increases, 3^n becomes larger, making the fraction 30/3^n approach zero. Therefore, the sequence {A_n} is convergent, and the limit of convergence is 0.


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The limit of the given sequence is π/4, and the sequence converges to this value.

The given sequence is {tan^−1(4n/(4n+5))}. To determine if the sequence converges or diverges, we can analyze the limit of the function as n approaches infinity.

As n goes to infinity, the function behaves like tan^−1(4n/4n), which simplifies to tan^−1(1). Since the arctangent function has a range of (-π/2, π/2), tan^−1(1) falls within this range, and it is equal to π/4 (or 45° in degrees).

Now, let's consider the difference between the given function and the simplified one: (4n+5) - 4n = 5. As n becomes larger, the effect of the constant term 5 becomes negligible. Consequently, the function approaches tan^−1(1) as n approaches infinity.

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Answer:

52.7 g

Step-by-step explanation:

We are given;

Initial mass of the element is 310 g

Rate of decay 8.9% per minute

Time for the decay 19 minutes

We are required to determine the amount of the element that will remain after 19 minutes.

We can use the formula;

New mass = Original mass × (1-r)^n

Where n is the time taken and r is the rate of decay.

Therefore;

Remaining mass = 310 g × (1-0.089)^19

                           = 52.748 g

                           = 52.7 g (to the nearest 10th)

Thus, the mass that will remain after 9 minutes will be 52.7 g

Answer:

Step-by-step explanation:

To find the probability that a student is a science major given that they are a graduate student, we need to use Bayes' theorem:

P(Science | Graduate) = P(Graduate | Science) * P(Science) / P(Graduate)

We know that P(Science) = 0.45 and P(Liberal Arts) = 0.55, and that P(Graduate | Science) = 0.35 and P(Graduate | Liberal Arts) = 0.25. We also know that the total probability of being a graduate student is:

P(Graduate) = P(Graduate | Science) * P(Science) + P(Graduate | Liberal Arts) * P(Liberal Arts)

Plugging in the values, we get:

P(Graduate) = 0.35 * 0.45 + 0.25 * 0.55 = 0.305

Now we can calculate the probability of being a science major given that the student is a graduate student:

P(Science | Graduate) = 0.35 * 0.45 / 0.305 = 0.515

Therefore, the probability that a student is a science major, given they are a graduate student, is approximately 0.515.

Answer:

0.72

Step-by-step explanation:

trust me

hope this helps you .

We can define the vector field as:F(x,y) = v = ⟨−y,x⟩/√(x²+y²).

This vector field satisfies the conditions that all vectors have length 1 and are perpendicular to the position vector at each point

Perpendicular lines are lines that intersect at a right angle (90 degrees).

Let's consider a two-dimensional vector field, denoted by F(x,y), where F is a vector function of two variables x and y. We want all vectors in this field to have length 1 and to be perpendicular to the position vector at each point.

The position vector at a point (x,y) is given by r = x, y , so we need to find a vector that is perpendicular to r and has length 1. One such vector is \ -y, x .

To make sure that all vectors in the field have length 1, we can normalize this vector by dividing it by its magnitude:

v = ⟨−y,x⟩/√(x²+y²).

Finally, we can define the vector field as:

F(x,y) = v = ⟨−y,x⟩/√(x²+y²).

This vector field satisfies the conditions that all vectors have length 1 and are perpendicular to the position vector at each point.

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We can define the vector field as:F(x,y) = v = ⟨−y,x⟩/√(x²+y²).

This vector field satisfies the conditions that all vectors have length 1 and are perpendicular to the position vector at each point

Perpendicular lines are lines that intersect at a right angle (90 degrees).

Let's consider a two-dimensional vector field, denoted by F(x,y), where F is a vector function of two variables x and y. We want all vectors in this field to have length 1 and to be perpendicular to the position vector at each point.

