Meghan spent the same amount of money, m, each day this week at the school cafeteria Monday through Friday. Which of the following expressions represents the amount of money
Meghan spent this week?

A) the error term is 0.16039
B) number of households required to get a 95% confidence interval, with an error of ± 0.09 is 21.
C) The probability of 12 random households recycling more than 45% of their waste is approximately 0.045 or 4.5%.

Error term = z√(p(1-p)/n)

Error term = 1.75 √(0.30(1-0.30)/25)
= 0.16039

Thus, the error term is 0.16039

2 ) to find the ample size required to estimate the proportion of recycling in poor neighborhoods with an error of +/-0.09 and a 95% confidence level,

n = (z² * p * (1-p)) / E²

n = (1.96² * 0.5 * (1-0.5)) / 0.09²

= 21

So the number of households required to get a 95% confidence interval, with an error of ± 0.09 is 21.

3)  To calculate the probability of 12 random households recycling more than 45% of their waste,

P(X > 12) = 1 - P(X <= 12)

P(X > 12) = 1 - P(X <= 12)

= 1 - 0.955

= 0.045

So the the probability of 12 random households recycling more than 45% of their waste is 0.045.

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Full Question:

In a small town, households recycle on average 30% of their waste. The new recycling committee wants to increase this proportion and study the relationship between recycling and income. They select 25 households from the two wealthier neighborhoods and estimate a 92% confidence interval for the true proportion of recycling. What is the error term of this interval?

The same recycling committee also focuses on poor neighborhoods. How many households do they need to sample to get a 95% confidence interval, with an error of +/-0. 09?

Finally, the same recycling committee wants to know the probability of 12 random households recycling more than 45% of their waste. This probability is: ?

The length from least to greatest is √3, √5, π, 2√3.

We have the lengths as π, √3, 2√3, √5.

Now, writing the decimals for each

π= 3.14

√3 = 1.732

2√3 = 3.464

√5= 2.23

So, arranging the length from least to greatest is √3, √5, π, 2√3.

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The congruence postulate that applies is C. Congruent - SSS

When the properties (i.e length of sides and/or internal angles) of two or more triangles are equal, then it can be said that they are congruent. This congruent relations can be expressed using either of the following postulates: Angle-Side-Side, Side-Angle-Side (SAS), Angle-Angle-Side (AAS), Side-Side-Side (SSS) etc.

Considering the given triangles ABC and DEF, it is was given that:

AB ≅ DE

BC ≅ EF

CA ≅ FD

Therefore, it can be concluded that the two triangles are congruent by Side-Side-Side (SSS) relations. So that the required congruence postulate that applies is C. Congruent - SSS

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The area under the standard normal curve to the right of z = 0.69 is approximately 0.2451.

Using a standard normal table:

Locate the row for 0.6 in the left-hand column of the table and the column for 0.09 along the top row of the table.

The intersection of the row and column gives the area to the left of z = 0.69, which is 0.7549.

Subtract this area from 1 to find the area to the right of z = 0.69:

area to the right of z = 0.69 = 1 - 0.7549 = 0.2451

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A) A steer weighing 1000 pounds is 2.78 standard deviations below the mean.

B) A z-score of 1.85 is closer to the mean than a z-score of -2.78, we can conclude that a steer weighing 1000 pounds is more unusual than one weighing 1250 pounds.

A) To find how many standard deviations below the mean a steer weighing 1000 pounds is, we need to use the formula for standard score (or z-score):

z = (x - μ) / σ

where x is the weight of the steer, μ is the mean weight, and σ is the standard deviation. Substituting the values we have:

z = (1000 - 1150) / 54

z = -2.78

B) To determine which is more unusual, we need to compare the z-scores for a steer weighing 1000 pounds and one weighing 1250 pounds. Using the same formula as before:

For a steer weighing 1000 pounds:

z1 = (1000 - 1150) / 54

z1 = -2.78

For a steer weighing 1250 pounds:

z2 = (1250 - 1150) / 54

z2 = 1.85

A positive z-score means the weight is above the mean, while a negative z-score means the weight is below the mean. Therefore, a steer weighing 1250 pounds is 1.85 standard deviations above the mean, while a steer weighing 1000 pounds is 2.78 standard deviations below the mean.

