Sparx 4: Item A
Bookwork code: C20
Find the size of the angle marked n. Give your answer in degrees (°).
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The inverse of the function [tex]y = x^2 + 4x + 4 is f^(-1)(x) = -2 ± √(x - 2).[/tex]C

is correct answer

To find the inverse of the function y = x^2 + 4x + 4, we can follow the steps for finding the inverse of a function:

Step 1: Replace y with x and x with y:

[tex]x = y^2 + 4y + 4[/tex]

Step 2: Solve the equation for y:

[tex]x = y^2 + 4y + 4[/tex]

Step 3: Rearrange the equation:

[tex]y^2 + 4y + 4 - x = 0[/tex]

Step 4: Solve the quadratic equation for y using factoring or the quadratic formula. In this case, the equation can be factored:

[tex](y + 2)(y + 2) - x = 0(y + 2)^2 - x = 0[/tex]

Step 5: Expand and rearrange the equation:

[tex]y^2 + 4y + 4 - x = 0y^2 + 4y = x - 4y^2 + 4y = x - 2^2y^2 + 4y = (x - 2^2)[/tex]

Step 6: Complete the square on the left side of the equation:

[tex](y^2 + 4y + 4) = (x - 2^2) + 4(y + 2)^2 = (x - 2)[/tex]

Step 7: Take the square root of both sides:

[tex]y + 2 = ±√(x - 2)[/tex]

Step 8: Solve for y:

[tex]y = -2 ± √(x - 2)[/tex]

The inverse function of y = x^2 + 4x + 4 is given by:

[tex]f^(-1)(x) = -2 ± √(x - 2)[/tex]

Therefore, the inverse of the function y = [tex]x^2 + 4x + 4 is f^(-1)(x) = -2 ± √(x - 2).[/tex]

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The fraction represented by point A, it is crucial to have information about the scale, intervals, and the relative position of point A on the number line.

With these details, you can calculate the fraction accurately.

Point A on the number line represents a fraction that requires additional information to determine its precise value.

I can provide you with a general approach to finding the fraction corresponding to a given point on the number line.

A number line represents a range of values, typically with a specific scale or interval.

The scale can be divided into equal parts, such as units or fractions.

To determine the fraction represented by point A, we need to know the scale and the position of point A relative to that scale.

Let's consider a number line that ranges from 0 to 1, with evenly spaced tick marks representing tenths.

If point A is located halfway between the tick marks representing 0.4 and 0.5, then it represents the fraction 0.45 (or 9/20 in simplified form).

If the number line has a different scale or is divided into different intervals, the fraction represented by point A will be different.

Without specific details about the number line and the position of point A, it is impossible to determine the fraction precisely.

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Answer: 268

Step-by-step explanation:

the statement “Zero is included in the range of the relation” is correct and the statement “5 is included in the domain of this relation” is also correct.

if we observe the given ordered pairs we can easily determine that,The domain is the set of all x-values from the ordered pairs in the relation.

So, the domain of the given relation is {−4,0,5} because we have only three unique x-values from the given ordered pairs.

The range is the set of all y-values from the ordered pairs in the relation.

So, the range of the given relation is {−7,−3,2,6} because we have four unique y-values from the given ordered pairs.

Also, we can conclude that the given relation is not a function since we have the same x-value (5) corresponding to two different y-values (−3,2).

And it is true that the zero is included in the range of the relation since we have the ordered pair (0,6) which gives the y-value of 6.

So, the statement “The given relation is a function” is incorrect. But the statement “Zero is included in the range of the relation” is correct and the statement “5 is included in the domain of this relation” is also correct.

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Answer:4

Step-by-step explanation:1+1 is 2, so if you take 2+2 it is 4, same as 2×2 is 4.

A 15-lb mixture, she should use 1.256 pounds of dried-fruit mixture and 13.744 pounds of nuts.

