Hannah and Becky are learning to type on a computer keyboard. Hannah's
typing speed is represented by the equation y = 11x where y is the number of
words she types and x is the number of minutes. Becky's typing speed is
given by the graph.
Number of words
60
48
36
24
12
2
3
Time (minutes)
Choose the statement that correctly compares their unit rates.
Adr
A. Hannah's unit rate is 2 more words per minute than Becky's unit
rate.
B. Hannah's unit rate is 1 more word per minute than Becky's unit
rate.
C. Hannah's unit rate is equal to Becky's unit rate.
D. Hannah's unit rate is 1 fewer word per minute than Becky's unit
rate

The rates of change of the two functions that compare correctly is A. The rate of change of function A is 6, and the rate of change of function B is -10/9.

To compare the rates of change of the two functions, we can calculate the slope of each function. The slope represents the rate of change of a linear function.

For Function A, the equation is y = 6x - 1. The coefficient of x, which is 6, represents the slope. The rate of change of Function A is 6.

For Function B, we are given three points: (13, 0), (4, 10), and (6, 15). We can calculate the slope using the formula: slope = (change in y) / (change in x). Taking the first two points, we have: slope = (10 - 0) / (4 - 13) = 10 / (-9) = -10/9.

Comparing the rates of change, we have:

A. The rate of change of function A is 6.

The rate of change of function B is -10/9.

The correct option is A. The rate of change of function A is 6, and the rate of change of function B is -10/9.

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Answer:

Below, rises, more, 2, 5

Step-by-step explanation:

Its an positive exponential growth function which means it increases sharply to the right it also is above the x-axis.

The graph is 2 times the graph of 5^x so its steeper or rises more

Plug in 0 to get 1*2=2

Plug in 1 to get 5*1=5

Answer:

Step-by-step explanation:

0

Answer:

0

Step-by-step explanation:

25 ÷ 5+7-(4 x 3)

25 ÷ 5+7-(4 x 3)

5+7-(4 x 3)

5+7-(4 x 3)

5+7-12

12-12

0

I wrote in bold the steps you need to follow using PEMDAS (Parentheses, Exponents, Multiplication and Divison left to right, and Addition and Subtraction left to right).

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]

Answer:

Step-by-step explanation:

Let the centre be C.

Since TR is a straight line,  

∠SCT + ∠SCR = 180

∠SCT = 180 - 53

∠SCT = 127

The angle of a semicircle is 180°. Minor arcs are arcs less than a semicircle i.e. less than 180° and major arcs are arcs greator than a semicircle i.e. greater than 180°.

a) arc(SPT) has measure of 90 + 65 + 25 + 53 = 233° > 180° and hence a major arc

Also 1° = π/180 radians

233° = 233 * π/180 = 1.29π radians

b) arc(ST) has measure of 127° < 180° and hence a minor arc

127° = 127 * π/180 = 0.71π radians

c) arc(RST) has a measure of  53 + 127 = 180° which is a semicircle

180° = 180* π/180 = π radians

d) arc(SP) has a measure of 53 + 25 + 65 = 143° < 180° and hence a minor arc

143° = 143* π/180 = 0.79π radians

e) arc(QST) has a measure of 25 + 53 + 127 = 205° > 180° and hence a major arc

205° = 205 * π/180 = 1.14π radians

f) arc(TQ) has a measure of 90 + 65 = 155° < 180° and hence a minor arc

155° = 155 * π/180 = 0.86π radians

Answer:

[tex]\text{a.} \quad \text{Major arc}:\;\;\overset{\frown}{SPT}=233^{\circ}[/tex]

[tex]\text{b.} \quad \text{Minor arc}:\;\;\overset{\frown}{ST}=127^{\circ}[/tex]

[tex]\text{c.} \quad \text{Semicircle}:\;\;\overset{\frown}{RST}=180^{\circ}[/tex]

[tex]\text{d.} \quad \text{Minor arc}:\;\;\overset{\frown}{SP}=143^{\circ}[/tex]

[tex]\text{e.} \quad \text{Major arc}:\;\;\overset{\frown}{QST}=205^{\circ}[/tex]

[tex]\text{F.} \quad \text{Minor arc}:\;\;\overset{\frown}{TQ}=155^{\circ}[/tex]

Step-by-step explanation:

A major arc is an arc in a circle that measures more than 180°.

It is named with three letters: two endpoints and a third point on the arc.

A minor arc is an arc in a circle that measures less than 180°.

It is named with two letters: its two endpoints.

A semicircle is a special case of an arc that measures exactly 180°.  

The endpoints of the semicircle are located on the diameter, and the semicircle divides the circle into two equal parts.

The measure of an arc of a circle is equal to the measure of its corresponding central angle.

[tex]\hrulefill[/tex]

a)  Arc SPT is a major arc since it is named with three letters.

It begins at point S, passes through point P, and ends at point T.

    [tex]\begin{aligned}\overset{\frown}{SPT}&=\overset{\frown}{SR}+\overset{\frown}{RQ}+\overset{\frown}{QP}+\overset{\frown}{PT}\\&=53^{\circ}+25^{\circ}+65^{\circ}+90^{\circ}\\&=233^{\circ}\end{aligned}[/tex]

[tex]\hrulefill[/tex]

b)  Arc ST is a minor arc since it is named with two letters.

It is measured in a counterclockwise direction from point S to point T.

    [tex]\begin{aligned}\overset{\frown}{ST}&=360^{\circ}-\overset{\frown}{SPT}\\&=360^{\circ}-233^{\circ}\\&=127^{\circ}\end{aligned}[/tex]

[tex]\hrulefill[/tex]

c)  Arc RST is a semicircle.

Arc RST is a semicircle since its endpoints are located on the diameter of the circle, RT.