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The lowest angle of elevation at which the astronomer can observe the star is 20.8°

We know that the angle of elevation is nothing but the angle between the horizontal line of sight and the straight line to a given object.

In this scenario, let us ssume that θ be an angle of elevation.

Since an astronomer is in the middle of the field,the 100 value is split in half.

The field is surrounded by the trees 20 m tall and the tripod holding the telescpoe 1 m above the ground.

This means that the vertical distance is 19 m and the horizontal distance is 50 m.

Consider the tangent of angle θ

tan(θ) = opposite side of angle θ / adjacent side of angle θ

tan(θ) = 19/50

tan(θ) = 0.38

θ = arctan(0.38)

θ = 20.8°

This is the angle of elevation.

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Yes, it is true that if A is a 3×3 matrix and K is a scalar, then det(KA) = [tex]K^3[/tex] det(A).

To see why this is true, let's use the definition of the determinant of a matrix. For a 3×3 matrix A with entries[tex]ka_{11}, ka_{12}, ka_{13}, ka_{21}, ka_{22}, ka_{23}, ka_{31}, ka_{32}, ka_{33}[/tex], the determinant det(A) is given by:

[tex]det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)[/tex]

Now consider the matrix KA, where K is a scalar. The entries of KA are simply the entries of A multiplied by K. In other words, the entries of KA are:

[tex]ka_{11}, ka_{12}, ka_{13}, ka_{21}, ka_{22}, ka_{23}, ka_{31}, ka_{32}, ka_{33}[/tex]

Using the same formula as above, we can calculate the determinant of KA:

[tex]det(KA) = (K a_{11})(K a_{22})(K a_{33}) - (K a_{11})(K a_{23})(K a_{32}) - (K a_{12})(K a_{21})(K a_{33})+ (K a_{12})(K a_{23})(K a_{31}) + (K a_{13})(K a_{21})(K a_{32}) - (K a_{13})(K a_{22})(K a_{31)[/tex]

Substituting in the entries of KA, we get:

[tex]det(KA) = (K a_{11})(K a_{22})(K a_{33}) - (K a_{11})(K a_{23})(K a_{32}) - (K a_{12})(K a_{21})(K a_{33})+ (K a_{12})(K a_{23})(K a_{31}) + (K a_{13})(K a_{21})(K a_{32}) - (K a_{13})(K a_{22})(K a_{31)[/tex]

Simplifying this expression, we get:

[tex]det(KA) = K^3 (a_{11} a_{22}a_{33} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31})[/tex]

But this is just the same as[tex]K^3[/tex] times the determinant of A! Therefore, we have shown that det(KA) =[tex]K^3[/tex]det(A) for any 3×3 matrix A and scalar K.

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Question

Is it true that If A is a 3×3 matrix, then det (KA) = K^3 * det(A)

The side lengths for this problem are given as follows:

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

For segment d, we have that it is opposite to the angle of 60º, while 7 is the adjacent segment, hence:

tan(60º) = d/7

[tex]\sqrt{3} = \frac{d}{7}[/tex]

[tex]d = 7\sqrt{3}[/tex]

For segment b, it is opposite/adjacent (does not matter as sin(45º) = cos(45º)) to an angle of 45º, while the hypotenuse is of 5 units, hence:

sin(45º) = b/5

[tex]b = 5 \times \frac{\sqrt{2}}{2}[/tex]

[tex]b = \frac{5\sqrt{2}}{2}[/tex]

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Answer:

Molly earned $55.44 in interest at the end of six years.

