calculate the sample mean and sample variance for the following frequency distribution of heart rates for a sample of american adults. if necessary, round to one more decimal place than the largest number of decimal places given in the data. heart rates in beats per minute class frequency 61 - 66 4 67 - 72 8 73 - 78 5 79 - 84 7 85 - 90 13
Sample Mean _______________
Sample Variance _________________

Context, it is not clear what is meant by "j(0,0)" or what the function's codomain being [tex]R^2[/tex] signifies.

It is not clear what you mean by "b" and "j". However, I can provide a general answer on how to find a holomorphic function.

A holomorphic function is a complex-valued function that is complex differentiable in a neighborhood of each point in its domain. This means that the function must satisfy the Cauchy-Riemann equations, which relate the partial derivatives of the function with respect to the real and imaginary parts of its input.

To find a holomorphic function, one approach is to use the power series representation of complex analytic functions. If a function f(z) is analytic at a point z0, then it has a power series expansion of the form:

f(z) = ∑n=0∞ [tex]c_n (z - z0)^n[/tex]

where [tex]c_n[/tex] are complex coefficients that depend on the function and z0. This power series converges in a neighborhood of z0, and the coefficients can be calculated using complex integration techniques.

Another approach is to use the Cauchy integral formula, which expresses the value of a holomorphic function at a point in terms of an integral over a closed curve that encloses the point. This formula allows one to compute the function at any point in its domain using complex integration techniques.

Without more information about "b" and "j", it is not possible to determine if a holomorphic function exists or what its properties might be. Similarly, without more context, it is not clear what is meant by "j(0,0)" or what the function's codomain being [tex]R^2[/tex] signifies.

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The population will decrease. Thus, the correct option is A.

Let a be the initial value and x be the power of the exponent function and b be the factor.

The exponent is given as

y = a(b)ˣ

The equation is modeled as,

M(r) = 945 × (0.935)ⁿ

Where 'n' is the number of years since 2010.

There are three conditions exist which are as follows:

On comparing, the value of 'b' is 0.0935 which is less than one. Then the population will decrease. Thus, the correct option is A.

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The missing options are given below.

A. The population is decreasing.

B. The population is increasing.

C. The population remains constant.

D. None of the above.

The maximum sustainable rate of harvest which is possible for the growth of fish P(t)  according to the differential equation " P' = 4P - 4P^2 - h " will be at the harvest rate of h=3.

In order to calculate the maximum sustainable harvest rate possible, we are required to find the equilibrium solutions of the differential equation firstly. Equilibrium solutions occur when P' = 0, so we can set the right-hand side of the differential equation at 0.

4P - 4P² - h = 0

Now a quadratic equation is formed in terms of P, which can be used to obtain P.

P = (1 ± √(1 + h))/2

There are two equilibrium solutions, obtained by these two values of P.

Now, in order to obtain a maximum sustainable harvest rate, we are required to have both equilibrium solutions to as positive ( as negative populations don't make any sense in this context).

Therefore, we have to find the largest value of h for which both of these equations have positive solutions:

(1 + √(1 + h))/2 > 0

(1 - √(1 + h))/2 > 0

On camparing both the inequalities, it can be concluded that the first inequality is always true, since the square root of a positive number is always positive. The second inequality is true when h < 3.

Hence, the maximum sustainable harvest rate is is found to be at h = 3. If the harvesting rate is greater than 3, then there will be no positive equilibrium solutions and the population will eventually be perished.

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The points (0, 0), (0, 3), (4, 0): A. points are vertices of a right triangle.

The points (1, 2), (3, 0), (3, 3): B. points are not vertices of a right triangle.

The midpoint between the two points (2, 2) and (6, 4) is [4, 3].

The midpoint between the two points (2, 3) and (4, 1) is [3, 2].

The company's sales in 1997 is equal to $610,000.

In order to determine the straight-line distance from your starting point, we would apply Pythagorean's theorem. Mathematically, Pythagorean's theorem is given by this mathematical expression:

c² = a² + b²

Where:

a, b, and c represents the side lengths of a right-angled triangle.

c² = 12² + 2²

c² = 144 + 4

c = √148

c = 12.17 miles.

In order to determine the midpoint of a line segment with two (2) endpoints, we would add each point together and divide by two (2).

Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint = [(6 + 2)/2, (4 + 2)/2]

Midpoint = [8/2, 6/2]

Midpoint = [4, 3]

Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint = [(2 + 4)/2, (3 + 1)/2]

Midpoint = [6/2, 4/2]

Midpoint = [3, 2]

By applying the midpoint formula, the sales for 1997 can be calculated as follows;

Sales = (1996 sales + 1998 sales)/Number of years

Sales = ($500,000 + $720,000)/2 years

Sales = $1,220,000/2

Sales = $610,000.

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The female volleyball player is taller in relation to the distribution of heights based on gender.

The mean height of a man is 69 inches and the standard deviation is 2.25 inches. The mean height of a woman is 64 inches and the standard deviation is 2 inches.

To compare the heights of the female volleyball player and the male soccer player, we need to convert their heights to inches.

6 feet 2 inches is equal to 74 inches.

The female volleyball player is 10 inches taller than the mean height for women in the United States, while the male soccer player is 5 inches taller than the mean height for men in the United States.

To determine who is taller in relation to the distribution of heights based on gender, we need to compare the difference between their heights and the mean height for their respective gender in terms of the standard deviation.

For the female volleyball player, the difference is 10 inches - 64 inches (the mean height for women) = 6 standard deviations (since the standard deviation for women is 2 inches).

For the male soccer player, the difference is 10 inches - 69 inches (the mean height for men) = 4.4 standard deviations (since the standard deviation for men is 2.25 inches).

Therefore, the female volleyball player is taller in relation to the distribution of heights based on gender.

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The data is analyzed and frequency table is created. The correct option for the histogram is the second option.

Here 30 individual samples are given. We are first categorizing the values in a frequency distribution table. It is shown as an image below.

Temperature          frequency            Relative frequency

    70-74                          5                        5/30 = .166 = 16.6%

    75-79                          4                        4/30 = .133 = 13.3%

   80-84                           7                        7/30 = 0.233 =23.3%

   85-89                          6                        6/30 = 0.2 = 20%

   90-94                         8                        8/30 = 0.266 = 2.66%

From the given histogram the second one represents the data given in the frequency table. It is marked in the image.

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The complete question is included as image

Answer:

3² +2³= 9+8=17

2¹ - 3²= 2-9=8

Step-by-step explanation:

Answer:

1-7

Step-by-step explanation:

[tex]3^{2}[/tex] + [tex]2^{2}[/tex] (3x3) + (2x2x2) = 9 + 8 = 17

[tex]2^{1}[/tex] - [tex]3^{2}[/tex] = 2 - (3x3) = 2 - 9 = -7

The system of equations representing this movie theater scenario is B:

A system of equations involves two or more equations whose solutions are determined concurrently or at the same time.

A system of equations is otherwise called simultaneous equations.

The total sales revenue = $63

The total number of tickets sold = 29

The cost per unit of children tickets = $3

The cost per unit of adult tickets = $2

The cost per unit of senior tickets = $2

Let the number of children tickets sold = 3c

Let the number of adult tickets sold = a

Let the number of senior tickets sold = s

c + a + s =29 Total number of tickets sold

3c+2a+2=63 Total amount realized from the ticket sales

3a=c

Thus, the correct system of equations is Option B.

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The equation of the relationship between glucose left (g) and time elapsed (t) is found and intercepts are also found.

What is an equation?

A formula known as an equation uses the equals sign (=) to express how two expressions are equal.

Given, the drip rate of glucose = 1.02 cc/min

Total glucose given = 120cc

Time taken to given 120cc glucose  = 120/1.02  = 117.6 minutes ≈ 118 minutes

a) We will use g for glucose left and t for the time in minutes.

When t = 0 ,  g = 120

When t = 20, g = 120 - (1.02 * 20) = 99.6cc

when t = 40, g = 120 - (1.02 * 40)  = 79.2cc

Similarly, we fill up the table as below:

t     0         20       40       60         80       100     120    

g    120    99.6    79.2     58.8     38.4       18        0

b) We know in 1 minute = 1.02 cc glucose is used

  So the glucose left after 1 minute = 120 - 1.02 * 1

From option (a), we can see that for 20 minutes

  g = 120 - (1.02 * 20)

So for t minutes, glucose left can be represented by the equation:

g = 120 - 1.02t

c) from the above equation, we can find :

       t intercept

g = 0

0 = 120 - 1.02t

120 = 1.02t

t = 117.6 minutes

   g- intercept

t = 0

 g = 120 - 1.02*0 = 120 cc

Now the graph is a straight line passing through (0,120) and (117.6,0).

