For the following exercises, determine whether each of the following relations is a function_ 1.y =2x + 8 2. {(2, 1), (3,2), (-1,1), (0, -2)}
For the following exercises, evaluate the function f(x) = -3x2 + 2x at the given input: 3. f(-2) 4. f(a)

The common volume of the two spheres is 5πr³/6. Since the two spheres are identical, the volume of the common portion is twice the volume of the spherical cap.

Since the distance between the centers of the two spheres is r/2, a cross-section of the two spheres and the plane perpendicular to their centers of symmetry will form an isosceles triangle with height h=r/2 and base 2√(r²-(r/2)²). The intersection of the two spheres will be a spherical cap with height h and radius r. The volume of this spherical cap is

V_cap = (1/3)π[tex]h^2[/tex](3r-h).

Since the two spheres are identical, the volume of the common portion is twice the volume of the spherical cap, so

Substituting h=r/2, we get

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Answer:

$1400 divided by 10 = $140

$1400 - $140 = $1260

That hypotenuse is the radius of the second circle which is equal to 15.

The segment joining the centers of the circles bisects the common chord.

So in the circle with radius 13, the common chord, the segment joining the centers of the circles, and the radius to an endpoint of the common chord form a right triangle with hypotenuse 13 and one leg 12; that makes the distance from the center of that circle to the common chord 5.

Since the length of the segment joining the two circles is 14, the distance from the center of the other circle to the common chord is 14-5=9.

Then in that other circle, we have a right triangle with legs 9 and 12, making the hypotenuse 15.

And that hypotenuse is the radius of the second circle which is equal to 15.

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Answer:

7)  216 ft²

8)  106 yd²

9)  200 m²

Step-by-step explanation:

To calculate the area of each given figure, subtract the area of the cut-out rectangle (marked in blue on the attached diagram) from the area of the larger rectangle.

[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]

Note: The height of the larger rectangle is not complete. I have assumed it is 12 ft.

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&=21 \cdot 12- 6\cdot 6\\&=252-36\\&=216\; \sf ft^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 17\cdot 10- (17-9)\cdot 8\\&= 17\cdot 10- 8\cdot 8\\&=170-64\\&=106\; \sf yd^2\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 15\cdot 20- 10\cdot 10 \\&=300-100\\&=200\; \sf m^2\end{aligned}[/tex]

If a is negative, the parabola opens down. Option C is the correct option.

What is a parabola?

Any point on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a fixed straight line. A parabola is a U-shaped plane curve. The topic of conic sections includes parabola, and all of its concepts are covered here.

A polynomial function of the second degree is a quadratic function. A quadratic function has this general form:

y = ax² + bx + c

A parabola is a name for a quadratic function's graph.

A parabola resembles the letter "U" or an upside-down "U" in general shape.

If the leading coefficient is greater than zero, the parabola opens upward; if the leading coefficient is less than zero, the parabola opens downward. This simple rule can be used to determine whether the graph of a quadratic function opens upward or downward.

Given equation is

y = ax²

The leading coefficient is a and it is a negative number. Thus the parabola opens down.

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The general rule of the arithmetic sequence is given as follows:

[tex]a_n = -1 + 2(n - 1)[/tex]

The terms are given as follows:

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The nth term of an arithmetic sequence is given by the following rule:

[tex]a_n = a_1 + (n - 1)d[/tex]

The difference between the 3 terms is of 6, hence the common difference is given as follows:

3d = 6

d = 2.

As the 2nd term is of 1, the first term is of:

[tex]a_1 = 1 - 2 = -1[/tex]

Hence the rule is given as follows:

[tex]a_n = -1 + 2(n - 1)[/tex]

The terms are given as follows:

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The correlation coefficient of X and Y is 1, since their joint density is always positive for x ≤ y.

The correlation coefficient of two continuous random variables X and Y is a measure of how similar the two variables are in terms of their behavior. In this case, X and Y have a joint density of e−y, for 0 < x ≤ y < ∞ and 0 otherwise. This implies that the correlation coefficient of X and Y is 1, since their joint density is always positive for x ≤ y. This means that there is a perfect linear relationship between the two variables, meaning that as one variable increases, the other will also increase by the same amount.

To calculate the correlation coefficient, we need to calculate the covariance and variances of X and Y. The covariance is calculated by integrating the joint density of X and Y over the given range. The variance of X and Y is then calculated by taking the square root of the expected value of the squares of the differences between the expected value of X and Y and their respective observed values. Since the joint density of X and Y is always positive for x ≤ y, the correlation coefficient of X and Y will always be 1.