Step-by-step explanation:

To calculate the interest earned by Molly's savings account, we can use the formula:

Interest = Principal x Rate x Time

where:

Principal is the initial amount of money deposited ($220.00)

Rate is the annual interest rate (4.2% or 0.042 as a decimal)

Time is the number of years the money is invested (6 years)

Plugging in the values, we get:

Interest = $220.00 x 0.042 x 6

Interest = $55.44

Therefore, Molly earned $55.44 in interest at the end of six years.

The segment bisector of QR is Ml.

The length of QR is equal to 32 units.

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

Since Ml is the midpoint and segment bisector of line segment QR, we have the following:

Line segment QM = Line segment MR

2x + 6 = 5x - 9

5x - 2x = 9 + 6

3x = 15

x = 15/3

x = 5.

Now, we can determine line segment QR as follows:

QM = 2x + 6 = 2(5) + 6 = 16 units.

MR = 5x - 9 = 5(5) - 9 = 16 units.

QR = QM + MR

QR = 16 + 16

QR = 32 units.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

For a = 1 and m = 1 does f(x) have a horizontal asymptote at y = 2 and a vertical asymptote at x = 1.

To find the values of a and m that give the function f(x) a horizontal asymptote at y = 2 and a vertical asymptote at x = 1, we need to analyze the behavior of the function as x approaches 1 and as x goes to infinity.

When x approaches 1 from the left and right sides, the denominator of f(x) approaches 0, so there is a vertical asymptote at x = 1. To have a vertical asymptote at x = 1, the numerator of f(x) cannot approach 0 as x approaches 1, so m must be greater than or equal to 1.

When x goes to infinity or negative infinity, the function f(x) approaches 2, which means there is a horizontal asymptote at y = 2. To have a horizontal asymptote at y = 2, the degree of the numerator must be equal to or less than the degree of the denominator. The degree of the numerator is m, and the degree of the denominator is 1.

So, the values of a and m that give f(x) a horizontal asymptote at y = 2 and a vertical asymptote at x = 1 are:

a = 1, m = 1

Substituting a = 1 and m = 1 into f(x), we get:

f(x) = 2x/(x+1)

which has a vertical asymptote at x = 1 and a horizontal asymptote at y = 2.

Therefore, the answer is a = 1, m = 1. The other values of a and m do not give a vertical asymptote at x = 1 and/or a horizontal asymptote at y = 2.

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Answer:

CCCCCCCC

Step-by-step explanation:

Edg 2023

The length of AB is 10m.

Let's assume that the length of AC = x,

According to the problem statement, AB= 2x ...........(1)

Also, BC is 10 cm shorter than AB, so,

BC=AB-10 .......(2)

The perimeter of the triangle is the sum of its three sides:

therefore,

1.95=AB+AC+BC

We can put the value from eq(1) & eq(2)

1.95=2x+x+(AB-10)

AB= 3x+8.05............... (3)

Now, we need to find the value of x

1.95= 2x+x+(AB-10)

or, 1.95 = 3x+ AB-10

Adding 10 to both sides

11.95 = 3x+AB.......(4)

Substituting the expression, we found for AB:

11.95=3x+3x=8.05

11.95= 6x

x= 0.65

So, the value of AB is

AB = 3x+8.05

AB = 3(0.65)+8.05

AB = 10 m

Therefore, The length of AB is 10m.

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For a jar contains 5 red discs, 10 blue discs and m green discs,

a) The probability that the first disc selected is red is equals to

[tex]\frac{5}{15 + m}.[/tex]

b) The smallest value of m for which Var(X)< 0.6 is equals to 13.

