Find the area of the figure. (Sides meet at right angles.)
3 in
B in
2 in
2 in
.
in
2 in
in
3 in

The approximation using the Trapezoidal Rule and the exact answer using the Fundamental Theorem of Calculus both make sense.

Using the Trapezoidal Rule, we have:

h = (pi - 0)/6 = pi/6

cos(0) + 2(cos(pi/6) + cos(pi/3) + cos(pi/2) + cos(2pi/3) + cos(5pi/6)) + cos(pi)

= 1 + 2(0.866 + 0.5 + 0 - 0.5 - 0.866) + (-1)

= 0

Using the Fundamental Theorem of Calculus, we have:

∫ cos 2x dx = [sin 2x / 2] from 0 to pi

= (sin 2pi / 2) - (sin 0 / 2)

= 0

Since both methods give us an answer of 0, the answer makes sense. The integral of a periodic function over one period, such as cos 2x over [0, pi], evaluates to 0.

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The answers make sense since the integral of cos 2x over [0, pi] is negative and our approximations are also negative. Additionally, the Fundamental Theorem of Calculus confirms our approximation.

Using the Trapezoidal Rule with n=6, we have:

delta_x = (pi - 0) / 6 = pi/6

x_0 = 0, x_1 = pi/6, x_2 = 2pi/6, x_3 = 3pi/6, x_4 = 4pi/6, x_5 = 5pi/6, x_6 = pi

f(x_0) = cos(20) = 1

f(x_1) = cos(2pi/6) = sqrt(3)/2

f(x_2) = cos(22pi/6) = 0

f(x_3) = cos(23pi/6) = -1

f(x_4) = cos(24pi/6) = 0

f(x_5) = cos(25pi/6) = -sqrt(3)/2

f(x_6) = cos(2*pi) = 1

Using the Trapezoidal Rule formula, we have:

integral cos 2x dx = (delta_x/2) * [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6)]

= (pi/36) * [1 + 2sqrt(3)/2 + 2(0) + 2(-1) + 2(0) + 2(-sqrt(3)/2) + 1]

= (pi/36) * [-2 + sqrt(3)]

≈ -0.471

To check our solution, we can use the Fundamental Theorem of Calculus:

F(x) = (1/2) * sin(2x)

F(pi) - F(0) = (1/2) * (sin(2pi) - sin(20)) = 0

F(pi) - F(0) ≈ -0.471

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Answer: yes

Step-by-step explanation: 64 is a factor of 4 because you can multiply it by 16 to get 64. Remember if you divide and get a whole number it is a factor .

Answer: \(y=\frac{2}{5}, \frac{6}{5}\)

Step-by-step explanation:

When answering a problem like this, you first isolate the absolute value.  TO do this, first subtract 8 from both sides, to get –2|4–5y|=-4.  Then divide both sides of the equation to get |4–5y|=2.  The next thing you do is split the equation into 4-5y=2 and 4-5y=-2, because the contents of the absolute value could be negative or positive, and simplifying both into y = 2/5, and y = 6/5y.

Hope it helps <3

Answer:

34.54

Step-by-step explanation:

To find the circumference of a circle, you need the formula 2π r^2. The radius is 5.5 so substitute that in for r (radius).

The calculated value of x in the circle is 42

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

The circle

The angle in a semicircle is a right angle

So, we have

2x + 6 = 90

Evaluate the like terms

So, we have

2x = 84

Divide both sides by 2

x = 42

Hence, the value of x is 42

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Using the Euclidean algorithm, we can find the greatest common divisor (GCD) of 280 and 588. By repeatedly dividing the larger number by the smaller number and taking the remainder, we can determine the GCD.

To find the GCD of 280 and 588, we start by dividing the larger number, 588, by the smaller number, 280:

588 ÷ 280 = 2 remainder 28

Next, we divide 280 by the remainder, 28:

280 ÷ 28 = 10 remainder 0

Since the remainder is now 0, we stop the algorithm. The last nonzero remainder, 28, is the GCD of 280 and 588.