Based on the given information, we can set up the following system of equations:
Equation 1: x + y = 15 (since the total weight of the mixture is 15 pounds)
Equation 2: (5.90 * x) + (14.75 * y) = 9.44 * 15 (since the total cost of the mixture is the cost per pound multiplied by the weight of each component)
Now, let's convert this system of equations into matrix form:
| 1  1 |   | x |   | 15 |
| 5.90  14.75 |   | y | = | 9.44 * 15 |
To solve for x and y, we can use matrix algebra. We can multiply the inverse of the coefficient matrix by the right-hand side matrix to get the solution matrix:
| x |   | 1  1 |^-1   | 15 |
| y | = | 5.90  14.75 |      | 9.44 * 15 |
Using matrix calculations, we find that the inverse of the coefficient matrix is:
| 0.0877  -0.0576 |
| -0.0059   0.0678 |
Now, multiplying the inverse matrix by the right-hand side matrix gives us:
| x |   | 0.0877  -0.0576 |   | 15 |
| y | = | -0.0059   0.0678 | * | 9.44 * 15 |
Simplifying the multiplication gives us:
x = 1.256
y = 13.744
Therefore, the saleswoman should use 1.256 pounds of dried-fruit mixture and 13.744 pounds of nuts to get a 15-pound mixture.

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Rounded to the nearest tenth of a percent, the probability is approximately 27.8%.

To determine if the event of rolling a pair of dice and getting doubles or a sum of 6 is mutually exclusive, we need to assess whether the two outcomes can occur simultaneously or if they are mutually exclusive.

Let's analyze the two outcomes separately:

1. Getting doubles: When rolling a pair of dice, getting doubles means that the numbers on both dice are the same. The possible outcomes for getting doubles are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). There are a total of 6 outcomes that satisfy this condition.

2. Getting a sum of 6: To get a sum of 6, the numbers on the two dice should add up to 6. The possible outcomes for this condition are (1,5), (2,4), (3,3), (4,2), and (5,1). There are a total of 5 outcomes that satisfy this condition.

Now, to determine if the two outcomes are mutually exclusive, we need to check if there are any outcomes that satisfy both conditions. In this case, the outcome (3,3) satisfies both conditions (doubles and sum of 6).

Since there is one outcome that satisfies both conditions, we can conclude that the event of rolling a pair of dice and getting doubles or a sum of 6 is not mutually exclusive.

To find the probability of this event, we need to determine the total number of possible outcomes when rolling a pair of dice. Each die has 6 sides, so there are 6 possible outcomes for each die, resulting in a total of 6 * 6 = 36 possible outcomes when rolling a pair of dice.

The probability is calculated by dividing the number of favorable outcomes (satisfying the condition) by the total number of possible outcomes.

Favorable outcomes = 6 (for doubles) + 5 (for sum of 6) - 1 (for the overlapping outcome) = 10

Probability = Favorable outcomes / Total number of outcomes = 10 / 36 ≈ 0.2778

Rounded to the nearest tenth of a percent, the probability is approximately 27.8%.

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The equation W = 32 - 0.05n represents the volume of water used, the glass tank's volume will be equal to W plus the volume occupied by the marbles.

In the given equation W = 32 - 0.05n, W represents the volume of water (in liters) Andrei uses, and n represents the number of marbles used. We can interpret this equation as follows: as Andrei increases the number of marbles used (n), the volume of water required (W) decreases.

To determine the glass tank's volume, we need to consider the relationship between the volume of water used and the number of marbles used. We can rearrange the equation to solve for n:

W = 32 - 0.05n

0.05n = 32 - W

n = (32 - W) / 0.05

The volume of the glass tank can be inferred from the equation, as the total volume will be the sum of the volume occupied by marbles and the volume occupied by water.

If we assume that the volume occupied by the marbles is constant, then the remaining volume in the tank will be filled with water. Therefore, the glass tank's volume can be represented as:

Glass Tank Volume = Volume of Water (W) + Volume of Marbles

The given equation W = 32 - 0.05n represents the volume of water used based on the number of marbles, but it does not directly provide information about the glass tank's volume. Further information about the marbles or the tank's dimensions is required to determine the glass tank's volume accurately.