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The CPU (Central Processing Unit) performs all arithmetic operations (for example, addition and subtraction) and all logic operations (such as sorting and comparing numbers).

The acronym that fills in the blank is CPU, which stands for Central Processing Unit. The CPU is essentially the brain of a computer, responsible for processing all the information that flows through the system.

Arithmetic operations are those that involve mathematical calculations, such as addition, subtraction, multiplication, and division. These are relatively straightforward tasks for the CPU, which can perform them quickly and accurately.

Logic operations, on the other hand, involve comparing and manipulating data based on various conditions.

CPU has the ability to perform logic operations is a crucial aspect of computing, as it allows computers to make decisions and process information based on a set of rules or conditions.

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The relationship between the two variables negative linear association.

A straight-line link between two variables is referred to statistically as a linear relationship (or linear association).

Linear relationships can be represented graphically or mathematically as the equation y = mx + b.

Given:

The data table is organized as follows:

1         2          3         4          5

32       26        20       14        8

We see that more the number of members in the team increase, the more the times decrease.

The correlation is a negative linear association, we conclude.

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P(success I at least 60 years old) = 0.125.

The possibility of an event in time is known as a probability in mathematics. How frequently does the incidence occur over the course of a specific time period, in plain English?

Given:

Over 1000 people try to climb mt Everest every year.

Of those who try to climb Everest, 31 percent succeed.

The probability that a climber is at least 60 years old is 0.04.

The probability that a climber is at least 60 years old and succeeds in climbing Everest is 0.005.

P(success I at least 60 years old) = 0.005/0.04 = 0.125

Therefore, the probability is 0.125.

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Answer:

Step-by-step explanation:

The set {0,1,2,3,4,5,6,7,8,9} has 2^10 = 1024 subsets. To see why, note that each element can be either in or out of a given subset, so there are 2 choices for each of the 10 elements, giving 2x2x2x2x2x2x2x2x2x2 = 1024 possible subsets.

The correct expression for P(X = k) will be [tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^k[/tex]

Given,

The geometric distribution arises as the distribution of the number of times we flip a coin until it comes up heads. consider now the distribution of the number of flips until the -th head appears, where each coin flip comes up heads independently with probability.

The negative binomial distribution describes the probability of getting the -th success (in this case, heads) on the th trial, given a probability p of success on each trial.

We can write this as:

[tex]P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}[/tex]

where k is the number of flips needed to get r successes, including the rth success.

In this case, we want the distribution of the number of flips until the rth head appears,

so we can set p = 1/2,

since the coin is fair.

We also need to adjust the formula to count the number of flips, including the flip with the rth head.

Since we need r heads to appear, the last head must appear on the rth flip, so we can write:

[tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^r \left(\frac{1}{2}\right)^{k-r+1}[/tex]

Simplifying the expression,

we get:

[tex]P(X = k) = \binom{k-1}{r-1} \left(\frac{1}{2}\right)^k[/tex]

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From the given data the probability that a respondent 18-29 years and 30+ years are 0.536 and 0.654 espectively. The probability that they answer yes (i.e. global warming will have an impact on them during their lifetime) is 0.265. There also appears to be a difference between ages 18-29 and 30+ regarding concern over global warming.

The probability that a respondent 18-29 years of age thinks that global warming will not pose a serious threat during his/her lifetime is:

P(18-29 and No) = 293/(134 + 293 + 117) = 0.536

The probability that a respondent 30+ years of age thinks that global warming will not pose a serious threat during his/her lifetime is:

P(30+ and No) = 432/(131 + 432 + 178) = 0.654

For a randomly selected respondent, the probability that they answer yes (i.e. global warming will have an impact on them during their lifetime) is:

P(Yes) = (134 + 131)/(134 + 131 + 293 + 432 + 117 + 178) = 0.265

Yes, there appears to be a difference between ages 18-29 and 30+ regarding concern over global warming. The probability of a respondent 30+ years of age thinking that global warming will not pose a serious threat during their lifetime (0.654) is higher than the corresponding probability for respondents 18-29 years of age (0.536). This suggests that older respondents are less concerned about global warming than younger respondents. However, a formal statistical test would be needed to confirm this difference.

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Answer:

The correct graph of the given function is option C. Y.

Step-by-step explanation:

The correct graph is option B (X) because it satisfies all the conditions specified in the function definition.