Therefore, the GCD of 280 and 588 is 28.

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Answer: 134

Step-by-step explanation:

Answer:

19cm

Step-by-step explanation:

lets assume its a triangle

perimeter = 5cm + 6cm +8cm

Step-by-step explanation:

if you want to find volume of cone you must you this formula: hπr² so 10*π(3)*r²(4)= 120ft² and

if you want to find volume of cylinder you should use this formule: πr²*h/3 =π(3)*r²(4)*h(3)/3= 12 the figure has 132 volume in totally.

The value of x is 3

Algebraic expressions are expressions made up of variables, terms, factors, constants and coefficients.

They are also made up of mathematical operations like addition, subtraction, multiplication, division, etc

It is important to note that the absolute value of a number takes a positive sign.

It is denoted with the sign ' | | '

Given the expression:

4- | 2x -1 | = -3

First, take the absolute value

4 - 2x + 1 = - 3

collect like terms

2x = -3 - 4 + 1

2x = - 7 + 1

2x = -6

Make 'x' the subject of formula

x = 6/2

x = 3

Thus, the value of x is 3

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The coefficient of \((x - 1)^k\) in the Taylor series for f(x) centered at 1 is (-1/2) for k > 1 and \(f^{(k)}(1) = -k!/(2^k)\) for k > 1

What is coefficient?

In mathematics, a coefficient is a numerical or constant factor that is applied to a variable or term. Coefficients are used in various mathematical contexts, including algebra, calculus, and statistics.

Since the first derivative of f(x) is \(f'(x) = -1/(x^2 * \sqrt{(x^2 - 1)})\), we have f'(1) = -1/0, which is undefined. Hence, we cannot use the Taylor series formula for f(x) centered at 1 directly.

However, we are given that \(f^{(k)}(1) = -k!/(2^k)\) for k > 1. Using this information, we can write the Taylor series formula for f(x) centered at 1 as:

\(f(x) = f(1) + f'(1)(x - 1) + (1/2!)f''(1)(x - 1)^2\)\(+$\sum_{k=2}^{\infty} \frac{1}{k!}f^{(k)}(1)(x-1)^k$\)

Substituting f(1) = 1/2 and f'(1) = -1/2, we get:

\($f(x) = \frac{1}{2} - \frac{1}{2}(x-1) + \frac{1}{2!} \left(-\frac{2}{2^2}\right) (x-1)^2 + \sum_{k=2}^{\infty} \frac{1}{k!} \left(-\frac{k!}{2^k}\right) (x-1)^k$\)

Simplifying the expression, we get:

\($f(x) = \frac{1}{2} - \frac{1}{2}(x-1) - \frac{1}{4}(x-1)^2 + \sum_{k=2}^{\infty} \left(-\frac{1}{2}\right)(x-1)^k$\)

Hence, the coefficient of \((x - 1)^k\) in the Taylor series for f(x) centered at 1 is (-1/2) for k > 1.

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The surface area of the sphere is about 50.27 square feet.

The formula for the surface area of a sphere is:

Surface area = 4πr²

The circumference of the sphere as C = 4r ft.

The radius of the sphere as follows:

C = 2πr

4r = 2πr

r = 2 ft

The radius of the sphere can use the formula for surface area to find the answer:

Surface area = 4πr²

Surface area = 4π(2²)

Surface area ≈ 50.27 square feet

Rounding the answer to the nearest hundredth, we get:

Surface area ≈ 50.27 square feet (rounded to two decimal places)

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The cardioid is recognized as a particular instance of an epicycloid and is formed by (x2 + y2 + an x)2 = a2(x2 + y2).

A two-dimensional plane figure called a cardioid features a heart-shaped curvature. The Greek word "cardioid," which meaning "heart," is the source of the word "cardioid." Thus, it is referred to as a heart-shaped curve.

The polar pattern of a cardioid microphone gives it its name. It is classified as a directional microphone since only noises from the front and sides may be picked up by it, and the direction has an impact on the sound image that is recorded.

here the explanation:-

The equation for the heart is (x2 + (9/4)*(y2 + z2 -1)3 - (x2*(z3) -(9/200)*(y2*(z3)).

