if f(x) = f(g(x)), where f(−4) = 9, f ′(−4) = 3, f ′(5) = 3, g(5) = −4, and g ′(5) = 4, find f ′(5). f '(5) =

There is no curve that describes the level curve of value c for the given surface, as the condition c < 0 makes it impossible to find a real solution.

In two- or three-dimensional space, a curve is a mathematical object that symbolises a continuous, smooth path. Curves can be derived from geometric operations, parametric equations, or mathematical equations. They are commonly used to simulate real-world processes in physics, engineering, mathematics, and many other disciplines.

To find the level curve of value c for the given surface [tex]z = f(x, y) = (x^2/4) + (y^2/25) = c[/tex], where c < 0, follow these steps:

Step 1: Write down the equation of the surface.
[tex]z = f(x, y) = (x^2/4) + (y^2/25)[/tex]

Step 2: Replace z with the constant c.
[tex]c = (x^2/4) + (y^2/25)[/tex]

Step 3: Rearrange the equation to isolate [tex]y^2[/tex].
[tex]y^2 = 25(c - (x^2/4))[/tex]

However, note that we're given that c < 0. This means that the value inside the parentheses (c - ([tex]x^2/4[/tex])) must also be negative. Since y^2 can't be negative, there's no real solution for this equation.

In conclusion, there is no curve that describes the level curve of value c for the given surface, as the condition c < 0 makes it impossible to find a real solution.

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There are 14.25 liters in 15 quarts based on the expression and data given.

To find the number of liters in 15 quarts, we can use the conversion factor given in the table for quarts to liters. The table states that 1 quart is equal to 0.95 liters.

To convert 15 quarts to liters, we can set up the following expression:

Number of liters = (Number of quarts) × (Conversion factor)

In this case:
Number of liters = 15 quarts × 0.95 liters/quart

Now, you can simply multiply 15 by 0.95 to find the number of liters:
Number of liters = 15 × 0.95 = 14.25 liters

So, there are 14.25 liters in 15 quarts.

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The experimental probability of winning a prize in the contest is 1/28 or approximately 0.0357.

To calculate the experimental probability of winning a prize in the contest, we need to divide the number of winning caps found by the total number of caps examined.

Here are the steps to follow:

Calculate the total number of caps examined:

Total number of bottles bought x Number of caps per bottle = Total number of caps examined

56 bottles x 1 cap per bottle = 56 caps examined

Calculate the number of winning caps found:

Given: 2 winning caps were found

Calculate the experimental probability of winning a prize:

Experimental probability = Number of winning caps found / Total number of caps examined

Experimental probability = 2 / 56

Experimental probability = 1 / 28

Explanation: Out of 56 caps examined, only 2 were found to be winning caps. Therefore, the probability of finding a winning cap is 2/56, which can be simplified to 1/28. This means that on average, for every 28 caps examined, one is expected to be a winning cap.

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When csc(θ)<0, it means that the cosecant of angle θ is negative. Recall that the cosecant of an angle is the reciprocal of its sine. Therefore, csc(θ)<0 when sin(θ)<0.

The sine function is negative in the third and fourth quadrants of the unit circle, where the y-coordinate of the point on the circle is negative. Therefore, if csc(θ)<0, angle θ could lie in Quadrant III or Quadrant IV. To summarize, when csc(θ)<0, angle θ could lie in Quadrant III or Quadrant IV. It cannot lie in Quadrant I or Quadrant II because the sine function is positive in those quadrants.

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Answer:the 3rd option

Step-by-step explanation:

Sam needs to study approximately 4.6 hours per week to earn a 4.0 GPA. The answer is (d) 4.6 hours.

We know that Sam wants to earn a 4.0 GPA, and his class attendance hours are fixed at 4. Therefore, we can solve for the number of hours he needs to spend studying to achieve this goal by setting the equation equal to 4.0 and solving for the hours spent studying:

4.0 = (0.50 x hours spent studying) + (0.25 x 4) + (0.25 x 2.75)

4.0 = (0.50 x hours spent studying) + 1 + 0.6875

2.3125 = 0.50 x hours spent studying

hours spent studying = 4.625

The correct option is (d) 4.6 hours.