The first segment of the graph starts at x=-2 and goes to the left, passing through the point (-4,-4) and representing the equation 2x+4. The second segment of the graph starts at x=-2 and goes to the right, passing through the point (2,3) and representing the equation 4+1/2x. The third segment of the graph starts at x=2 and goes to the right, passing through the point (5,0) and representing the equation -x+5.

Option A (W) has a line that passes through the point (2,6), which is not on the graph of the function, and option C (Y) has a line that passes through the point (2,5), which is also not on the graph of the function. Option D (Z) has an open circle at (-2,3), which should be a closed circle according to the function definition, and a closed circle at (2,5), which should be an open circle according to the function definition.

Answer:

11

Step-by-step explanation:

[tex]\sqrt{64}[/tex] -10 + 13

8 - 10 + 13

-2 + 13

11

The volume of the solid is [tex]4a^2h[/tex].

Each square cross-section has a side length equal to the diameter of the circle, which is 2a.

Therefore, the area of each cross-section is

[tex](2a)^2 = 4a^2[/tex]

The volume of the solid can be found by integrating the area of the cross sections over the height of the solid.

Since the cross-sections are perpendicular to the base, the height of each cross-section is equal to the thickness of the solid, which we can denote as dx.

Thus, the volume of the solid is given by the integral of the cross-sectional area with respect to x, from x = 0 to x = h, the height of the solid:

V = ∫ (0 to h) [tex]4a^2 dx[/tex]

V = [tex]4a^2[/tex] ∫(0 to h) dx

V = [tex]4a^2[/tex] (0 to h)

V = [tex]4a^2[/tex] (h - 0)

V = [tex]4a^2h[/tex]

Therefore, the volume of the solid is [tex]4a^2h[/tex]

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If the production is increased by 10%, the new production levels for sport utility vehicles and pickup trucks are:

Factory 1: 110 sport utility vehicles

Factory 2: 99 sport utility vehicles

Factory 3: 77 sport utility vehicles, 66 pickup trucks

Factory 4: 33 sport utility vehicles, 66 pickup trucks

To find the production levels if production is increased by 10%, we need to multiply each entry of the matrix by 1.1:

A' = 1.1A = | 110 99 77 33 || 44 22 66 66 |

The matrix A' represents the new production levels. The entry in the first row and first column of A' (110) represents the number of sport utility vehicles produced at the first factory after the 10% increase in production. Similarly, the entry in the second row and third column of A' (66) represents the number of pickup trucks produced at the third factory after the 10% increase in production.

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The percentage decrease in the temperature is 87.13 %.

The percentage formula is dividing the required value by the total value and multiplying by 100.

Example:

Required percentage value = a

total value = b

Percentage = a/b x 100

Example:

50% = 50/100 = 1/2

25% = 25/100 = 1/4

20% = 20/100 = 1/5

10% = 10/100 = 1/10

We have,

The temperature of a mixture changes from 121(1/4)°F to 15(3/5)°F.

This means,

Initial temperature = 121(1/4) = 485/4

Final temperature = 15(3/5) = 78/5

The change in temperature.

= 78/5 - 485/4

=  (312 - 2425)/20

= 2113/20

= - 105(13/20)

This means,

There is a decrease in temperature.

Now,

The percentage decreased.

= 105(13/20) / (485/4) x 100

= (2113/20) / (485/4) x 100

= 2113/20 x 4/485 x 100

= 2113/2425 x 100

= 87.13%

Thus,

The percentage decrease in the temperature is 87.13 %.

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The expression that represents the number of inches the plant grew is 8 - x.

An expression is combination of variables, numbers and operators.

A plant measured x inches tall last week and 8 inches tall this week.

We have to find the expression which represent the plant grew this week.

The number of inches the plant grew can be represented by the difference between its height this week and its height last week.

Eight minus x is the required expression.

8 - x

Hence,  the expression that represents the number of inches the plant grew is 8 - x.

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Complete question

A plant measured x inches tall last week and 8 inches tall this week. Represent the number of inches the plant grew

A. X - 8

B. 8 - X

For the expression not chosen, describe a situation that the expression might represent.

The value of x is 24. The angle measures of each vertex of the yard are m∠A = 160°, m∠B = 142°, m∠C = 120°, m∠D = 156°, m∠E = 31°, m∠F = 111°.

What is a polygon?

A polygon is a closed polygonal chain made up of a finite number of straight line segments and is a type of plane figure in geometry. A polygon is a region that is bounded by a bounding circuit, a bounding plane, or both. A polygonal circuit's segments are referred to as its edges or sides.