The cardioid is recognized as a particular instance of an epicycloid and is formed by (x2 + y2 + an x)2 = a2(x2 + y2).

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Answer:

because, for any n by n matrix, the sum of the dimension of the column space and the dimension of the null space must equal n. If the two dimensions are the same, their sum is an even number.

Step-by-step explanation:

It is impossible for an n-by-n matrix, where n is odd, to have a null space equal to its column space because the dimensions of the two spaces cannot be the same.

The null space of a matrix A is the set of all solutions to the equation Ax=0, where x is a column vector of appropriate dimensions. The column space of A is the span of the columns of A, which is the set of all linear combinations of the columns of A.

If the null space and column space of A are equal, then the dimension of the null space must be equal to the dimension of the column space. By the Rank-Nullity Theorem, the sum of the dimensions of the null space and the column space is equal to the number of columns in A.

Therefore, if n is odd, the dimensions of the null space and column space cannot be equal since their sum is even. Therefore, it is impossible for an n-by-n matrix, where n is odd, to have a null space equal to its column space.

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There are 15,625 ways a student can fill in the answers for the SAT math test.

For each of the 6 questions on the SAT math test, there are 5 possible answers.

To determine the total number of ways a student can fill in the answers for the test, we need to calculate the total number of possible combinations of answers for all 6 questions.

Since each question has 5 possible answers, there are 5 choices for the first question, 5 choices for the second question, and so on.

To find the total number of ways to answer all 6 questions, we can use the multiplication principle of counting.

That is, the total number of ways to answer all 6 questions is the product of the number of choices for each question:

Total number of ways = 5 x 5 x 5 x 5 x 5 x 5

= 5⁶

= 15,625

Therefore, there are 15,625 ways a student can fill in the answers for the SAT math test.

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The statement If a simple random sample is chosen with replacement, each individual has the same chance of selection on every draw is True.

If a simple random sample is chosen with replacement, it means that after each selection, the chosen individual is placed back into the population before the next selection is made. In this case, each individual in the population has the same chance of being selected for every draw.

The process of replacing the selected individual ensures that the selection probabilities remain constant throughout the sampling process. As a result, each individual has an equal probability of being chosen on each draw, making the statement true.

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The correct answer is False because, If the sample size is large enough (n greater than or equal to 30) the sampling distribution is approximately normal regardless of the shape of the population.

The sample fraction, often called the "p-hat", is the ratio of the number of sample successes to the size of the sample. The standard deviation of (p) decreases as the sample size n increases. This is because n is included in the denominator of the standard deviation formula. That is, (p has) has fewer variables for larger samples. The P-hat (must be a lowercase p with a caret (^) circumflex) indicates the percentage of the sample (this is the x-bar, the average of the samples).

A p-hat (proportion) sampling distribution is a collection of equal-sized repeated sample proportions drawn from the same population to represent it. According to the central limit theorem, the sampling distribution of p-hat is approximately normally distributed for large sample sizes.

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Answer:

let Greg's age be X

therefore;

4X - 3 = 3X + 7

4X - 3X = 7 + 3

X = 10

Greg is 10 years old

hope it's helpful

Answer:

\(\large\boxed{\sf 64\ cm }\)

Step-by-step explanation:

Here it is given that the side of a square is 16cm , and we need to find the perimeter of the square .

As we know that ,

\(\sf\qquad\longrightarrow \bigg\lgroup Perimeter_{square} = 4(side) \bigg\rgroup\)

• On substituting the respective values ,

\(\sf\qquad\longrightarrow Perimeter = 4(side)\\ \\\)

\(\sf\qquad\longrightarrow Perimeter = 4\times 16cm \\\\\)

\(\sf\qquad\longrightarrow \pink{ Perimeter = 64cm } \)

Perimeter:-

\(\\ \rm\hookrightarrow 4a\)

\(\\ \rm\hookrightarrow 4(16)\)

\(\\ \rm\hookrightarrow 64cm\)

Answer:

14 portion

Step-by-step explanation:

Given:

Total mass of potatoes = 49 pound

Mass of each small bag = 3.5 pound

Find:

Total number of bags

Computation:

Total number of bags = Total mass of potatoes / Mass of each small bag

Total number of bags = 49 / 3.5

Total number of bags = 14 bags

Total 14 portion can me made from big bag

Based on the given segments measuring 4.7 m, 1.6 m, and 2.9 m, no triangles can be formed.