Therefore the equation of the relationship between g and t is found and intercepts are found.

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Answer:

Caca

Step-by-step explanation:

Caca

After evaluating f (-2), where f(x )is the piecewise function graphed given, the value of f(-2) = -3.

What is the piecewise function?

A function that is defined piecemeal is one that has numerous subfunctions, each of which applies to a distinct interval in the domain. Instead of being a property of the function itself, piecewise definition is a means to express the function.

Here, we have

The problem asks for f(-2), which is the value of y, that is, the vertical axis, when x, that is, the horizontal axis is -2.

We can see that at x = -2, the function changes the definition, thus it is called a piece-wise function.

To find the numeric value at x = -2, we have to look at the definition with the closed circle, which is at y = -3.

Hence, the value of f(-2) = -3.

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Answer:

x = 30, -40

Step-by-step explanation:

This a right triangle, so you can use the Pythagorean Theorem.

[tex]a^{2}+ b^{2}=c^{2}[/tex], now in place of the a, b and c use x, x +10 and 50.

so a = x, b =x+10 and c = 50

[tex]x^{2} + (x+10)^{2} =50^{2}[/tex]

[tex]x^{2} +x^{2} +20x +100 = 2500[/tex]

Between 7.7 weeks and 68.5 weeks search times does chebyshev's theorem guarantee that we will find at least 89% of the graduates.

Chebyshev's theorem states that for any distribution, at least 1 - 1/k^2 of the data will fall within k standard deviations from the mean.

To find the bounds for at least 89% of the graduates, we need to find the value of k that satisfies this condition.

Using Chebyshev's theorem, we have:

1 - 1/k^2 = 0.89

Solving for k, we get:

k = sqrt(1/0.11) ≈ 3.3

Therefore, at least 89% of the graduates will have job search times between:

38.1 - 3.3 x 10.1 ≈ 7.7 weeks

38.1 + 3.3 x 10.1 ≈ 68.5 weeks

Rounding to the nearest tenth, we get:

between 7.7 weeks and 68.5 weeks.

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The summation of permutations of the n-1 rolls that contain at least one 6.

Permutation refers to the arrangement of a set of objects in a specific order or sequence. It is a mathematical concept that involves counting the number of ways in which a set of objects can be ordered or arranged.

However, we need to exclude the sequences that already contain a 6 before the nth roll. These sequences do not belong in event E n . Therefore, we need to find the number of permutations of the n-1 rolls that contain at least one 6.

Let's first consider the case where there is exactly one 6 in the sequence. We have n-1 slots for the rolls, and we need to choose one of them to be a 6. There are (n-1) ways to choose the position of the 6, and the remaining (n-2) slots can be filled with any of the remaining 5 numbers. Therefore, there are

=> [tex](n-1) * 5^{n-2}[/tex]

ways to have exactly one 6 in the sequence.

Now let's consider the case where there are two 6s in the sequence. We have n-1 slots for the rolls, and we need to choose two of them to be 6s. There are (n-1) choose 2 ways to choose the positions of the 6s. For the remaining (n-3) slots, we can fill them with any of the remaining 4 numbers. Therefore, there are

=> [tex](n-1 choose 2) * 4^{n-3}[/tex]

ways to have exactly two 6s in the sequence.

We can continue this pattern for the case where there are k 6s in the sequence. There are (n-1) choose k ways to choose the positions of the k 6s. For the remaining (n-k-1) slots, we can fill them with any of the remaining 6-(k+1) = 5-k numbers. Therefore, there are

=> [tex](n-1 choose k) * (5-k)^{n-k-1}[/tex]

ways to have exactly k 6s in the sequence.

To get the total number of outcomes in event E n , we need to sum up the number of outcomes for each possible number of 6s in the sequence. Therefore, we have:

[tex]|E n | = 6^{(n-1) - (n-1)} * 5^{(n-2) + (n-1 choose 2)} * 4^{(n-3)} - ... + (-1)^{(n-1) * (n-1 choose n-1)} * 1^{(n-n)}[/tex]

This is a formula for the total number of outcomes in event E n .

We use permutations to count the number of ways to arrange the rolls, and we use the principle of inclusion-exclusion to exclude the sequences that contain a 6 before the nth roll.