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The sequence of transformations that maps ABC to A’B’C is a reflection across the line y = x followed by a translation of 10 units right and 4 units up.

The equivalent vertices C and C' are given as follows:

Considering the change in the orientation of the figure, it was reflected over the line y = x, for which the rule is given as follows:

(x,y) -> (y,x).

Hence:

(-4,2) -> (2,-4)

To go from (2,-4) to (12,0), the translation rule is given as follows:

(x,y) -> (x + 10, y + 4)

Hence the translation is defined as follows:

10 units right and 4 units up.

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the period of y = 3cot(x) remains as π. the period of y = 3cot(4x - 3x) is also π.

To find the period of the function y = 3cot(4x - 3x), we need to determine the length of one complete cycle of the function.

First, let's simplify the expression inside the cotangent function: 4x - 3x = x.

Now, the cotangent function has a period of π, which means it repeats every π units.

To find the period of the given function y = 3cot(x), we need to determine how the variable x affects the period. Since the coefficient of x is 1, it means that the period is not affected by any scaling or stretching.

In summary, the period of the function y = 3cot(4x - 3x) is π, which means the function repeats every π units along the x-axis.

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Answer: C

Step-by-step explanation:

Point D is at -14 on the number line.

In Mathematics, the midpoint of a line segment with two end points can be calculated by adding each end point on a line segment together and then divide by two (2).

Since E is the midpoint of line segment CD, we can logically deduce the following relationship:

Line segment CD = Line segment C + Line segment D

Midpoint E = (point C + point D)/2

By substituting the given points into the equation above, we have the following:

-3 = (8 + D)/2

-6 = 8 + D

D = -6 - 8

D = -14

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The diagram to the right BC is parallel to DE.

Here, AD=15,

DB=12,

AE=10, and

EC=8.

We need to determine if BC||DE or not.

angle ADE = 180 - 90  = 49.106 degree.

Now, using the triangle BDC, we can find the value of sin[angle CDB] as follows.

angle CDB = angle DAB = angle ABD = 180 - angle ADE - angle EDC

To determine if BC is parallel to DE, we can use the concept of proportional sides of the triangle.

From the information given, AD = 15, DB = 12, AE = 10, and EC = 8.

If BC is parallel to DE, then by the intercept theorem (also known as Thales' theorem), the Die segments AD/DB and AE/EC must be the same.

Let's calculate these ratios:

AD/DB = 15/12 = 5/4

AE/EC = 10/8 = 5/4

AD/DB = AE/EC = 5/ 4, so we can conclude that: BC is parallel to DE.

The reason for this is that if two ratios are equal, it means that the corresponding sides of the triangle formed by the intersecting lines are proportional, which is a characteristic of parallel lines.

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The nearest tenth of a cubic foot, the volume of water remaining in Container A is approximately 138.2 cubic feet.

To find the volume of water remaining in Container A after it is completely filled and transferred to Container B, we need to calculate the difference in their volumes.

The volume of a cylinder can be calculated using the formula:

V = πr^2h

where V is the volume, r is the radius, and h is the height.

Container A:

Diameter = 10 feet

Radius (r) = 10/2 = 5 feet

Height (h) = 5 feet

Container B:

Diameter = 6 feet

Radius (r) = 6/2 = 3 feet

Height (h) = 9 feet

Now, let's calculate the volumes of both containers:

Volume of Container A = π(5^2)(5) = 125π cubic feet

Volume of Container B = π(3^2)(9) = 81π cubic feet

To find the volume of water remaining in Container A, we subtract the volume of Container B from the volume of Container A:

Volume remaining in Container A = Volume of Container A - Volume of Container B

= 125π - 81π

= 44π cubic feet

To get the volume to the nearest tenth of a cubic foot, we can use an approximation for the value of π, which is approximately 3.14:

Volume remaining in Container A ≈ 44(3.14) ≈ 138.16 cubic feet

Therefore, to the nearest tenth of a cubic foot, the volume of water remaining in Container A is approximately 138.2 cubic feet.