The formula of the sum of interior angles of a polygon is (n - 2)×180°, where n is number of sides.

The number of sides of the given irregularly shaped garden is 6.

The sum of the interior angles of the garden is (6 - 2)×180° = 720°

Also given:

m∠A = (7x - 8)°, m∠B = (4x + 46)°, m∠C = (5x)° , m∠D = (6x + 12)°, m∠E = (x + 7)°, m∠F = (5x - 9)°.

The sum of the interior angles of the garden is

(7x - 8)° + (4x + 46)° + (5x)° + (6x + 12)° +  (x + 7)° + (5x - 9)°

Therefore,

(7x - 8)° + (4x + 46)° + (5x)° + (6x + 12)° +  (x + 7)° + (5x - 9)° = 720°

Combine like terms:

(7x +4x + 5x + 6x + x + 5x) + (-8 +46 + 12 + 7 - 9) = 720

28x + 48 = 720

Subtract 48 from both sides:

28x = 672

Divide both sides by 28:

x = 24

Putting x = 28 in the given angles:

m∠A = (7×24 - 8)° = 160°

m∠B = (4×24 + 46)°= 142°

m∠C = (5×24)° = 120°

m∠D = (6×24 + 12)°= 156°

m∠E = (24 + 7)° = 31°  

m∠F = (5×24 - 9)° = 111°

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The additional factors that might have an effect on the stress levels of tea drinkers are called confounding variables. Confounding variables are variables that are not the main focus of the study but may have an impact on the results.

The term used to describe these additional factors that might have an effect on the stress levels of tea drinkers is "confounding variables." Confounding variables are extraneous factors that can affect the relationship between two variables of interest and can lead to incorrect conclusions about the relationship. In this case, the higher proportion of yoga practitioners and the increased exercise among tea drinkers could be considered confounding variables because they may be influencing the observed relationship between tea drinking and lower stress levels, rather than tea consumption alone.

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The probability that a woman is actually pregnant given that the test is positive is 0.974 (or 97.4%).

a) To find the probability that a woman is actually pregnant given that the test is positive, we can use Bayes' theorem:

P(p|+) = P(+|p) P(p) / P (+)

where P(p) is the prior probability of a woman being pregnant (which is 0.82), P (+) is the probability of a positive test result, and P(+|p) is the conditional probability of a positive test result given that the woman is pregnant.

We are given that the test has a false positive rate of 0.02, which means that P(+|~p) = 0.02 (where ~p denotes "not pregnant"). We are also given that the test has a correct positive rate of 0.95, which means that P (+|p) = 0.95. Therefore, we can calculate:

P (+) = P(+|p) P(p) + P(+|~p) P(~p)

= 0.95 * 0.82 + 0.02 * (1 - 0.82)

= 0.802

Substituting into Bayes' theorem, we get:

P(p|+) = 0.95 * 0.82 / 0.802

= 0.974

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Answer:

2:5

Step-by-step explanation:

We know

There are 9 girls and 6 boys taking golf lessons.

So, there are a total of 15 taking a golf lesson.

Write the ratio that compares the number of girls taking golf lessons to the total number of students taking golf lessons.

The ratio is

6:15 = 2:5

So, the ratio is  2:5

Answer:

[tex]\dfrac{4a^6b^5}{c^3}[/tex]

Step-by-step explanation:

Given expression:

[tex]\dfrac{8a^4b^{-3}c^{12}}{2a^{-2}b^{-8}c^{15}}[/tex]

Separate like terms:

[tex]\dfrac{8}{2} \cdot \dfrac{a^4}{a^{-2}}\cdot \dfrac{b^{-3}}{b^{-8}} \cdot \dfrac{c^{12}}{c^{15}}[/tex]

Divide the numbers:

[tex]4\cdot \dfrac{a^4}{a^{-2}}\cdot \dfrac{b^{-3}}{b^{-8}} \cdot \dfrac{c^{12}}{c^{15}}[/tex]

[tex]\textsf{Apply the exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]

[tex]4 \cdot a^{4-(-2)} \cdot b^{-3-(-8)} \cdot c^{12-15}[/tex]

Simplify the exponents:

[tex]4 \cdot a^6 \cdot b^5 \cdot c^{-3}[/tex]

[tex]\textsf{Apply the exponent rule} \quad a^{-n}=\dfrac{1}{a^n}:[/tex]

[tex]4 \cdot a^6 \cdot b^5 \cdot \dfrac{1}{c^3}[/tex]

Simplify:

[tex]\dfrac{4a^6b^5}{c^3}[/tex]