We have a jar contains different colour discs. Number of red discs = 5

Number of green discs = m

Number of blue discs = 10

Total number of discs = 5 + 10 + m

= 15 + m

The process is performed four times, so, possible value n = 4. We have to determine the probability that the first disc selected is red.

a) Probability is defined as the ratio of favourable outcomes to the total possible outcomes. So, the probability that the first disc selected is red,[tex] P(red) = \frac{ 5}{(15 + m)}[/tex]

(b) The variance is calculated as

[tex]σ_{X²} = Var (X)[/tex]

[tex]= ∑_i (x_i − μ)² p(x_i) \\ [/tex]

= np(1 - p)

= [tex] 4(\frac{5}{15 + m})×(1- \frac{5}{15 + m}) < 0.6[/tex]

[tex] \frac{20( 10 - m)}{(15 + m)²} < 0.6[/tex]

=> 200 - 20m < 0.6( 15 + m)²

=> 200 - 20m < 9.0 + 0.6 m² + 18

=> 0.6m² + 20m - 200 + 27 > 0

=> 0.6m² + 20m - 173 > 0

=> m > 12.2075, so, m = 13

Hence, the smallest value of m for which Var(X)<0.6 is 13.

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The unit rate of the juice is 0.5 carrot juice per green apple juice

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

This means that

Carrot juice = 2

Green apple juice = 4

Using the above as a guide, we have the following:

Unit rate = Carrot juice/Green apple juice

Substitute the known values in the above equation, so, we have the following representation

Unit rate = 2/4

Evaluate

Unit rate = 0.5

Hence, the unit rate is 0.5 carrot juice per green apple juice

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The solutions of the equation f(x) = g(x) include the following: B. -3, 0, 2, -4.

In Mathematics and Geometry, a point of intersection simply refers to the location on a graph where two (2) lines intersect or cross each other, which is primarily represented as an ordered pair containing the point that corresponds to the x-coordinate (x-axis) and y-coordinate (y-axis) on a cartesian coordinate.

By critically observing the graph of the lines representing the given system of equations, we can reasonably and logically deduce that the correct solution set lies in both Quadrant II and IV and it is denoted by the point of intersection of both the x-coordinate (x-axis) and y-coordinate (y-axis), which is given by this ordered pair (-3, 0) and (2, -4).

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Answer:  Hence, the length of swimming pool is 52 m and its breadth if 25 m.  

Step-by-step explanation:

The perimeter of rectangular swimming pool is 154 m and its length is 2 meter more than twice its breadth.

The submarine depth after the dive is determined as 17 feet.

The current depth of the submarine is calculated as follows;

If the submarine is currently at a depth of -15 feet and then dives an additional 32 feet down, the new depth can be calculated by adding the two depths.

-15 feet + 32 feet = 17 feet

Therefore, the submarine is at a depth of 17 feet after the dive.

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The probability that high school A will lose some state funding is approximately 0.0287 or 2.87%.

Probability is a way to gauge how likely something is to happen. Many things are difficult to predict with absolute certainty.

We can use the normal distribution to approximate the probability that high school A will lose some state funding. Let X be the number of absent students on the selected school day. We know that X follows a normal distribution with mean µ = 120 and standard deviation σ = 10.5.

We need to find the probability that X is greater than 140. To do this, we standardize X by subtracting the mean and dividing by the standard deviation:

Z = (X - µ) / σ = (140 - 120) / 10.5 = 1.90

We can now use a standard normal distribution table or calculator to find the probability that a standard normal variable is greater than 1.90. The result is approximately 0.0287.

Therefore, the probability that high school A will lose some state funding is approximately 0.0287 or 2.87%.

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If another school is considering changing their pizza vendor, they may also select separate random samples of 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors to gather feedback from the students. By doing so, they can obtain a representative sample of the entire student population and ensure that each grade level is equally represented.

This will provide valuable insight into the preferences and opinions of the students, allowing the school to make an informed decision on which pizza vendor to choose. It is important to consider the feedback of all students in the decision-making process to ensure that the majority of the student body is satisfied with the food options provided by the school.