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1) 10% stock dividend: Increase in shares, no change in equity value.

2) 2 for 1 stock split: Doubles shares, no change in equity value.

3) 1 for 5 reverse stock split: Decreases shares, no change in equity value.

The decisions made by Mahmud Mahtab regarding stock dividend, stock split, and reverse stock split would result in the following changes to the accounts:

1) 10% Stock Dividend:

A stock dividend is a distribution of additional shares of stock to existing shareholders. In this case, a 10% stock dividend means that for every 10 shares held, shareholders will receive an additional share. As a result, the following changes would occur:

- Common stock: The number of shares would increase by 10% (3,000 additional shares).

- Additional paid-up capital: There would be no change.

- Retained earnings: There would be a reduction of the amount equal to the market value of the shares issued as dividends.

2) 2 for 1 Stock Split:

A stock split involves dividing existing shares into a larger number of shares. In a 2 for 1 stock split, each existing share would be split into two shares. The following changes would take place:

- Common stock: The number of shares would double (60,000 shares).

- Additional paid-up capital: There would be no change.

- Retained earnings: There would be no change.

3) 1 for 5 Reverse Stock Split:

A reverse stock split combines a number of existing shares into a smaller number of shares. In a 1 for 5 reverse stock split, every five shares would be consolidated into one share. The accounts would be affected as follows:

- Common stock: The number of shares would decrease by 80% (24,000 shares).

- Additional paid-up capital: There would be no change.

- Retained earnings: There would be no change.

Note: The above changes would affect the number of shares and their par value, but the total shareholders' equity (1,00,00,000 TK) would remain unchanged.

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The amount that is equal to 24 pints can be determined by using the conversion factors between different units of volume. The following amounts are equal to 24 pints:

A. 48 cups: There are 2 cups in 1 pint. Therefore, 24 pints = 24 × 2 cups = 48 cups.

B. 192 fluid ounces: There are 8 fluid ounces in 1 cup, and 2 cups in 1 pint. Therefore, 24 pints = 24 × 2 cups = 48 cups = 48 × 8 fluid ounces = 384 fluid ounces. Hence, 192 fluid ounces are equal to 24 pints.

C. 3 gallons: There are 8 pints in 1 gallon. Therefore, 24 pints = 24/8 gallons = 3 gallons.

D. 48 quarts: There are 2 pints in 1 quart. Therefore, 24 pints = 24/2 quarts = 12 quarts. Hence, 48 quarts are not equal to 24 pints.

Therefore, the correct answers are A, B, and C.

To find the value of y, we can use the fact that opposite sides of a parallelogram are equal in length. Therefore, we can set up the following equations:

10 = 10 + 2x

2y + 4 = 18

Simplifying the first equation, we get:

2x = 0

Solving for x, we get:

x = 0

Substituting x = 0 into the second equation, we get:

2y + 4 = 18

Simplifying this equation, we get:

2y = 14

Dividing both sides by 2, we get:

y = 7

Therefore, the value of y is 7.

In the first row, the ordered pair can be expressed as f(8) = -6 in function notation.

Of(3) is 18.

Of(x) = -5 when x is -2.

The given table represents a function, and we need to express the values in function notation and determine specific inputs that correspond to certain outputs.

In the table's first row, the ordered pair are shown which can be expressed as function notation:

f(8) = -6

This means that when the input (x) is 8, the corresponding output (f(x)) is -6.

f(x) = -5 when x is -2.

This means that when the input (x) is -2, the value of the function (f(x)) is -5.

In the table, the function is defined by the equation f(x) = 8x - 6. Each input value (x) is multiplied by 8 and then subtracted by 6 to determine the corresponding output (f(x)).

