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The exponential decay function that satisfies the given conditions is:

[tex]f(x) = 4 * (1/2)^x[/tex].

In this equation, the y-intercept is 4, which means that when x = 0, the function value is 4. As x increases by 1, the function decreases by a factor of one-half. This behavior is captured by raising 1/2 to the power of x in the equation.

The base of the exponent, 1/2, ensures that the function decreases exponentially. When x = 1, the exponent becomes 1, and[tex]1/2^1[/tex] equals 1/2. This means that the function value decreases to half of its previous value. Similarly, when x = 2, the exponent becomes 2, and[tex]1/2^2[/tex] equals 1/4. The function value decreases to one-fourth of its previous value, and so on.

By multiplying the exponential term by 4, we ensure that the y-intercept is 4. This scaling factor allows us to control the initial value of the function and match the given condition.

The exponential decay function[tex]f(x) = 4 * (1/2)^x[/tex] represents a decaying process where the y-values decrease exponentially as x increases, while starting at a y-intercept of 4.

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The angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

To find the angle the slide forms with the tower, we can use trigonometric ratios. Let's consider the right triangle formed by the height of the tower (18 m), the length of the slide (20 m), and the angle we want to find.

Using the tangent function, we have:

tan(angle) = opposite/adjacent

In this case, the opposite side is the height of the tower (18 m) and the adjacent side is the length of the slide (20 m). Therefore:

tan(angle) = 18/20

To find the angle, we can take the inverse tangent (arctan) of both sides:

angle = arctan(18/20)

Using a calculator, we find that arctan(18/20) is approximately 40.56 degrees.

Therefore, the angle the slide forms with the tower is approximately 41 degrees (rounded to the nearest degree).

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The cube root of 30 would be between 3 and 4, but closer to 3.

Cube root is the number that needs to be multiplied three times to get the original number.

The cube root of a number can be determined by using the prime factorization method. In order to find the cube root of a number:

Step 1: Start with the prime factorization of the given number.

Step 2: Then, divide the factors obtained into groups containing three same factors.

Step 3: After that, remove the cube root symbol and multiply the factors to get the answer. If there is any factor left that cannot be divided equally into groups of three, that means the given number is not a perfect cube and we cannot find the cube root of that number.

We have to find the cube root of 30.

Prime factorization of 30 = [tex]2\times3\times5[/tex].

Therefore the cube root of 30 = [tex]\sqrt[3]{ (2\times3\times5)}= \sqrt[3]{30}[/tex].

As [tex]\sqrt[3]{30}[/tex] cannot be reduced further, then the result for the cube root of 30 is an irrational number as well.

So here we will use approximation method to find the cube root of 30 using Halley's approach:

Halley’s Cube Root Formula:

[tex]{\sqrt[3]{\text{a}} = \dfrac{\text{x}[(\text{x}^3 + 2\text{a})}{(2\text{x}^3 + \text{a})]}}[/tex]

The letter “a” stands in for the required cube root computation.

Take the cube root of the nearest perfect cube, “x” to obtain the estimated value.

Here we have a = 30

and we will substitute x = 3 because 3³ = 27 < 30 is the nearest perfect cube.

Substituting a and x in Halley's formula,

[tex]\sqrt[3]{30} = \dfrac{3[(3^3 + 2\times30)}{(2\times3^3 + 30)]}[/tex]

      [tex]= \dfrac{3[(27+60)}{(54+30)]}[/tex]

      [tex]= 3\huge \text(\dfrac{87}{84} \huge \text)[/tex]

      [tex]= 3\times1.0357[/tex]

[tex]\bold{\sqrt[3]{30} = 3.107}[/tex].

Hence, the cube root of 30 is 3.107.

Therefore, we can conclude that the cube root of 30 would be between 3 and 4, but closer to 3.

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Complete question:

Describe in words where cube root of 30 would be plotted on a number line.

A. Between 3 and 4, but closer to 3

B. Between 3 and 4, but closer to 4

C. Between 2 and 3, but closer to 2

D. Between 2 and 3, but closer to 3

The correct answer is option (a): Var(X) = E(X^2) - (E(X))^2.