The complete question is:

Sam wants to improve his GPA. to earn a 4.0 this semester. His prior GPA was a 2.75. Imagine that there is an equation that says his new GPA could be calculated based on the number of hours he spends studying, his class attendance, and his prior GPA. Written as an equation, it is Grade = (0.50 x hours spent studying) + (0.25 x class attendance hours) + (0.25 x prior GPA). Sam plans to attend class for 4 out of 4 hours each week. Use the equation to determine approximately how many hours per week Sam needs to study to earn a 4.0.

Select one:

a. 2.3 hours

b. 4.0 hours

c. 1.7 hours

d. 4.6 hours

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The probability that a family has more than 3 cars among the 100 families polled is approximately 0.11 or 11%.

From the given probability distribution, we can add up the probabilities of owning 4 or 5 vehicles, which are 0.36 and 0.3, respectively. Thus, the probability of a family owning 4 or 5 vehicles is 0.36 + 0.3 = 0.66. To find the probability of a family owning more than 3 cars, we subtract the probability of owning 0, 1, 2, or 3 vehicles from 1, which is the total probability of owning any number of vehicles.

Thus, the probability of owning more than 3 cars is 1 - (0.5 + 0.45 + 0.4 + 0.36) = 0.11 or 11%.

It is important to note that this calculation assumes that the given probability distribution accurately represents the population of interest and that the sample of 100 families is a representative sample.

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The statement is true. If a function f is differentiable at a point a, then the line tangent to f at x = a can be used to approximate values of f near x = a.

The line tangent to f at x = a is the best linear approximation to the function f at x = a. It is the line that passes through the point (a, f(a)) and has a slope equal to the derivative of f at x = a, denoted f'(a). This line is also known as the linearization of f at x = a.

To approximate the value of f at a nearby point x = a + h, where h is a small number, we can use the equation of the tangent line:

y = f(a) + f'(a) * (x - a)

Substituting x = a + h into this equation gives:

y = f(a) + f'(a) * (a + h - a)

y = f(a) + f'(a) * h

Therefore, an approximation for the value of f at x = a + h is given by f(a) + f'(a) * h. This is known as the linear approximation or tangent line approximation of f at x = a.

However, it is important to note that this approximation is only accurate when h is small, and the function f is differentiable at x = a. If h is large or the function is not differentiable at x = a, the approximation may not be accurate.

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The solution to the equation is a = z / -4

This means that if we know the value of "z," we can plug it into this equation to find the value of "a" that satisfies the equation.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

Sure, I can explain this equation for you!

The equation is in the form of "z equals -4a for a," which means we're trying to solve for the variable "a" in terms of "z."

Starting with the equation:

z = -4a

To isolate "a" on one side of the equation, we want to get rid of the coefficient of "-4" that's multiplied by "a".

We can do this by dividing both sides of the equation by "-4":

z / -4 = (-4a) / -4

On the right side, the "-4" in the numerator and the "-4" in the denominator cancel out, leaving only "a":

z / -4 = a

hence, the solution to the equation is a = z / -4

This means that if we know the value of "z," we can plug it into this equation to find the value of "a" that satisfies the equation.

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Answer:

  A

Step-by-step explanation:

You want the figure that is not a polyhedron.

A polyhedron is a solid figure with plane faces. The curved side of figure A means it is not a polyhedron.

Figure A is not a polyhedron.

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The solution to the differential equation [tex]dy/dt = 7y[/tex] satisfying y(3) = 2 is [tex]y(t) = (2/e^(21))e^(7t)[/tex].

A differential equation is a type of mathematical equation that quantifies the pace at which a quantity changes over time. It connects an unknown function to its derivatives and can be used to simulate a variety of real-world occurrences, including fluid movement, disease transmission, and item motion.

We have the differential equation [tex]dy/dt = 7y[/tex] and the initial condition y(3) = 2. Let's find the solution satisfying this condition.