By substituting different values for x into the function, we can find the corresponding values for f(x) in the table.

when x = 3, we have:

f(3) = 8(3) - 6

f(3) = 24 - 6

f(3) = 18

So, f(3) is 18.

Similarly, when x = -2, we have:

f(-2) = 8(-2) - 6

f(-2) = -16 - 6

f(-2) = -22

Thus, when x is -2, f(x) is -22.

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Among the 80 customers, 64 individuals choose to order drinks at the restaurant.

To find the number of people who order drinks at the restaurant, we can start by calculating the percentage of customers who do not order drinks.

Given that 20% of the customers do not order drinks, we can calculate the number of customers who do not order drinks by multiplying 80 (total customers) by 0.20 (20% expressed as a decimal):

Number of customers who do not order drinks = 80 * 0.20 = 16.

Since the total number of customers is 80, and 16 customers do not order drinks, we can subtract the number of customers who do not order drinks from the total number of customers to find the number of customers who do order drinks:

Number of customers who order drinks = 80 - 16 = 64.

Therefore, 64 people order drinks at the restaurant.

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True because corresponding sides and angles of the triangles are congrent.

The lines c and d are not parallel.

The correct answer is A. No, because 26 and 21 are not congruent.

In order for two lines to be parallel, the corresponding angles formed by a transversal should be congruent. However, in this case, angles 26 and 21 are not congruent, as indicated by their different measures of 40° and 140°, respectively.

Therefore, we can conclude that lines c and d are not parallel. The congruence of angles 22 and 23, mentioned in option B, or the congruence of angles 22 and 26, mentioned in option D, does not provide sufficient information to determine the parallelism of lines c and d.

The key factor in determining parallel lines is the congruence of corresponding angles, which is not met in this scenario.

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The missing values for the inverse matrix A⁻¹ are:

The solution is:

In Mathematics and Geometry, a square matrix refers to a type of matrix that is composed of an equal number of both rows and columns. Generally speaking, m × m matrix is typically referred to as a square matrix of order m.

Next, we would convert the system of three linear equations in standard form to matrix form as follows;

[tex]\left[\begin{array}{ccc}1&1&1\\10&-20&98\\5&10&-10\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] =\left[\begin{array}{ccc}200\\250\\500\end{array}\right][/tex]

By using technology, the solutions to this system of three linear equations include the following:

x = -9200

y = 7025

z = 2375.

By using technology, the inverse of this matrix is given by;

[tex]A^{-1}=\left[\begin{array}{ccc}-78&2&11.8\\59&-1.5&-8.8\\20&-0.5&-3\end{array}\right][/tex]

Therefore, the missing values are as shown in the inverse of this matrix A⁻¹ above.

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Based on the Pythagorean theorem, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side. So, we can use this to find the possible values of the third side of the triangle.

The two given sides are 15 cm and 17 cm. We can label the unknown side as "x". Then, we can set up an equation:

15^2 + 17^2 = x^2

Simplifying this equation, we get:

225 + 289 = x^2

514 = x^2

x = sqrt(514)

Therefore, the possible value of the third side that would make it a right triangle is approximately 22.72 cm.

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{18}\\ a=\stackrel{adjacent}{x}\\ o=\stackrel{opposite}{12} \end{cases} \\\\\\ x=\sqrt{ 18^2 - 12^2}\implies x=\sqrt{ 324 - 144 } \implies x=\sqrt{ 180 }\implies x\approx 13.4[/tex]

The missing side of a right-angled triangle, assuming it to be the hypotenuse with other sides as 12 and 18, can be calculated using the Pythagorean theorem. The square root of the sum of 12^2 and 18^2 gives the length of the hypotenuse, which is approximately 21.6 when rounded to the nearest tenth.

This question relates to the Pythagorean theorem which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: a² + b² = c².

Assuming that the missing side in your question (x) is the hypotenuse and the other two sides are 12 and 18, you can calculate it using the aforementioned theorem. Therefore, you find x by calculating the square root of (12² + 18²), or √(144 + 324) which equals to √468. This value, rounded to the nearest tenth, is 21.6.