The position vector at a point (x,y) is given by r = x, y , so we need to find a vector that is perpendicular to r and has length 1. One such vector is \ -y, x .

To make sure that all vectors in the field have length 1, we can normalize this vector by dividing it by its magnitude:

v = ⟨−y,x⟩/√(x²+y²).

Finally, we can define the vector field as:

F(x,y) = v = ⟨−y,x⟩/√(x²+y²).

This vector field satisfies the conditions that all vectors have length 1 and are perpendicular to the position vector at each point.

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The value of the integral ∫∫R 2xy^2 dA over the given region R is 64/3.

To evaluate the integral ∫∫R 2xy^2 dA over the region R given by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 4, we integrate with respect to x first, and then with respect to y:

∫∫R 2xy^2 dA = ∫[0,4] ∫[0,1] 2xy^2 dx dy

Integrating with respect to x, we get:

∫[0,4] ∫[0,1] 2xy^2 dx dy = ∫[0,4] (y^2) [x^2]0^1 dy

Simplifying the expression inside the integral, we get:

∫[0,4] (y^2) [x^2]0^1 dy = ∫[0,4] y^2 dy

Integrating with respect to y, we get:

∫[0,4] y^2 dy = [y^3/3]0^4

Substituting the limits of integration and simplifying, we get:

[y^3/3]0^4 = (4^3/3) - (0^3/3) = 64/3

Therefore, the value of the integral ∫∫R 2xy^2 dA over the given region R is 64/3.

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We can determine the population in 1820 was 8.5753 using a linear equation.

A linear equation is one that has just a constant and a first order (linear) component, like y=mx+b, where m is the slope and b is the y-intercept.

When x and y are the variables, the aforementioned is sometimes referred to as a "linear equation of two variables."

dp/dt = [tex]1.37^{t}[/tex]

Integrate both sides.

p[h] = ( [tex]1.37^{t}[/tex])/In (1.37)   + c

1810 ⇒ t = 0

7.4 = 1/In (1.37) + C

C = 4.2235

p(H) = ( [tex]1.37^{t}[/tex])/In (1.37) + 4.2235

P (1) =  [tex]1.37^{t}[/tex]In (1.37) + 4.2235

= 8.5753

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Answer:

The probability of the 4th donation being Type O or Type B is:

P(Type O or B) = P(Type O) + P(Type B) = 0.49 + 0.20 = 0.69

The probability of the 4th donation being safe for someone with Type B blood is the probability that it is Type O or Type B, which is 0.69. Therefore, the probability that the 4th donation selected at random can be safely used in a blood transfusion on someone with Type B blood is:

P(safe for Type B) = 0.69

Answer: (0.69)³(0.31)

Hi! Based on the information provided, using homework 10 data with a significance level (α) of 0.05 and a p-value of 0.038, your conclusion is that you would reject the null hypothesis.

This is because the p-value (0.038) is less than the significance level (0.05), indicating that there is significant evidence to suggest that the alternative hypothesis is true. Therefore, the conclusion is made based on the evidence to suggest that there is a statistically significant difference between the groups being compared in the study analyzed in homework 10.

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The point Q after dilation with a scale factor of 1.5 about the point (0, -2) is (6, -2). So, correct option is C.

To find the new coordinates of point P after dilation with a scale factor of 1.5 about the point (0, -2), we can use the following formula:

Q(x, y) = S(x, y) = (1.5(x - 0) + 0, 1.5(y + 2) - 2)

Substituting the coordinates of point P (4, -2), we get:

Q(x, y) = S(4, -2) = (1.5(4 - 0) + 0, 1.5(-2 + 2) - 2)

Q(x, y) = S(4, -2) = (6, -2)

Therefore, the new point after dilation is Q(6, -2).

To check which of the given points A, B, C, and D match the new point Q, we can compare their coordinates. Only point C(6, -2) matches the new point Q, so that must be the answer. Points A, B, and D do not match the new point.

So, correct option is C.

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