Step 1: Separate the variables. Divide both sides by y to isolate dy:[tex]y(t) = (2/e^(21))e^(7t)[/tex]
[tex](dy/dt)/y = 7[/tex]

Step 2: Integrate both sides with respect to t:
[tex]\int\limits{x} \, (1/y) dy = \int\limits{x} \, 7 dt[/tex]

Step 3: Solve the integrals:
[tex]ln|y| = 7t + C₁[/tex]

Step 4: Solve for y by taking the exponent of both sides:
[tex]y(t) = e^(7t + C₁)[/tex]
Step 5: Rewrite the equation using the constant C:
[tex]y(t) = Ce^(7t)[/tex]

Step 6: Apply the initial condition y(3) = 2 to find C:
[tex]2 = Ce^(7*3)[/tex]

Step 7: Solve for C:
[tex]C = 2/e^(21)[/tex]

Step 8: Write the final solution:
[tex]y(t) = (2/e^(21))e^(7t)[/tex]

So, the solution to the differential equation [tex]dy/dt = 7y[/tex] satisfying y(3) = 2 is [tex]y(t) = (2/e^(21))e^(7t)[/tex].

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The correct answer is the statements for tangent line are (b) I and III only. I) f-1 (10) = 12 (II) (f-1)(12) = 1/7

The tangent line at x=10 is given by y = 7(x-10) + 12, which has a slope of 7. This means that the derivative of f(x) at x=10, f'(10), is equal to 7.

We can use the inverse function theorem to find the derivative of the inverse function f^(-1)(x) at x=12, denoted as (f^(-1))'(12). This is given by:

(f^(-1))'(12) = 1/f'(f^(-1)(12))

Since the tangent line at x=10 is given by y=7(x-10)+12, we know that f(10) = 12. Therefore, f^(-1)(12) = 10. Substituting this into the above equation, we get:

(f^(-1))'(12) = 1/f'(10) = 1/7

So, statement IV is false.

To check the other statements, we can use the fact that f(f^(-1)(x)) = x. Substituting x=10, we get:

f(f^(-1)(10)) = 10

Since f(10) = 12, this implies that f^(-1)(10) = 10/12 = 5/6. Therefore, statement I is true.

Similarly, substituting x=12, we get:

f(f^(-1)(12)) = 12

Since f(10) = 12, this implies that f^(-1)(12) = 10. Therefore, statement II is false, and statement III is true.

In summary, the correct statements are I and III only, so the answer is (b).

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The required, ratio of sides that is equivalent to tan V is SU/VS.

In the given figure,
Consider the triangles VSU and VUT. By applying the tangent function to both triangles, we can establish the following relationships:

The tangent of angle V is equal to the ratio of side SU to side VS, i.e., tanV = SU/VS.

Similarly, the tangent of angle V is also equal to the ratio of side UT to side VU, i.e., tanV = UT/VU.

By utilizing the tangent function in these two triangles, we can derive these equations.

Thus. the required, ratio of sides that is equivalent to tan V is SU/VS.

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The percentag.e of female college runners between 114 - 138 pounds is 82%

Given that X is normally distributed with mean μ = 122 pounds and standard deviation σ = 8 pounds.

We want to find [tex]P(114 < X < 138)[/tex]

To get this, we will standardize X first:

[tex]P(114 < X < 138) = P((114 - 122)/8 < (X - 122)/8 < (138 - 122)/8)[/tex]

= P(-1 < Z < 2)

Using standard normal table, we find that probability of Z falling between -1 and 2 is:

= 0.8186

That means:

[tex]P(114 < X < 138)[/tex] = 0.8186

[tex]P(114 < X < 138)[/tex] = 81.86%

[tex]P(114 < X < 138)[/tex] = 82%

Missing options:

(A)18% (B) 32% (C) 68% (D) 82% (E)95%

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The sampling distribution of sample means for a random sample of 40 dogs will be approximately normally distributed with a mean of 15 pounds and standard deviation of 4 pounds divided by the square root of 40.

This is due to the central limit theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the population distribution. In this case, the large enough sample size (n=40) will allow us to assume normality for the sampling distribution of sample means.
The standard deviation of the sampling distribution (also known as the standard error) is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the standard error is  [tex]\frac{4}{\sqrt{40}} = 0.632[/tex].
Therefore, option C is the correct answer. Option A is incorrect because the sampling distribution is not necessarily strongly skewed right, as the central limit theorem will cause the distribution to approach normality. Option B is incorrect because the standard deviation of the sampling distribution is not 0.632 pounds, but rather the standard error is 0.632 pounds. Option D is incorrect because the standard deviation of the sampling distribution is not the same as the standard error.