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the sequence g is a decreasing sequence.

Here is the step-by-step solution to the problem based on the information given in the question:

Wanahton purified a portion of water with 900 grams of contaminants. Each hour, a third of the contaminants was filtered out.Let g(n) be the amount of contaminants (in grams) that remained by the beginning of the nth hour. g is a sequence. We need to determine the kind of sequence.The amount of contaminants in the beginning

= 900 grams

Amount of contaminants filtered out

= (1/3) * 900 = 300

grams Amount of contaminants left after first

hour = 900 - 300 = 600

gramsAmount of contaminants filtered out in the second

hour = (1/3) * 600 = 200

grams Amount of contaminants left after the second

hour = 600 - 200 = 400 grams

Amount of contaminants filtered out in the third

hour = (1/3) * 400 = 133.33 grams

(rounded to two decimal places)Amount of contaminants left after the third

hour = 400 - 133.33 = 266.67 grams

(rounded to two decimal places)In this way, the sequence will continue. We can observe that the amount of contaminants in the beginning is decreasing and the sequence is decreasing as well

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700 divided by 3.14 is approximately 2.23.

To divide a whole number by a decimal without using a calculator, you can follow these steps:

Convert the decimal divisor into a whole number by multiplying both the numerator and denominator by a power of 10 that eliminates the decimal. In this case, we have 3.14, so we can multiply it by 100 to get 314 as the new divisor.

700 ÷ (3.14 × 100)

Perform the division using the converted divisor. Divide 700 by 314:

700 ÷ 314 = 2.229...

Round the result to the desired number of decimal places, if necessary. Since the question does not specify the level of precision required, we will round the result to two decimal places:

2.229... ≈ 2.23

Therefore, 700 divided by 3.14 is approximately 2.23.

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The triangles EFG and WVU are similar by SSS

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

Triangle EFG and WVU

The triangles in this figure are

EFG and WVU

These triangles are similar because:

The triangles do have similar corresponding sides

i.e. Ratio = 525/252 = 875/420 = 825/396

Evaluate

Ratio = 2.083 = 2.083 = 2.083

Hence, the triangles EFG and WVU are similar

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The difference between the observed and expected number of late departures is statistically significant, we need to perform a binomial test by calculating the probability of observing 16 or more late departures out of 300 flights assuming the null hypothesis is true. If the resulting p-value is less than 0.05, we can conclude that the difference is statistically significant.

Based on the given information, an airline with a 95% on-time departure rate has 16 out of 300 flights with late departures. The question asks whether the difference between what occurred (16 late departures) and what you would expect by chance is statistically significant.
To determine if the difference is statistically significant, we need to conduct a hypothesis test. In this case, we can use a binomial test since we are dealing with two possible outcomes: on-time departure or late departure.
The null hypothesis (H0) assumes that the observed number of late departures is equal to what would be expected by chance. The alternative hypothesis (Ha) assumes that there is a significant difference between the observed and expected values.
In this case, the expected number of late departures can be calculated by multiplying the total number of flights (300) by the expected on-time departure rate (95%). This gives us an expected value of 300 * 0.05 = 15.
To perform the binomial test, we can calculate the probability of observing 16 or more late departures out of 300 flights, assuming the null hypothesis is true. This probability can be calculated using the binomial probability formula or by using statistical software.
If the resulting probability (also known as the p-value) is less than the significance level (usually 0.05), we can reject the null hypothesis and conclude that the difference between what occurred and what would be expected by chance is statistically significant. Otherwise, if the p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that the difference is not statistically significant.
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Answer: All of the choices in the response would be correct.

Step-by-step explanation:

Hello!

Answer:

11/12

Step-by-step explanation:

To solve this problem, we would need to find the denominator of the 2 numbers. In this case, it is 12. We will now convert the fractions to twelfths.

2/3 = 8/12

1/4 = 3/12

After we add 8/12 and 3/12, we get our answer 11/12.