Step 1: Select random samples of students from each grade level.
Step 2: Provide the new pizza to each student in the samples.
Step 3: Ask the students to evaluate the pizza based on taste and quality.
Step 4: Collect the feedback from all 200 students.
Step 5: Analyze the data to determine if the majority of students prefer the new pizza vendor.

Based on the analysis, the school can make an informed decision about whether to change their pizza vendor or not.

another school is also cosidering changing thier pizza vendor. the school selects seperate random samples of 50 freshman, 50 sophiomres, 50 jjuniors, and 50 seniors. each student tries

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This is an example of interviewer bias where the gender of the interviewer may have impacted the replies of the unmarried males in the poll.

This is an example of interviewer bias, where the gender of the interviewer may have influenced the responses of the unmarried men in the survey. The results may not accurately reflect the true opinions of the participants as their answers could have been affected by their desire to impress or please the interviewer.

This situation is an example of response bias, specifically interviewer bias, which occurs when the respondent's answer is influenced by the gender or characteristics of the interviewer, rather than their true preferences or opinions.

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The area of the base of Barbara's woodshed is 72 square feet.

To find the area of the base of Barbara's woodshed, we need to know the shape of the base. In this case, the base is in the shape of a rectangle. A rectangle is a four-sided figure with opposite sides parallel and equal in length.

To calculate the area of a rectangle, we need to multiply its length by its width. In this case, the length of the rectangle is 12 feet, and its width is 6 feet. So, the area of the base of the woodshed is:

Area = Length x Width

Area = 12 ft x 6 ft

Area = 72 square feet

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Complete Question:

Barbara built a woodshed. She made the base of the woodshed in the shape of the drawing. What is the area of the base of Barbara's woodshed? Include the unit. When the dimensions are given as 12 ft,6 ft, 11 ft and 7 ft.

If the price of corn feed is $4 and corn feed is on the horizontal axis, and the price of soybean feed is $5 and soybean feed lies on the vertical axis, then the expansion path is C =5S/2 (option a).

To find the expansion path, we need to find the optimal combination of inputs that will maximize pig production while keeping the cost of production at a minimum. This can be achieved by calculating the ratio of the prices of the two inputs, which is given by:

Price ratio = Price of soybean-based feed/Price of corn-based feed

Price ratio = 5/2

Now, we can use this price ratio to find the optimal combination of inputs that will minimize the cost of production while maximizing pig production. This can be done by solving for the quantity of soybean-based feed used in terms of the quantity of corn-based feed used:

S = (5/2)C

This equation represents the expansion path, which shows the optimal combination of inputs that will minimize the cost of production while maximizing pig production. We can prove this by substituting the value of S into the production function:

P = 8C + 25((5/4)C)

P = 8C + 31.25C

P = 39.25C

Hence the correct option is (a)

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The circumference of a circle is given by $C = \pi d$, where $d$ is the diameter of the circle. In this case, the diameter of Annie's bicycle wheels is $20$ inches, so the circumference of each wheel is:

$C = \pi d = \pi (20\text{ in}) \approx 62.83 \text{ in}$

Since each wheel makes 3 full revolutions every second, the distance that Annie travels in one second is:

$distance = 2C \cdot \text{revolutions per second} = 2 \cdot 62.83 \text{ in} \cdot 3 \approx 377 \text{ in}$

Converting inches to feet, we get:

$distance = 377 \text{ in} \cdot \frac{1 \text{ ft}}{12 \text{ in}} \approx 31 \text{ ft}$

Rounding to the nearest foot, Annie travels approximately $\boxed{31}$ feet in one second.

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Step-by-step explanation:

Let's call the shorter base of the trapezoid "b1", the longer base "b2", and the height "h". We can use the formula for the surface area of a right prism to set up an equation:

Surface area of prism = 2(base area) + (lateral area) = 1280

The base area is the area of the trapezoid, which is given by:

(base area) = (1/2)(b1 + b2)h

The lateral area is the area of the four rectangular faces of the prism, which are all congruent. Each face has an area equal to the product of the height and the length of one of the legs of the trapezoid, so the lateral area is:

(lateral area) = 4hl

where l is the length of one of the legs of the trapezoid.