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The first partial derivatives of the function f(x, y) = [tex]x^4 - 4xy^9[/tex] are fx(x, y) = 4x³ and fy(x, y) = [tex]-36xy^8[/tex].

To find the first partial derivatives of the function f(x, y) = [tex]x^4 - 4xy^9[/tex], we need to take the partial derivative with respect to each variable separately while treating the other variable as a constant.
The partial derivative of f(x, y) with respect to x (fx) is obtained by differentiating [tex]x^4[/tex] with respect to x, which gives [tex]4x^3[/tex]. The second term [tex]-4xy^9[/tex] does not involve x, so it drops out in the differentiation process. Therefore, fx(x, y) = [tex]4x^3[/tex].
Similarly, the partial derivative of f(x, y) with respect to y (fy) is obtained by differentiating [tex]-4xy^9[/tex] with respect to y, which gives [tex]-36xy^8[/tex]. The first term x^4 does not involve y, so it drops out in the differentiation process. Therefore, fy(x, y) = [tex]-36xy^8[/tex].
In summary, the first partial derivatives of the function f(x, y) = [tex]x^4 - 4xy^9[/tex] are fx(x, y) = 4x³ and fy(x, y) = [tex]-36xy^8[/tex].

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The component that is the reason why AE (Aggregate Expenditure) may be different from GDP (Gross Domestic Product) is unplanned inventory investment.

Unplanned inventory investment occurs when actual sales differ from expected sales, leading to unplanned changes in inventory levels. When firms produce more output than what consumers are willing to buy, the unsold goods accumulate as inventory. On the other hand, when the demand for goods exceeds the production levels, firms may run out of inventory.

The difference between actual inventory levels and planned inventory levels can lead to unplanned changes in inventory investment, which affects GDP. If actual inventory levels are greater than planned inventory levels, this indicates that firms have produced more than what consumers are willing to buy. Therefore, firms will reduce production in the future, leading to a decrease in GDP. Conversely, if actual inventory levels are lower than planned inventory levels, this indicates that firms have produced less than what consumers are willing to buy. Therefore, firms will increase production in the future, leading to an increase in GDP. Thus, unplanned inventory investment plays a significant role in the difference between AE and GDP

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Yes, if x and y are rational numbers, then 3x + 2y is also a rational number. This can be proven using the definition of rational numbers and the closure properties of addition and multiplication.

A rational number is defined as any number that can be expressed as the ratio of two integers, where the denominator is not zero. For example, 3/4, 7/2, and -5/6 are all rational numbers.

Now, let's assume that x and y are rational numbers. Then, by definition, we can write x = p/q and y = r/s, where p, q, r, and s are integers and q and s are not zero.

Using this notation, we can write:

3x + 2y = 3(p/q) + 2(r/s)
= (3p/q) + (2r/s)
= (3ps + 2rq) / qs

Since p, q, r, and s are all integers and qs is not zero, (3ps + 2rq) / qs is also a ratio of two integers where the denominator is not zero. Therefore, 3x + 2y is a rational number.

In conclusion, we can say that if x and y are rational numbers, then 3x + 2y is also a rational number. This result follows directly from the definition of rational numbers and the closure properties of addition and multiplication.

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The Experimental probability of rolling either a 4 or 5 is 0.25 or 25%. To interpret experimental probabilities with caution and to repeat

experiments under different conditions to confirm the results.

The experimental probability of rolling either a 4 or 5, we need to add up the number of times that a 4 or 5 was rolled and divide by the total number of rolls. From the given table, we can see that a 4 was rolled 3 times and a 5 was rolled 5 times. Therefore, the total number of times that either a 4 or 5 was rolled is:

3 (for 4) + 5 (for 5) = 8

The total number of rolls is:

9 (for 1) + 9 (for 2) + 3 (for 3) + 3 (for 4) + 5 (for 5) + 3 (for 6) = 32

Therefore, the experimental probability of rolling either a 4 or 5 is:

8/32 = 1/4 = 0.25

So the experimental probability of rolling either a 4 or 5 is 0.25 or 25%. This means that if the experiment were repeated many times under similar conditions, we would expect to get either a 4 or 5 approximately 25% of the time.