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The school newspaper reporter wants to find out how many students will attend Tet festivities. She randomly surveys 13 students and knows that 24% of students typically attend the festivities. We are interested in the probability of the number of students who will attend.

The probability that at most 4 students will attend is 0.4936 and the probability that more than 3 students will attend is 0.5064. Calculating these probabilities can be done using the binomial probability formula. First, calculate the probability of exactly 4 students attending. This probability is [tex]0.24^4 * 0.76^9 = 0.0038[/tex]. Then, calculate the probability of 3 students or fewer attending by adding the probabilities of 0, 1, 2, and 3 students attending, which is 0.0013 + 0.0358 + 0.1045 + 0.2696 = 0.4113. Finally, subtract 0.4113 from 1 to get 0.4936. We are interested in the probability of the number of students who will attend. To calculate the probability of more than 3 students attending, subtract 0.4113 from 1 to get 0.5064.

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The form of the function that most quickly reveals the y-intercept is m(c) = 2 + 6(x + 2), and the y-intercept is 14.

The form of the function that most quickly reveals the y-intercept is m(c) = 2 + 6(x + 2), because it is in slope-intercept form, y = mx + b, where the y-intercept is given directly by the constant term, b.

To find the y-intercept of this function, we can substitute x = 0, since the y-intercept occurs when x = 0:

m(0) = 2 + 6(0 + 2) = 2 + 12 = 14

Therefore, the y-intercept of the function is 14.

Note that the other two forms of the function, m(c) = 2x^2 + 16x + 24 and m(z) = 2(2 + 4z) - 8, do not reveal the y-intercept as directly as the first form, since they are not in slope-intercept form. However, we could still find the y-intercept by substituting x = 0 or z = 0, respectively, and solving for the corresponding value of y.

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one possible explanation for why there are two different ways of adding up the symbols on the tablet is given below.

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.

The ancient Egyptian number system was based on hieroglyphs, and it used a decimal system with hieroglyphs representing the powers of 10. However, without seeing the specific symbols on the tablet, it is difficult to provide an accurate answer.

Assuming that the tablet is using a basic arithmetic operation such as addition, and each symbol represents a different number,

it is possible that there are two different ways to add up the numbers represented by the symbols on the tablet.

One way to approach this problem is to try to find two different sets of numbers that when added, produce the same result.

For example, if the symbols represent the numbers 2, 3, and 5, then one way to add them up would be:

2 + 3 + 5 = 10

Another way to add them up would be:

3 + 7 = 10

In this case, the symbol for 2 would represent the number 7, and the symbol for 5 would represent the number 3. This would be one possible explanation for why there are two different ways of adding up the symbols on the tablet.

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sorry but without seeing the actual tablet then we cant work it out to get a correct answer

We clearly seen that a little sign mistake here in 'j'.

In the given question  you have computed the cross product of v=i+2j−3k and w=2i−j+2k. You got v×w=i+8j−5k Without actually redoing v×w and we have to tell how can we spot a mistake in my work.

cross product of v and w i.e v×w is orthogonal to v and w

i.e (v×w)×v = 0 = (v×w)×w

Now v×w = i+8j₋5k

v = i+2j−3k

w = 2i−j+2k

then,

(v×w)×v = ( i+8j₋5k)×(i+2j−3k)

(v×w)×v = 1+16+15

(v×w)×v = 32≠0

and

(v×w)×w = ( i+8j₋5k)×(2i−j+2k)

(v×w)×w = 2₋8₋10

(v×w)×w = ₋16≠0

hence given cross product is wrong

Also we clearly seen that a little sign mistake here in 'j'

i.e if (v×w) = ( i₋8j₋5k)

(v×w)×v = ( i₋8j₋5k)×(i+2j−3k)

(v×w)×v = 1₋16+15

(v×w)×v = 0

and

(v×w)×w = ( i₋8j₋5k)×(2i−j+2k)

(v×w)×w = 2+8₋10

Hence, (v×w) = ( i₋8j₋5k)

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we can purchase a maximum of 23 hats if we buy them for $3.50 each, or a maximum of 27 hats if we buy them for $2.97 each.

Two or more expressions with an Equal sign is called as Equation.

Function to represent the total cost of the order in terms of the number of hats ordered:

C(n) = {3.50n, if n < 25;

2.97n, if n >= 25}

where n is the number of hats ordered, and C(n) is the cost of the order.