Substituting these expressions into the formula for the surface area of the prism, we get:

2[(1/2)(b1 + b2)h] + 4hl = 1280

Simplifying and rearranging, we get:

h(b1 + b2) + 2hl = 1280

We also know that the sum of the lengths of the legs of the trapezoid is 52 feet, which means:

l1 + l2 = 52

But we can express l1 and l2 in terms of b1 and b2 using the formula for the area of a trapezoid:

(base area) = (1/2)(b1 + b2)h = (1/2)(l1 + l2)h

Simplifying, we get:

b1 + b2 = (l1 + l2)h

Substituting this into the previous equation, we get:

h[(l1 + l2)h] + 2hl = 1280

Simplifying, we get:

h^2(l1 + l2) + 2hl = 1280

Substituting l1 + l2 = 52, we get:

h^2(52) + 2hl = 1280

This is a quadratic equation in h. We can solve it using the quadratic formula:

h = [-2l ± sqrt(4l^2 + 4h^2(52)(1280 - 2hl))] / 2(52)

Simplifying and factoring out a 2, we get:

h = [-l ± sqrt(l^2 + h^2(1280 - 2hl))] / 52

We have two possible solutions for h, but one of them is negative, which doesn't make sense in the context of the problem. So we can discard the negative solution and focus on the positive one:

h = [-l + sqrt(l^2 + h^2(1280 - 2hl))] / 52

We don't know the exact value of h yet, but we can use this equation to set up a system of equations that we can solve for h. Specifically, we can use the fact that the legs of the trapezoid add up to 52 feet to solve for l in terms of b1 and b2:

l = 52 - (b1 + b2)

Substituting this into the equation for h, we get:

h = [-l + sqrt(l^2 + h^2(1280 - 2hl))] / 52

h = [-52 + (b1 + b2) + sqrt((52 - (b1 + b2))^2 + h^2(1280 - 2h(b1 + b

Answer:

Approximately 20.33 (rounded to two decimal places)

Step-by-step explanation:

To find the standard deviation, we will first find the mean (average) of the scores, then find the variance, and finally take the square root of the variance to get the standard deviation.

Calculate the mean:

(80 + 30 + 34 + 32 + 55 + 27 + 38 + 41 + 84) / 9 = 421 / 9 = 46.78 (rounded to two decimal places)

Calculate the variance:

a. Find the difference between each score and the mean, and then square the differences (rounded to two decimal places).

(80 - 46.78)^2 = 1103.57

(30 - 46.78)^2 = 281.57

(34 - 46.78)^2 = 163.33

(32 - 46.78)^2 = 218.45

(55 - 46.78)^2 = 67.57

(27 - 46.78)^2 = 391.25

(38 - 46.78)^2 = 77.09

(41 - 46.78)^2 = 33.41

(84 - 46.78)^2 = 1385.33

b. Find the sum of the squared differences.

1103.57 + 281.57 + 163.33 + 218.45 + 67.57 + 391.25 + 77.09 + 33.41 + 1385.33 = 3721.57

c. Divide the sum of the squared differences by the number of scores (n = 9).

3721.57 / 9 = 413.51 (rounded to two decimal places)

Calculate the standard deviation:

Take the square root of the variance.

√413.51 = 20.33 (rounded to two decimal places)

The standard deviation of the scores is approximately 20.33 (rounded to two decimal places).

There will be 1, 2, 6, 21 non-isomorphic simple graphs for 2, 3, 4 and 5 vertices.

What are non-isomorphic simple graphs?

A non-isomorphic simple graph is a graph that is distinct from another graph, even if the two graphs have the same number of vertices and edges, and the same connectivity pattern. In other words, two graphs are non-isomorphic if they cannot be transformed into each other by a relabeling of their vertices.