In order for a triangle to be formed, the sum of the lengths of any two sides must be greater than the length of the third side. Let's check if this condition is met with the given segments.

The sum of the lengths of the shortest two sides (1.6 m and 2.9 m) is 4.5 m, which is less than the longest side measuring 4.7 m. Since the sum of the two shorter sides is not greater than the length of the longest side, it is not possible to form a triangle with these given segments. Therefore, no triangles can be formed in this case.


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The answer is C. 400 times more likely. The more likely is a manual filer to make an error than an electronic filer if 20% of all manually filed returns contain %-errors, and 0.05% of all electronically filed returns contain errors is 400 times using percentages.

It's important to note that percentages are ratios of numbers in relation to 100.

So, we must first convert 20% and 0.05% to decimals.

20% = 20/100

       = 0.200.0

5% = 0.05/100

     = 0.0005

Therefore, we know that if 20% of all manually filed returns contain errors, then 0.2 of every 1 return contains errors. Similarly, if 0.05% of all electronically filed returns contain errors, then 0.0005 of every 1 return contains errors.

To determine how much more likely a manual filer is to make an error than an electronic filer, we need to divide the probability of a manual filer making an error by the probability of an electronic filer making an error:(Probability of manual filer making an error) / (Probability of electronic filer making an error)= 0.2/0.0005= 400

Therefore, a manual filer is 400 times more likely to make an error than an electronic filer.

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Answer:

106°

Step-by-step explanation:

AOD = COB (because of vertical angles)

COB = COF + FOB

COF = 90

FOB = 16

COB = 90 + 16 = 106

AOD = 106

Hope this helps :)

Have a nice day!

Simply put, probability is the likelihood that something will occur.When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.Statistics is the study of events that follow a probability distribution.

Calculate probability ?

Therefore,

p = 0.43

if, in fact, 43% of shoppers make a purchase before leaving a store;

p = 78/200

=  0.39

if, in a sample of 200 shoppers, 78 make a purchase.

The sample proportion is a random variable that can't be predicted with certainty because it fluctuates from sample to sample.

It will be represented as a random variable by the letter P.

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GIVEN:

We are given a circle with center A and diameter CB.

The length of arc CDB is 14;

Required;

We are required to calculate the lenght of the radius.

Step-by-step solution;

The length of an arc with the central angle measured in radians is;

The length of the arc is given, so also is the angle theta (180 degrees, that is half the circle).

We can convert the degree measure of the angle into radians as follows;

Now we substitute the value of d (degree measure);

Now we take the formula for the length of an arc as follows;

Divide both sides by pi;

Therefore, the radius is

ANSWER:

he indefinite integral of (x - 1)/(x² + 2x)² dx is (1/8) ln|x + 2| - (1/8) ln|x| - 1/(4(x + 1)²) + C, where C is the constant of integration.

To evaluate the indefinite integral ∫(x - 1)/(x² + 2x² + 2x)² dx, we can proceed with a substitution. Let u = x² + 2x. Then, du = (2x + 2) dx.