The variance of a random variable X is defined as the average of the squared differences between each value of X and its expected value (E(X)). Mathematically, it can be expressed as Var(X) = E((X - E(X))^2).

Expanding the squared term, we have Var(X) = E(X^2 - 2XE(X) + (E(X))^2). Distributing and rearranging, we get Var(X) = E(X^2) - 2E(X)E(X) + (E(X))^2. Simplifying, we obtain Var(X) = E(X^2) - (E(X))^2.

The given sequence can be generated by multiplying each term by 2, starting from the initial term of 6.

The pattern that fits the given sequence 6, 12, 24, 48, 96, ... is that each term is 2 times the previous term.

In the sequence 6, 12, 24, 48, 96, ... there are multiple possible patterns, each resulting from a different rule applied to generate the next term. Let's examine each of the proposed patterns:

Each term is 6 more than the previous term:

Starting with 6, if we add 6 to each term, we get:

6 + 6 = 12

12 + 6 = 18

18 + 6 = 24

24 + 6 = 30

30 + 6 = 36

...

This pattern does not match the given sequence since it does not produce the subsequent terms.

Each term is 12 more than the previous term:

Starting with 6, if we add 12 to each term, we get:

6 + 12 = 18

18 + 12 = 30

30 + 12 = 42

42 + 12 = 54

54 + 12 = 66

...

This pattern also does not match the given sequence.

Each term is 1/2 the previous term:

Starting with 6, if we multiply each term by 1/2, we get:

6 [tex]\times[/tex] 1/2 = 3

3 [tex]\times[/tex] 1/2 = 1.5

1.5 [tex]\times[/tex] 1/2 = 0.75

0.75 [tex]\times[/tex] 1/2 = 0.375

0.375 [tex]\times[/tex] 1/2 = 0.1875

...

This pattern does not match the given sequence.

Each term is 2 times the previous term:

Starting with 6, if we multiply each term by 2, we get:

6 [tex]\times[/tex] 2 = 12

12 [tex]\times[/tex] 2 = 24

24 [tex]\times[/tex]2 = 48

48 [tex]\times[/tex]2 = 96

96 [tex]\times[/tex]2 = 192

This pattern perfectly matches the given sequence. Each term is indeed 2 times the previous term, resulting in the next term.

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Answer:

x = 8

Step-by-step explanation:

The given diagram shows a triangle with the length of two sides and its included angle.

To find the value of the missing variable x, we can use the Law of Cosines.

[tex]\boxed{\begin{minipage}{6 cm}\underline{Law of Cosines} \\\\$c^2=a^2+b^2-2ab \cos C$\\\\where:\\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides.\\ \phantom{ww}$\bullet$ $C$ is the angle opposite side $c$. \\\end{minipage}}[/tex]

From inspection of the given triangle:

Substitute the values into the formula and solve for x:

[tex]\begin{aligned}x^2&=18^2+21^2-2(18)(21)\cos 22^{\circ}\\x^2&=324+441-756\cos 22^{\circ}\\x^2&=765-756\cos 22^{\circ}\\x&=\sqrt{765-756\cos 22^{\circ}}\\x&=8.00306228...\\x&=8\end{aligned}[/tex]

Therefore, the value of the missing variable x is x = 8, rounded to the nearest hundredth.

The calculated value of x in the triangle is 15

From the question, we have the following parameters that can be used in our computation:

The triangles ABC and EDC

Since the triangles are similar, then we have

(3x - 5)/(5x - 5) = 32/56

This gives

32(5x - 5) = 56(3x - 5)

When solved for x, we have

x = 15

Hence, the value of x is 15

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If there are 200 high school students in the district, the number of high school students expected to be in Chemistry is 60 because the percentage who offer Chemistry in the district is 30%.

The number of high school students who offer Chemistry in the district can be determined by multiplying the total number of high school students and the percentage of students who offer Chemistry.

The result of a multiplication operation (multiplicand and multiplier), which is one of the basic mathematical operations, is known as the product.