For a small number of vertices, we can enumerate all non-isomorphic simple graphs by hand.

For n = 2, there is only one possible graph, which is the edge connecting the two vertices.

For n = 3, there are only two possible graphs: a triangle (complete graph on 3 vertices) and a single edge with an isolated vertex.

For n = 4, there are six possible graphs:

Complete graph on 4 vertices

Cycle graph on 4 vertices

Complete bipartite graph K2,2

Graph with a central vertex adjacent to all other vertices

Graph with two vertices of degree 3 and two vertices of degree 1

Graph with one vertex of degree 3 and three vertices of degree 1

For n = 5, there are 21 possible graphs, which can be generated by adding edges to the graphs for n = 4.

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The inevitable difference between the mean of a sample and the mean of a population based on chance alone is known as sampling error.

Sampling error is a result of the random nature of sampling from a population, meaning that any given sample is unlikely to perfectly represent the entire population. This is where probability comes into play, as the likelihood of obtaining a certain sample is dependent on the probability of each member of the population being selected.

Therefore, in order to minimize sampling error, researchers often use random sampling techniques to ensure that each member of the population has an equal probability of being selected for the sample.

The inevitable difference between the mean of a sample and the mean of a population based on chance alone is a) sampling error. This occurs because a random sample may not perfectly represent the entire population, leading to slight variations in the mean.

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The names of the radius , chord and diameter of a circle are as follow,

Radius of the circle are PQ, PM , And PR.

Diameter of the circle is QM

Chord of the circle is ON.

Length of QM in the circle = 4units.

In the attached figure of the circle,

Center of the circle is P.

radius of the circle is a distance from the center of the circle to its circumference.

Radius = PQ, PM , And PR.

Diameter of the circle passing through center P is QM.

Chord of the circle representing a line segment having endpoints on the circumference of the circle.

Chords are ON and MQ.

Diameter is the longest chord.

length of PR is 2 units,

PR is radius

QM is diameter

QM = 2(PR)

length of QM = 2(2)

                      = 4 units.

Therefore, for the given circle we have,

Radius are PQ, PM , And PR.

Diameter is QM

Chord is ON.

Length of QM = 4units.

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The above question is incomplete, the complete question is:

A circle with center P is shown in the figure below.

(a) Name a diameter:

(b) Name a radius:

(c) Name a chord:

(d) If the length of PR is 2 units, what is the length of QM

Attached figure.

The number of respondents needed in a poll to adequately represent a larger population depends on several factors, such as the size of the population, the level of confidence desired, and the margin of error tolerated.

The general rule of thumb is that a sample size of at least 400 respondents is needed to provide a representative sample for a population of 100,000 or less. However, for larger populations, a sample size of at least 1,000 respondents may be required.

It is also important to note that a larger sample size does not always guarantee more accurate results. Other factors such as the sampling method, the wording of the questions, and the response rate can also impact the accuracy of poll results. Therefore, it is essential to carefully design and conduct polls to ensure their results are adequately representative of the larger population.

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There are 4 incongruent primitive roots and 6 has order 12 and it is a primitive root.

There are 12 elements of the group     \(U_{13}\)   , namely all the positive integers less than 13, as these are relatively prime to 13. Now, if there are primitive roots, there are     \(\phi (\phi (n))\)    of them. So we must compute     \(\phi (12) = \phi (4\times 3) = \phi (4) \phi (3) = 2\times 2 = 4\)   . There are 4 incongruent primitive roots.

To find them, take the powers each element in turn:

1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1  (2 has order 12, it is a primitive root)

Of course, the higher powers of 2 cannot be.

Proceeding this way, we get next get that 6, 7, and 11 are also primitive roots.

For example, the powers of 6 give: 6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1. We see 6 has order 12 and it is a primitive root. So 2, 6, 7, 11.

Therefore, There are 4 incongruent primitive roots and 6 has order 12 and it is a primitive root.

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