Now, let's rewrite the integral in terms of u:

∫(x - 1)/(x² + 2x² + 2x)² dx = ∫(x - 1)/(u²)² * (1/(2x + 2)) du

= ∫(x - 1)/(4u²) * (1/(2x + 2)) du

= ∫(x - 1)/(8u²) * (1/(x + 1)) du

We can simplify the expression further

∫(x - 1)/(8u²) * (1/(x + 1)) du = ∫(x - 1)/(8(x² + 2x)²) du

Now, we can split the integrand into partial fractions

(x - 1)/(8(x² + 2x)²) = A/(x² + 2x) + B/(x² + 2x)²

To find the values of A and B, let's find the common denominator:

A(x² + 2x)² + B(x² + 2x) = x - 1

Expanding and combining like terms:

(A + B) x³ + (4A + 2B) x² + (4A) x + (4A) = x - 1

Matching coefficients, we have the following equations

A + B = 0 (coefficient of x³ terms)

4A + 2B = 0 (coefficient of x² terms)

4A = 1 (coefficient of x terms)

4A = -1 (constant terms)

From the first equation, B = -A.

Substituting this into the second equation: 4A + 2(-A) = 0, we get A = 0.

From the fourth equation: 4A = -1, we find A = -1/4.

Therefore, A = -1/4 and B = 1/4.

Now, we can rewrite the integral with the partial fractions:

∫(x - 1)/(8(x² + 2x)²) du = ∫(-1/4)/(x² + 2x) dx + ∫(1/4)/(x² + 2x)² dx

Let's evaluate each integral separately

∫(-1/4)/(x² + 2x) dx = (-1/4) ∫1/(x(x + 2)) dx

We can apply the method of partial fractions again for this integral. Let's find the values of C and D such that:

1/(x(x + 2)) = C/x + D/(x + 2)

Multiplying both sides by x(x + 2), we get:

1 = C(x + 2) + Dx

Expanding and combining like terms:

1 = (C + D) x + 2C

Matching coefficients, we have:

C + D = 0 (coefficient of x terms)

2C = 1 (constant terms)

From the first equation, D = -C.

Substituting this into the second equation: C - C = 0, we get C = 1/2.

Therefore, C = 1/2 and D = -1/2.

Now, let's rewrite the integral:

∫(-1/4)/(x² + 2x) dx = (-1/4) ∫(1/2x - 1/2(x + 2)) dx

= (-1/8) ln|x| - (-1/8) ln|x + 2| + K1

= (1/8) ln|x + 2| - (1/8) ln|x| + K1

Next, let's evaluate the second integral

∫(1/4)/(x² + 2x)² dx = (1/4) ∫1/(x² + 2x)² dx

We can apply a u-substitution here. Let u = x² + 2x + 1. Then, du = (2x + 2) dx.

Rewriting the integral in terms of u:

(1/4) ∫1/(x² + 2x)² dx = (1/4) ∫1/u² du

= (1/4) (-1/u) + K₂

= -1/(4u) + K₂

Substituting back u = x₂ + 2x + 1:

-1/(4u) + K₂ = -1/(4(x² + 2x + 1)) + K₂

= -1/(4(x + 1)²) + K2

Finally, we can combine the results of both integrals

∫(x - 1)/(8(x² + 2x)²) dx = (1/8) ln|x + 2| - (1/8) ln|x| - 1/(4(x + 1)²) + C

So, the indefinite integral of the given expression is (1/8) ln|x + 2| - (1/8) ln|x| - 1/(4(x + 1)²) + C, where C is the constant of integration.

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--The given question is incomplete, the complete question is given below " Evaluate the indefinite integral. (Use C for the constant of

integration.) ∫(x - 1)/(x² + 2x² + 2x)²dx "--

The x-intercept is 5 and the y-intercept is 2.

The standard form of the line is 2x + 5y = 10.

The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis. In other words, the x -intercept is the value of x when y = 0 and the y-intercept is the value of y when x = 0.

Hence, the x-intercept is 5 and the y-intercept is 2.

Let's find the equation in standard form.

Ax + By = C

where

Using slope intercept form,

y = mx + b

where

Hence, using (5, 0) and (0, 2)

m = 2 - 0 / 0 - 5

m = - 2 / 5

Therefore.

y = - 2 / 5x + 2

Hence, let's multiply through by 5 to eliminate the fraction

5y = -2x + 10

Therefore, the standard form is as follows:

2x + 5y = 10

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Main Answer:The circumstances under which the shape of the sampling distribution of pn is approximately normal.