    [tex]\overset{\frown}{RST}=180^{\circ}[/tex]

[tex]\hrulefill[/tex]

d)  Arc SP is a minor arc since it is named with two letters.

It is measured in a clockwise direction from point S to point P.

(If it was measured in a counterclockwise direction, it would be a major arc, as it would be more than 180°, and therefore would be named using three letters).

    [tex]\begin{aligned}\overset{\frown}{SP}&=\overset{\frown}{SR}+\overset{\frown}{RQ}+\overset{\frown}{QP}\\&=53^{\circ}+25^{\circ}+65^{\circ}\\&=143^{\circ}\end{aligned}[/tex]

[tex]\hrulefill[/tex]

e)  Arc QST is a major arc since it is named with three letters.

It begins at point Q, passes through point S, and ends at point T.

    [tex]\begin{aligned}\overset{\frown}{QST}&=\overset{\frown}{QR}+\overset{\frown}{RS}+\overset{\frown}{ST}\\&=25^{\circ}+53^{\circ}+127^{\circ}\\&=205^{\circ}\end{aligned}[/tex]

[tex]\hrulefill[/tex]

f)  Arc TQ is a minor arc since it is named with two letters.

It is measured in a counterclockwise direction from point T to point Q.

(If it was measured in a clockwise direction, it would be a major arc, as it would be more than 180°, and therefore would be named using three letters).

    [tex]\begin{aligned}\overset{\frown}{TQ}&=\overset{\frown}{TP}+\overset{\frown}{PQ}\\&=90^{\circ}+65^{\circ}\\&=155^{\circ}\end{aligned}[/tex]

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!

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:

a. 16

c. 120°

e. 60°

Step-by-step explanation:

Properties of Parallelogram:

For Question:

a.

DC= AB=16 Opposite side is congruent.

c.

m ∡DCB = m ∡DAB=120° Opposite angles are congruent.

e.

m ∡ABC= ​?

m ∡ABC+ m∡DAB =180° Consecutive angles are supplementary.

Substituting value

m ∡ABC + 120°=180°

m ∡ABC =180°-120°=60°

m ∡ABC=60°

Answer:

a)  DC = 16

c)  m∠DCB = 120°

e)  m∠ABC = 60°

Step-by-step explanation:

The opposite sides of a parallelogram are equal in length. Therefore, DC is the same length as AB.

As AB = 16, then DC = 16.

[tex]\hrulefill[/tex]

The opposite angles of a parallelogram are equal in measure. Therefore, m∠DCB is equal to m∠DAB.

As m∠DAB = 120°, then m∠DCB = 120°.

[tex]\hrulefill[/tex]

Adjacent angles of a parallelogram sum to 180°. Therefore:

⇒ m∠ABC + m∠DAB = 180°

⇒ m∠ABC + 120° = 180°

⇒ m∠ABC + 120° - 120° = 180° - 120°

⇒ m∠ABC = 60°

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

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

The interval notation (-3, 4) represents the set of real numbers r that are greater than -3 and less than 4, excluding -3 and 4.

The set builder notation {r | -3 < r < 4} can be expressed in interval notation as (-3, 4).

In interval notation, the parentheses indicate that the endpoints, -3 and 4, are not included in the set.

The interval (-3, 4) represents all the real numbers r that are greater than -3 and less than 4, but not including -3 and 4.

It can also be visualized on a number line as an open interval between -3 and 4, where the endpoints are not filled in.

The interval (-3, 4) can be interpreted as a range of values for r. Any real number between -3 and 4, excluding the endpoints, would satisfy the given set builder notations.

For example, -2, 0, and 3 are all included in the interval (-3, 4), but -3 and 4 themselves are not part of the set.

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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 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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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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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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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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Para representar gráficamente los vectores 2u, -3v y 1/4, necesitamos utilizar los vectores dados A(-1,3), B(-2,4), C(0,-2) y D(8,1).

Vector 2u:

El vector 2u se obtiene al multiplicar el vector u por 2. Si conocemos las coordenadas de u, podríamos multiplicar cada componente por 2. Sin embargo, no se proporciona información sobre las coordenadas de u, por lo que no podemos realizar este cálculo específico.

Vector -3v:

Similar al caso anterior, para obtener el vector -3v, debemos multiplicar el vector v por -3. Sin información sobre las coordenadas de v, no podemos realizar este cálculo específico.

Vector 1/4:

El vector 1/4 se obtiene al multiplicar cada componente de los vectores A, B, C y D por 1/4. Podemos calcular las nuevas coordenadas de estos vectores:A' = (-1/4, 3/4)

B' = (-1/2, 1)

C' = (0, -1/2)

D' = (2, 1/4)

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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?

The length of line segment IG of the circle using the chord-chord power theorem is 6.

Chord-chord power theorem simply state that "If two chords of a circle intersect, then the product of the measures of the parts of one chord is equal or the same as the product of the measures of the parts of the other chord".

From the figure:

Line segment FG = 12

Line segment GH = 4

Line segment GJ = 8

Line segment IG = ?

Now, usig the chord-chord power theorem:

Line segment FG × Line segment GH = Line segment GJ × Line segment IG

Plug in the values:

12 × 4 = 8 × Line segment IG

48 = 8 × Line segment IG

Line segment IG = 48/8

Line segment IG = 6

Therefore, the line segment IG measures 6 units.

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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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The given information does not provide any clear indication for determining the position that should be taken on 3/01 and 3/02. Without additional information, it is not possible to make a decision. The table only displays the closing prices of the July GBP Futures Contract on different days, and it is unclear what trading strategy or what scenario is being considered. Additional information about the goals and objectives, the market conditions, and other relevant factors would be necessary to make a decision about trading positions.

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 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

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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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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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.