Answer:

Quadrant II

Step-by-step explanation:

sin (∅) > 0 means sin (∅)is positive

sec (∅) < 0 means sec (∅) is negative

thus,

∅ lies in quadrant II

The flight takes a total of  3 hours 45 minutes

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

Inital time = 3:30 pm EST

Final time = 4:15 pm PST

When converted, we have

Inital time = 12:30 pm PST

Final time = 4:15 pm PST

So, we have

Difference = 4:15 pm PST -  12:30 pm PST

Evaluate the difference

Difference = 3 hours 45 minutes

Hence, the diration is 3 hours 45 minutes

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The position vector of a particle that has an acceleration, a(t) = 19t i + eᵗ j + e⁻ᵗ k, is equals to the 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k.

We have, The acceleration vector function of a particle is defined as, a(t)

= 19t i + eᵗ j + e⁻ᵗ k and intial velocity and position is, v(0) = k, r(0) = j + k.

We have to calculate the position vector of a particle. Now, as we know, the acceleration of a particle is equals to derivative of velocity of particle with time. In other words, velocity is integration of acceleration with respect to time.

Mathematically, v(t) = ∫a(t)dt , let C be integration constant ( vector).

v(t) = ∫a(t)dt = ∫[ (19t) i + (eᵗ) j + (e⁻ᵗ) k] dt

=> v(t) = 19(t²/2) i + eᵗ j - e⁻ᵗ k + C

At t = 0 , v(0) = k

=> k = 19(0²/2) i + e⁰j - e⁻⁰ k + C

=> k = 0 + j - k + C

=> C = 2k - j

so, v(t) = 19(t²/2) i + eᵗ j - e⁻ᵗ k + 2k - j

= 19(t²/2) i + (eᵗ - 1) j - (e⁻ᵗ - 2) k

Now, Position of a particle is determined by integrating the velocity of particle with respect to time, r(t) = ∫v(t)dt , let D be integration constant ( vector). So, r(t)

= ∫[19(t²/2) i + (eᵗ - 1) j - (e⁻ᵗ - 2) k ] dt

= 19 (t³/6) i + eᵗ j - t j - e⁻ᵗ k + 2t k + D

At t = 0, r(0) = j + k

=> j + k = 19 (0³/6) i +e⁰ j - 0j - e⁻⁰ k +2× 0k +D

=> j + k = 0 + j - k + D

=> D = 2k

so, r(t) = 19 (t³/6) i + eᵗ j - t j - e⁻ᵗ k + 2t k + 2k

= 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k

Hence, required position vector is 19 (t³/6)i + (eᵗ - t )j - (e⁻ᵗ - 2t - 2)k.

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Answer:

Step-by-step explanation:

250 per week base

15%of 1800 he made is = 270

270+250 =520

He made 520$ in the week he made 1800$ in sales

Jaxson's salary is a combination of a fixed $250 plus 15% commission on his weekly sales. For $1800 in sales, he will earn $520. If the sales amount is x dollars, then he will earn $250 + 0.15*x dollars in a week.

Jaxson's income consists of a guaranteed base salary of $250 per week and a commission of 15% on his total weekly sales. If Jaxson made $1800 in sales in a week, his total income for the week would be his base salary plus his commission i.e., $250 + 0.15*$1800 = $520.

Similarly, if Jaxson made x dollars in sales in a week, his total income for that week would be his base salary plus his commission on the sales i.e., $250 + 0.15*x dollars.

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Answer: Let's assume that each of the 12 collectors donates n paintings. Then the total number of paintings donated will be 12 * n. We know that the total number of paintings donated by all collectors is 48, so we have the equation:

12 * n = 48

Dividing both sides by 12, we find that each collector donates n = 4 paintings. So, if Sergio has p paintings in his collection, after his donation he will have p - 4 paintings in his collection. The final answer is:

p - 4

Step-by-step explanation:

In a survery of hospital employees here it is parameter or  statistic  data is given  in Question

Parameters are numeric values ​​that characterize the population as a whole. A statistic is a number that describes the characteristics of a sample.

For example, median income in the United States is a demographic parameter. Conversely, the median income for the US sample is a sample statistic. Both values ​​represent median income, but one is a parameter to the statistic. Easy to remember parameters and stats! Both are summary values ​​that describe the group and have a convenient mnemonic device to remind you which group is being described. Notice their initials.

Parameter = Population

Statistics = Sample

A population is the entire group of people, things, animals, trades, etc. that are being studied. A sample is part of a population.

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