The total number of high school students in the district = 200

The percentage of students who offer Chemistry in the district = 30%

The number of students likely to be offering Chemistry in the district = 60 (200 x 30%).

Thus, we can conclude that 60 high school students are in Chemistry based on the Chemistry percentage.

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The percentage of high school students in the district who offer Chemistry is 30%.  If there are 200 high school students in the district, how many would you expect to be in Chemistry?

One of the transformations undergone by Triangle 1 is a rotation, which involves turning the triangle around a fixed point while preserving its shape and size.

A rotation is a transformation that turns an object around a fixed point, known as the center of rotation. In the given results, if the triangle appears in a different orientation but retains its shape and size, it indicates a rotation.

During a rotation, each point of the triangle is moved along a circular path around the center of rotation. The distance from the center of rotation remains constant, and the angle between any two corresponding points on the original and rotated triangles is preserved. The direction of rotation can be clockwise or counterclockwise, depending on the given results.

To describe a rotation, we need to specify the angle of rotation and the direction. For example, "Triangle 1 underwent a counterclockwise rotation of 90 degrees" would indicate that the triangle was rotated by 90 degrees in the counterclockwise direction.

The specific rotation can be described by stating the angle of rotation and the direction.

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The correct representation for the joint probability of two dependent events Y and Z is P(Y) * P(Z|Y). Option E

The joint probability of two dependent events Y and Z can be written as the probability of Y occurring multiplied by the conditional probability of Z given Y. This can be represented as P(Y) * P(Z|Y).

Here's the justification:

P(Y) represents the probability of event Y occurring independently.

P(Z|Y) represents the conditional probability of event Z occurring given that event Y has already occurred.

When Y and Z are dependent events, the occurrence of Y affects the probability of Z happening. Therefore, we need to consider the probability of Y occurring first (P(Y)) and then the probability of Z occurring given that Y has already occurred (P(Z|Y)).

Multiplying these two probabilities together gives us the joint probability of both Y and Z occurring simultaneously, which is denoted as P(Y and Z).

Hence, the correct representation for the joint probability of two dependent events Y and Z is P(Y) * P(Z|Y). Option E.

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The sixth term of the geometric sequence is 2048.

The given geometric sequence is 2, 8, 32, 128. We can observe that each term is obtained by multiplying the previous term by 4. Therefore, the common ratio (r) of the sequence is 4.

The formula for the nth term (an) of a geometric sequence is given by:

an = a1 * r^(n-1)

where a1 is the first term and r is the common ratio.

For this sequence, a1 = 2 and r = 4. Plugging in these values into the formula, we get:

an = 2 * 4^(n-1)

To find a6, we substitute n = 6 into the formula:

a6 = 2 * 4^(6-1)

  = 2 * 4^5

  = 2 * 1024

  = 2048

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The Probable question may be:
Write an equation for the nth term of the geometric sequence 2, 8, 32, 128,

Then find a6. Round to the nearest tenth if necessary.

a = 5×4 X

a1 = n-1 X

We have created three different linear systems using the equation 20x + 12y = 24.

System 1 has infinitely many solutions, System 2 has no solution, and System 3 has a unique solution.

Let's create three different linear systems using the equation 20x + 12y = 24 and ensure that each system has a different number of solutions.

System 1:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 40x + 24y = 48

Explanation: In this system, we multiplied both sides of the given equation by 2 to create Equation 2.

By doing so, we have essentially created two equations that are multiples of each other.

Since the equations are equivalent, they represent the same line, and the system has infinitely many solutions.

Any values of x and y that satisfy the first equation will automatically satisfy the second equation as well.

System 2:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 20x + 12y = 48

Explanation: In this system, we changed the constant term in Equation 2 to 48.

By doing so, we have created two parallel lines with the same slope. Since the lines are parallel, they will never intersect, and the system has no solution.

There are no values of x and y that satisfy both equations simultaneously.

System 3:

Equation 1: 20x + 12y = 24 (given)

Equation 2: 40x + 24y = 48

Explanation: In this system, we multiplied both sides of Equation 2 by 2 to create Equation 2.