It is important to note that this probability is based on the results of a single experiment, and the true probability may differ if the experiment were repeated many times. Additionally, the results may be influenced by factors such as the quality of the die, the rolling surface, and the technique used to roll the die. Therefore, it is important to interpret experimental probabilities with caution and to repeat experiments under different conditions to confirm the results.

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Based on the characteristics of the line and parabola , the correct answer is: A. [tex]\(y = \begin{cases} x^2 + 2, ; x \leq 1 \\ -x + 2, ; x > 1 \end{cases}\)[/tex]

Based on the given information, let's analyze the characteristics of the line and parabola to determine the correct representation:

1. Line: In the context of graphing, a line appears as a straight line that can extend in any direction across the coordinate plane. It can have a positive or negative slope, or be horizontal or vertical.

- The line passes through the points [tex](1,1),(2,0) , (4, -2) , ( 8 , -6)[/tex]

- It extends along the first and fourth quadrants.

- A closed dot is shown at the point (1,1).

2. Parabola: In the context of graphing, a parabola appears as a curved line. It can open upward or downward and can be concave or convex. The vertex of the parabola represents the lowest or highest point on the curve, and the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric halves.

- The parabola passes through the points [tex](1,3) , (-2,6), (10,-3)[/tex]

- It extends along the first and second quadrants.

- An open dot is shown at the point (1,3).

- The vertex of the parabola lies at (0,2).

Given these characteristics, we can determine the correct representation:

The correct answer is:

A. [tex]\(y = \begin{cases} x^2 + 2, ; x \leq 1 \\ -x + 2, ; x > 1 \end{cases}\)[/tex]

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If function "z = x² - xy + 4y²" and (x, y) changes from interval (1, -1) to (1.03, -0.95), then the value of dz is 11.46, and Δz is 0.46.

The "multi-variable" function z = f(x,y) is given to be : x² - xy + 4y²;

Differentiating the function "z" with respect to "x",

We get,

dz/dx = 2x - y + 0

dz/dx = 2x - y,     ...equation(1)

Differentiating the function "z" with respect to "y",

We get,

dz/dy = 0 - x.1 + 8y,

dz/dy = 8y - x,      ...equation(2)

So, the "total-derivative" of "z" can be written as :

dz = (2x - y)dx + (8y - x)dy,

Given that "z" changes from (1, -1) to (1.03, -0.95);

So, we substitute, (x,y) as (1, -1), and (dx,dy) = (1.03, -0.95),

We get,

dz = (2(-1)-1)(1.03 + 1) + (8(-9) -1)(-0.95 -1),

dz = (-3)(2.03) + (-9)(-1.95),

dz = -6.09 + 17.55,

dz = 11.46.

Now, we compute Δz,

The z-value corresponding to (1,-1),

z₁ = (1)² - (1)(-1) + 4(-1)² = -2, and

The z-value corresponding to (1.03, -0.95),

z₂ = (1.03)² - (1.03)(-0.95) + 4(-0.95)² = -1.57.

So, Δz = z₂ - z₁ = -1.57 -(-2) = 0.46.

Therefore, the value of dz is 11.46, and Δz is 0.46.

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The given question is incomplete, the complete question is

If the function z = x² - xy + 4y² and (x, y) changes from (1, -1) to (1.03, -0.95), Compare the values of dz and Δz.

Therefore, the average value of the function f(x) = 25 - x^2 on the interval [0, 5] is 50/3.

To find the average value of the function f(x) = 25 - x^2 on the interval [0, 5], we need to calculate the definite integral of the function over the interval and divide it by the length of the interval.