Supporting Question and Answer:

What is the Central Limit Theorem and how does it relate to the shape of the sampling distribution of pn?

The Central Limit Theorem states that, under certain conditions, the sampling distribution of the sample mean (or sum) approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. This theorem is applicable to the sampling distribution of pn (proportion) because the proportion can be considered as the sample mean of a binary variable (success or failure). As the sample size increases, the Central Limit Theorem ensures that the sampling distribution of pn becomes more approximately normal, making it a useful approximation for making inferences about the population proportion.

Body of the Solution:The shape of the sampling distribution of pn (proportion) is approximately normal under certain circumstances. These circumstances are related to the properties of the population being sampled and the sample size. Here are the key factors:

1.Large sample size: The sampling distribution of pn tends to become more approximately normal as the sample size increases. This is known as the Central Limit Theorem. As the sample size grows larger, the distribution of sample proportions approaches a normal distribution regardless of the shape of the population distribution.

2.Random sampling: The sample should be selected randomly from the population to ensure that each member of the population has an equal chance of being included in the sample. Random sampling helps to ensure that the sample is representative of the population.

3.Independence assumption: The sampled observations should be independent of each other. This means that the selection of one observation should not influence the selection or behavior of other observations. Independence is crucial to ensure that the sampling distribution accurately reflects the population distribution.

4.Adequate population size: If the population size is sufficiently large,

relative to the sample size, the shape of the sampling distribution of pn is approximately normal. In practice, if the population is at least 10 times larger than the sample size, this condition is considered to be met.

5.Binomial distribution approximation: The shape of the sampling distribution of pn is also approximately normal when the underlying population distribution follows a binomial distribution. The binomial distribution is characterized by a fixed number of trials and two possible outcomes (success or failure) for each trial.

Final Answer: These circumstances increase the likelihood of the sampling distribution of pn being approximately normal, it does not guarantee it in all cases. In practice, checking the normality of the sampling distribution can be done using statistical tests or graphical methods, such as a histogram or a normal probability plot.  

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The circumstances under which the shape of the sampling distribution of pn is approximately normal.

The Central Limit Theorem states that, under certain conditions, the sampling distribution of the sample mean (or sum) approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. This theorem is applicable to the sampling distribution of pn (proportion) because the proportion can be considered as the sample mean of a binary variable (success or failure). As the sample size increases, the Central Limit Theorem ensures that the sampling distribution of pn becomes more approximately normal, making it a useful approximation for making inferences about the population proportion.

The shape of the sampling distribution of pn (proportion) is approximately normal under certain circumstances. These circumstances are related to the properties of the population being sampled and the sample size. Here are the key factors:

1.Large sample size: The sampling distribution of pn tends to become more approximately normal as the sample size increases. This is known as the Central Limit Theorem. As the sample size grows larger, the distribution of sample proportions approaches a normal distribution regardless of the shape of the population distribution.

2.Random sampling: The sample should be selected randomly from the population to ensure that each member of the population has an equal chance of being included in the sample. Random sampling helps to ensure that the sample is representative of the population.

3.Independence assumption: The sampled observations should be independent of each other. This means that the selection of one observation should not influence the selection or behavior of other observations. Independence is crucial to ensure that the sampling distribution accurately reflects the population distribution.

4.Adequate population size: If the population size is sufficiently large,

relative to the sample size, the shape of the sampling distribution of pn is approximately normal. In practice, if the population is at least 10 times larger than the sample size, this condition is considered to be met.

5.Binomial distribution approximation: The shape of the sampling distribution of pn is also approximately normal when the underlying population distribution follows a binomial distribution. The binomial distribution is characterized by a fixed number of trials and two possible outcomes (success or failure) for each trial.

These circumstances increase the likelihood of the sampling distribution of pn being approximately normal, it does not guarantee it in all cases. In practice, checking the normality of the sampling distribution can be done using statistical tests or graphical methods, such as a histogram or a normal probability plot.  

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