By doing so, we have created two equations that have the same slope but different y-intercepts.

Since the lines are not parallel and have different y-intercepts, they will intersect at a single point, and the system has a unique solution.

There will be one specific pair of values for x and y that satisfy both equations simultaneously.

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Statement that is always true about obtuse triangles:

An obtuse triangle always has one angle that measures more than 90 degrees.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

An obtuse angle is an angle that measures more than 90 degrees. Since an obtuse triangle is defined as having one obtuse angle, it will always have an angle that measures more than 90 degrees.

Therefore, the statement that an obtuse triangle always has one angle that measures more than 90 degrees is always true.

Statement that is sometimes true about obtuse triangles:

An obtuse triangle can have different side lengths.

In the given tree diagram, the "Obtuse" category represents triangles with at least one obtuse angle.

The "Scalene" category represents triangles with different side lengths. Therefore, it is possible for an obtuse triangle to have different side lengths, making the statement "An obtuse triangle can have different side lengths" sometimes true.

However, it is also possible for an obtuse triangle to have two or more sides with the same length, which would make it an isosceles or equilateral triangle.

Hence, the statement is only sometimes true and not always true.

In summary, an always true statement about obtuse triangles is that they always have one angle that measures more than 90 degrees.

A sometimes true statement about obtuse triangles is that they can have different side lengths.

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Answer:

(f + g)(1) = 5

Step-by-step explanation:

(f + g) means we are going to add f(x) and g(x). But also, the (1) part means we are going to let x be equal to 1. We're going to fill in 1 in place of x. You can do this in either order.

Generally speaking its "easier" to fill in the 1 for x first and then do the adding part.

f(x) = x + 6

f(1) = 1 + 6 = 7

and,

g(x) = x - 3

g(1) = 1 - 3 = -2

add the 7 and -2 together:

7 + - 2

= 5

It works out the same if you add first:

f(x) + g(x)

= x + 6 + x - 3

= 2x + 3

then put the 1 in:

= 2×1 + 3

= 2 + 3

= 5

Hope this helps!

Answer: substitute the given number for the variable in the expression and then simplify the expression using the order of operations

Step-by-step explanation:3a2 - 4b2 for a = -3/4 and b = 1/2

The regression equation is y = 17.1643X - 2.47977

To solve this problem, we have to calculate the equation of regression.

Sum of X = 2.97

Sum of Y = 28.66

Mean X = 0.33

Mean Y = 3.1844

Sum of squares (SSX) = 0.3552

Sum of products (SP) = 6.0959

Regression Equation = y = bX + a

b = SP/SSX = 6.1/0.36 = 17.1643

a = MY - bMX = 3.18 - (17.16*0.33) = -2.47977

y = 17.1643X - 2.47977

The line of best fit is y = 17.1643X - 2.47977

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By using the change of base formula: The solution to the equation log(base y) (X-2) = 5 is [tex]X = y^5 + 2.[/tex]

To solve the equation log(base y) (X-2) = 5 using the change of base formula, we can rewrite the equation as log(base b) (X-2) / log(base b) y = 5.

Using the change of base formula, we can choose any base for b.

Let's choose base 10 for simplicity.

So the equation becomes log(base 10) (X-2) / log(base 10) y = 5.

We know that log(base 10) (X-2) represents the logarithm of (X-2) to the base 10, and log(base 10) y represents the logarithm of y to the base 10.

Now, to solve for X, we can isolate it by multiplying both sides of the equation by log(base 10) y:

log(base 10) (X-2) = 5 [tex]\times[/tex] log(base 10) y.