The average value (AV) is given by the formula:

AV = (1 / (b - a)) * ∫[a to b] f(x) dx

In this case, a = 0 and b = 5, so the average value becomes:

AV = (1 / (5 - 0)) * ∫[0 to 5] (25 - x^2) dx

Simplifying, we have:

AV = (1/5) * ∫[0 to 5] (25 - x^2) dx

To evaluate the integral, we integrate term by term:

AV = (1/5) * [25x - (x^3 / 3)] evaluated from 0 to 5

AV = (1/5) * [(255 - (5^3 / 3)) - (250 - (0^3 / 3))]

AV = (1/5) * [(125 - (125 / 3)) - 0]

AV = (1/5) * [(375/3 - 125/3)]

AV = (1/5) * (250/3)

AV = 250/15

AV = 50/3

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Answer:

C. 40

Step-by-step explanation:

36/(5x+13)=30/(6x-2)

Cross multiply

36·(6x-2)=30·(5x+13)

216x-72=150x+390

216x-150x=390+72

66x=462

x=7

Substituting 7 in for x,

6(7)-2

42-2

40

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{4\% of 6000}}{\left( \cfrac{4}{100} \right)6000}\implies 240[/tex]

The fee for a private vehicle, v, is $2.

We have,

Let's set up the equation to find v, the fee for a private vehicle.

The given information states that during peak visiting time, the total earnings from entrance fees and reservations is $115,200, which is 3,600 times the sum of $30, and v.

We can write the equation as:

3,600(30 + v) = 115,200

To solve for v, we can begin by simplifying the equation:

108,000 + 3,600v = 115,200

Next, we isolate the term with v by subtracting 108,000 from both sides of the equation:

3,600v = 115,200 - 108,000

3,600v = 7,200

Finally, we solve for v by dividing both sides of the equation by 3,600:

v = 7,200 / 3,600

v = 2

Therefore,

The fee for a private vehicle, v, is $2.

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The answer d. To examine points in a control chart to check for nonrandom variability.

A random variable is a mathematical function that maps outcomes of a random event or experiment to numerical values. In other words, it assigns a numerical value to each outcome of a random event or experiment.

A run test is not typically used for acceptance sampling, but it can be used to examine points in a control chart to check for nonrandom variability. Control charts are used to monitor a process over time and detect any patterns or trends in the data that may indicate the presence of non-random variability, such as a shift, trend, or cycle.

A run test is a statistical test that examines patterns or runs of consecutive data points above or below the centerline on a control chart, which may indicate nonrandom variability.

If a significant run is detected, it may signal the need for further investigation and corrective action to address the underlying cause of the variation.

Therefore,

The answer d. To examine points in a control chart to check for nonrandom variability.

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Answer: I'm in 2nd grade

Step-by-step explanation: Fortnight is a acrobatic FIRST PERSON SHOOTER GAME

For consumption smoothers, the marginal propensity to consume out of anticipated changes in income is one. Option 4 is answer.

Consumption smoothers are individuals who smooth out their consumption patterns in the face of anticipated changes in income. In other words, they tend to spend a smaller portion of any additional income than those who do not smooth their consumption. Therefore, the marginal propensity to consume out of anticipated changes in income is one, meaning that for every additional unit of anticipated income, consumption increases by one unit. Option 4 is the correct answer.

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The planning value for the population standard deviation is estimated to be 13. The sample size needed for a margin of error of 47 at 95% confidence is 36, and the sample size needed for a margin of error of 27 at 95% confidence is 91.

The range of a data set is used to estimate the population standard deviation (σ) using the formula σ ≈ range/4. Therefore, in this case, the planning value for the population standard deviation is estimated to be 52/4 = 13.

To find the sample size needed to provide a margin of error of 47 at 95% confidence, we can use the formula n = (z^2 * σ^2)/E^2, where z is the z-score corresponding to the confidence level (1.96 for 95% confidence), σ is the estimated population standard deviation, and E is the margin of error. Substituting the given values, we get n = (1.96^2 * 13^2)/47^2 ≈ 36. Therefore, a sample size of 36 or more would be needed to provide a margin of error of 47 at 95% confidence.

To find the sample size needed to provide a margin of error of 27 at 95% confidence, we can use the same formula as above. Substituting the given values, we get n = (1.96^2 * 13^2)/27^2 ≈ 91. Therefore, a sample size of 91 or more would be needed to provide a margin of error of 27 at 95% confidence.

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