If the budget for hats is $80, we can set C(n) equal to 80 and solve for n:

3.50n = 80, if n < 25;

2.97n = 80, if n >= 25

For n < 25, we get:

n = 80/3.50 ≈ 22.86 =23

if the budget is $80 and the hats cost $3.50 each, we can buy a maximum of 23 hats.

For n >= 25, we get:

n = 80/2.97 ≈ 26.96

So if the budget is $80 and the hats cost $2.97 each, we can buy a maximum of 27 hats.

Therefore, we can purchase a maximum of 23 hats if we buy them for $3.50 each, or a maximum of 27 hats if we buy them for $2.97 each.

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Answer:

Step-by-step explanation:

10000

Answer:10000

Step-by-step explanation:

dimosure

Using a telephone directory as the sampling frame can lead to under-coverage and bias, and it is generally not considered a representative sampling method.

Other more representative sampling methods, such as random digit dialing, address-based sampling, or a combination of methods, may be used to obtain a more accurate representation of the population.

Selecting a sample of households from the telephone directory can result in under-coverage and bias for several reasons.

Concept: When random sampling is not used, bias or regular flaws in the way the sample mean the population can occur.

Voluntary reply to samples, in which respondents choose their own respondents, and benefit samples, in which people who live nearby are included in the sample, are particularly biased.

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At a significance level of 0.01, the hypothesis that the average number of dropouts in the high school is 21 is not refutable based on the available data.

To determine if the average number of dropouts in the high school is 21, we need to conduct a one-sample t-test. The null hypothesis is that the average number of dropouts is 21, and the alternative hypothesis is that it is not 21.

We can use statistical software or a calculator to find the test statistic and p-value. With a sample size of 17, the degree of freedom is 16. Assuming a significance level of 0.01, the critical t-value is ±2.921.

The calculated t-value for this sample is -0.294, and the p-value is 0.773. Since the calculated t-value is not in the rejection region, we fail to reject the null hypothesis. This means that we do not have sufficient evidence to claim that the average number of dropouts is different from 21.

In conclusion, at a significance level of 0.01, the hypothesis that the average number of dropouts in the high school is 21 is not refutable based on the available data.

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The probability that a randomly chosen fitness club member signs up for a free running session is 0.12 or 12%.

What is probability ?
Probability is a measure of the likelihood or chance of an event occurring. It is usually expressed as a number between 0 and 1, where 0 means that the event is impossible and 1 means that the event is certain to occur. For example, if the probability of an event is 0.5, it means that there is a 50% chance that the event will occur.

Given by the question:

There are a total of 100 members in the fitness club, and 30 members have signed up for the free session. Let's call the number of members who signed up for the weightlifting session "x", so the number of members who signed up for the running session and the swimming session each is "2x".

The total number of members who signed up for the free session is:

x + 2x + 2x = 5x

We know that 30 members signed up for the free session, so we can set up the following equation:

5x = 30

Solving for x, we get:

x = 6

Therefore, 6 members signed up for the weightlifting session, and 12 members signed up for each of the running and swimming sessions.

The probability of a randomly chosen member signing up for a free running session is the number of members who signed up for the running session divided by the total number of members:

P(Running) = 12/100

P(Running) = 0.12

Therefore, the probability that a randomly chosen fitness club member signs up for a free running session is 0.12 or 12%.

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Number of adult tickets that Paige sold is 350 and the number of children tickets that she sold is 105.

Linear equations involve one or more expressions including variables and constants and the highest exponent of the variable is 1.

System of linear equations involve two or more linear equations.

Let x be the number of adult tickets and y be the number of children tickets.

Total number of tickets sold = 455

So, x + y = 455

Cost of each adult ticket is $8 and cost of each children ticket is $5 and the total cost is $3,325.

8x + 5y = 3325

So we got a system of two linear equations.

x + y = 455 ⇒ y = 455 - x

8x + 5y = 3325

Substituting y = 455 - x in 8x + 5y = 3325,

8x + 5(455 - x) = 3325

Solving,

8x + 2275 - 5x = 3325

3x = 1050

x= 350

y = 455 - 350 = 105

Hence number of adult tickets Paige sold is 350 and that of children tickets is 105.

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Answer:

Step-by-step explanation:

1.