This simplifies to:

log(base 10) (X-2) [tex]= log(base 10) y^5.[/tex]

Since the logarithms on both sides have the same base, we can remove the logarithm and equate the arguments:

[tex]X - 2 = y^5.[/tex]

Now we can solve for X by adding 2 to both sides:

[tex]X = y^5 + 2.[/tex]

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Answer:

volume=60cm3, surface area=96cm2

Step-by-step explanation:

volume=1/3×(6×6)×5

=60cm3

surface area= 4(1/2×6×5)+(6×6)

=96cm2

The calculated vertex of the function y = 2(x + 4)(x - 2) is (-1, -18)

From the question, we have the following parameters that can be used in our computation:

y = 2(x + 4)(x - 2)

Expand the equation

So, we have

y = 2x² + 4x - 16

Differentiate the function and set to 0

So, we have

4x + 4 = 0

So, we have

4x = -4

Evaluate

x = -1

Next, we have

y = 2(-1 + 4)(-1 - 2)

Evaluate

y = -18

This means that the vertex is (-1, -18)

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Answer:

[tex](x+5)^2+(y+5)^2=25[/tex]

Step-by-step explanation:

Recall that the equation of a circle with center (h,k) and radius "r" is [tex](x-h)^2+(y-k)^2=r^2[/tex]

Since the center of the circle is (h,k)=(-5,-5) and the radius is r=5, then our equation will be [tex](x-(-5))^2+(y-(-5))^2=5^2[/tex] which can be simplified into [tex](x+5)^2+(y+5)^2=25[/tex]

Hola!

-4 - 6

= -10

the answer is -10

Given the equation f(x) = a · bx where a and b are constants. So, the answer to the given problem is g(x) = a · bx + h, and the explanation of the trigonometric function.

To find the equation of transformed function g in terms of function f is explained below: If f(x) = a · bx, then the transformed function g(x) can be represented by g(x) = a · bx + h, where h is the vertical shift (if h > 0, the graph shifts upward, and if h < 0, the graph shifts downward).

Now, we have to replace the value of 'a' to complete the equation of g(x). But, we don't have any value of 'a' provided in the question. Hence, we can't determine the equation of transformed function g in terms of function f for the given information.

Next, let's move to the trigonometric function. It is given that: R S 9 sin cos tan sin cos tan-¹ /ASin, Cos, Tan, Cosec, Sec, and Cot are six trigonometric functions. Let's see their definitions and their corresponding inverse functions:

1. Sine: It is defined as the ratio of the length of the side opposite the given angle to the length of the hypotenuse in a right-angled triangle. Its corresponding inverse function is sin⁻¹.

2. Cosine: It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. Its corresponding inverse function is cos⁻¹.

3. Tangent: It is defined as the ratio of the length of the side opposite the given angle to the length of the adjacent side in a right-angled triangle. Its corresponding inverse function is tan⁻¹.

4. Cosecant: It is defined as the ratio of the length of the hypotenuse to the length of the side opposite the given angle in a right-angled triangle. Its corresponding inverse function is cosec⁻¹.

5. Secant: It is defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. Its corresponding inverse function is sec⁻¹.

6. Cotangent: It is defined as the ratio of the length of the adjacent side to the length of the side opposite the given angle in a right-angled triangle. Its corresponding inverse function is cot⁻¹.

Hence, the answer to the given problem is g(x) = a · bx + h, and the explanation of the trigonometric function.

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Answer: y = 12.6

Step-by-step explanation:

Since x = 2 and y = 6.3 * x, y = 6.3 * 2.

6.3 * 2 is equal to 12.6, so y is 12.6.

Answer:

y = 12.6

Step-by-step explanation:

y = 6.3x                     x = 2

Solve for y.

y = 6.3(2)

y = 12.6

So, the answer is 12.6

Incremental cash flows associated with the donation of a valuable painting from a corporation's private collection may include It's important to note that any direct costs associated with the donation.

Tax benefits: The corporation may be eligible for tax deductions or credits for charitable donations, which could result in a reduction in its tax liability and generate cash flow savings.

Opportunity cost: If the corporation could have sold the painting instead of donating it, the incremental cash flow would be the potential proceeds from the sale.

Storage and maintenance cost savings: By donating the painting to the art museum, the corporation no longer has to incur expenses for storing, insuring, and maintaining the artwork, resulting in cost savings.

Public relations and marketing benefits: Donating the painting can enhance the corporation's reputation and generate positive publicity, potentially leading to increased customer goodwill and brand value, which can translate into future cash flows.

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The final answer after evaluating the expression 3.14([tex]a^{2}[/tex] + ab) (by putting the value a = 3 and b = 4) is 65.94.

When a = 3 and b = 4, we substitute the supplied values into the expression to assess 3.14([tex]a^{2}[/tex] + ab):

3.14([tex]3^{2}[/tex] + 3 * 4)

We begin by solving the exponent:

[tex]3^{2}[/tex] = 3 * 3 = 9

The values are then entered into the expression:

3.14(9 + 3 * 4)

Inside the brackets, multiply the result:

3.14(9 + 12)

The numbers in the brackets are added:

3.14(21)

The decimal number is now multiplied by 21:

3.14 * 21 = 65.94

The evaluated expression is 65.94 as a result.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

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The answer is:

Work/explanation:

We're asked to evaluate the expression [tex]\sf{3.14(a^2+ab)}[/tex] for a = 3 and b = 4.

Plug in the data:

[tex]\sf{3.14(3^2+3*4)}[/tex]

[tex]\sf{3.14(9+12)}[/tex]

[tex]\sf{3.14(21)}[/tex]

[tex]\bf{65.94}[/tex]

The amount of snow accumulated between 2 am and 5 am is: 1.25 inches

The general formula for the equation of a line in slope intercept form is:

y = mx + c

where:

m is slope

c is y-intercept

From the given graph attached, we see that the y-axis gives the amount of snow at different specific times.

Meanwhile the x-axis gives the time in hours after midnight

At 2am, the y-axis value is 1.25 inches, and as such at 2am snow accumulation was 1.25 inches.

At 5 am, the y-axis value reads 2.5 inches, and as such at 5am snow accumulation was 2.5 inches.

The difference in both snow accumulations is:  2.5 - 1.25 = 1.25

Hence, 1.25 inches snow accumulated between 2 am and 5 am.

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a. Mean = 3

Standard deviation = 2

b. The z-scores for each score in the sample are: 1, -1.5, 0.5, 1, -1, -0.5, 0.5.

a. To compute the mean and standard deviation for the sample, we follow these steps:

Calculate the mean (average)

Mean = (sum of all scores) / (number of scores)

Mean = (5 + 0 + 4 + 5 + 1 + 2 + 4) / 7

Mean = 21 / 7

Mean = 3

The mean of the sample is 3.

Calculate the standard deviation

The formula for standard deviation for a sample is given by:

Standard deviation = sqrt((sum of squared differences from the mean) / (number of scores - 1))

First, calculate the squared differences from the mean for each score:

(5 - 3)^2 = 4

(0 - 3)^2 = 9

(4 - 3)^2 = 1

(5 - 3)^2 = 4

(1 - 3)^2 = 4

(2 - 3)^2 = 1

(4 - 3)^2 = 1

Next, sum up these squared differences:

4 + 9 + 1 + 4 + 4 + 1 + 1 = 24

Now, divide this sum by (number of scores - 1):

24 / (7 - 1) = 24 / 6 = 4

Finally, take the square root of this result:

Standard deviation = sqrt(4) = 2

The standard deviation of the sample is 2.

b. To find the z-score for each score in the sample, we use the formula:

z = (X - Mean) / Standard deviation

For each score, we substitute the values into the formula:

X = 5, z = (5 - 3) / 2 = 2 / 2 = 1

X = 0, z = (0 - 3) / 2 = -3 / 2 = -1.5

X = 4, z = (4 - 3) / 2 = 1 / 2 = 0.5

X = 5, z = (5 - 3) / 2 = 2 / 2 = 1

X = 1, z = (1 - 3) / 2 = -2 / 2 = -1

X = 2, z = (2 - 3) / 2 = -1 / 2 = -0.5

X = 4, z = (4 - 3) / 2 = 1 / 2 = 0.5

The z-scores for each score in the sample are:

z = 1, z = -1.5, z = 0.5, z = 1, z = -1, z = -0.5, z = 0.5

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