The values of all 9 angles shown were found using the property of a kite they are:

m∠1=55°

m∠2=35°

m∠3=35°

m∠4=90°

m∠5=55°

m∠6=67°

m∠7=67°

m∠8=23°

m∠9=23°

What is an angle ?

In Kite figure, it is given that:

∠2+∠3 = 70°

Since they are equal so ∠2=∠3=35

∠8+∠9 = 46°

Since they are equal so ∠8=∠9=23°

∠5 = 180-∠3-∠4 = 180-35-90=55°

Similarly, ∠1 = 55°

∠6+∠7 = 180-∠8-∠9 = 180-46 = 134°

Since they are equal

∠6=∠7 = 67°

Therefore, all 9 angles were calculated using property of a kite.

m∠1=55°

m∠2=35°

m∠3=35°

m∠4=90°

m∠5=55°

m∠6=67°

m∠7=67°

m∠8=23°

m∠9=23°

In mathematics, an angle is a geometric figure formed by two rays or line segments that share a common endpoint, called the vertex. The rays or line segments are called the sides or legs of the angle, and the distance between the sides at the vertex is called the angle's measure.

Angles are typically measured in degrees or radians, and they can be classified by their measures as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees and less than 180 degrees), straight (exactly 180 degrees), reflex (greater than 180 degrees and less than 360 degrees), or full (exactly 360 degrees).

Kites have several properties related to their angles, including:

Two pairs of opposite angles in a kite are congruent. That is, the angles formed between the pairs of congruent sides are equal.

One diagonal of a kite bisects the other diagonal. This means that the diagonal that connects the non-congruent vertices of the kite divides the other diagonal into two equal segments.

The sum of the measures of the two non-congruent angles in a kite is 180 degrees. This is because the kite can be divided into two congruent triangles by drawing the diagonal that bisects the other diagonal, and the sum of the angles in a triangle is always 180 degrees.

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The formula for distance between two points d = √(c- a)² + (d -b)².

To derive the formula, let us consider two points in 2D plane A (a, b) and B(c, d) is d = √(c- a)² + (d -b)².

Given:

The Euclidean distance formula is another name for the formula used to calculate the separation between two points on a two-dimensional plane.

To derive the formula, let us consider two points in 2D plane A (a, b) and B(c, d)

By the Pythagoras theorem,

d²= (c- a)² + (d -b)²

d = √(c- a)² + (d -b)²

For example, Find the distance between the points (-2, 3), and (5, 6).

d = √(c- a)² + (d -b)²

d= √(5 -(-2))² + (6 -3)²

d= √7² + 3²

d= √49+ 9

d= √58

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10^-2/3 - 5 is the x-intercept of the given equation.

The x-intercept of a function is the point where the function f(x) is equivalent to zero.

Given the function below;

f(x)=3log(x+5)+2

3log(x+5)+2 = 0

Make x the subject of the formula:

3log(x+5) = -2

log(x+5) = -2/3

x + 5 = 10^-2/3

x = 10^-2/3 - 5

Hence the x-intercept of the graph of f(x)=3log(x+5)+2 is 10^-2/3 - 5

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The statements are classified as follows:

7. Never true.

8. Never true.

9. Never true.

The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.

Hence statement 9 is false.

In a triangle, the sum of the lengths of the two smaller sides has to be greater than the length of the greater side.

Hence statements 7 and 8 are false.

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When estimating the population average using the sample mean, adding observations to a sample will always decrease the standard error of your estimate is random sampling variability.

In statistical analysis, estimating the population average is a common task. One way to estimate this is by using the sample mean. However, as more observations are added to the sample, the standard error of the estimate decreases. Let's explore why this is the case.

The standard error of the mean (SEM) is a measure of the amount of variability in a sample mean from one sample to another. It can be calculated using the formula:

SEM = standard deviation / square root of sample size

As we can see, the SEM is inversely proportional to the square root of the sample size. This means that as we increase the sample size by adding more observations, the denominator of the SEM formula increases, causing the SEM to decrease.

Now, let's consider how this affects our estimate of the population average. The sample mean is an unbiased estimate of the population mean. This means that on average, the sample mean will equal the population mean. However, due to random sampling variability, the sample mean can differ from the population mean.

The standard error of the mean gives us an idea of how much the sample mean can vary from one sample to another.

Therefore, adding more observations to a sample decreases the SEM, making our estimate of the population average more precise. This is because the average of a larger sample is more representative of the population average, as it is less affected by random sampling variability.

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