What is the x-coordinate of the midpoint of line segment AB with points A(21, 5) and B(-9, -13).

Answers

Answer 1

Answer:

  6

Step-by-step explanation:

You want the x-coordinate of the midpoint of segment AB with endpoint coordinates A(21, 5) and B(-9, -13).

Midpoint

The midpoint of a line segment has coordinates that are the average of the endpoint coordinates:

  M = (A +B)/2

  M = ((21, 5) +(-9, -13))/2 = (12, -8)/2

  M = (6, -4)

The midpoint of AB is (6, -4).

The x-coordinate of the midpoint is 6.

What Is The X-coordinate Of The Midpoint Of Line Segment AB With Points A(21, 5) And B(-9, -13).

Related Questions

Jim takes out a mortgage for 30 years at an interest rate of 2. 49% and his monthly repayments are $986. 50. What is the principal loan amount? Round your answer to the nearest ten thousand dollars. Do NOT round until you have calculated the final answer

Answers

The principal mortgage amount is $220,000.

To compute the principal loan amount, we need to apply the compound interest formulation for a loan:

M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]

Where M is the monthly reimbursement, P is the primary mortgage amount, i is the month-to-month interest charge, and n is the range of months.

First, we need to calculate the month-to-month interest charge:

i = 2.49% / 12 = 0.02075

Subsequent, we want to calculate the range of months:

n = 30 years x 12 months/year = 360 months

Now we are able to plug in the given values:

$986.50 = P [ 0.02075(1 + 0.02075)^360 ] / [ (1 + 0.02075)^360 – 1]

Solving for P, we get:

P = $221,114.07

Rounding to the nearest ten thousand dollars, the principal mortgage amount is $220,000.

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Find the measurements of the numbered angles of this circle

Answers

98 Degrees and 82 degrees  are  the measurements of the numbered angles of this circle.

What is a circle?

With no sides or edges, a circle is a figure with a round shape. A circle can be characterized in geometry as a closed shape, a two-dimensional shape, or a curved shape.

                           The collection of all points in the plane that make up a circle are all equally spaced from a certain point known as the "centre," making the form a closed two-dimensional shape. The symmetry line of reflection is formed by each line that traverses the circle. Moreover, it possesses rotational symmetry around the center for each angle.

Measure of angle 1 = (33 + 131)/2

                         = 82 degrees

Measure of angle 2 = 180 - 82

                         = 98 Degrees

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the area of base of a cylindrical tank is 38.5m². Find the perimeter of base​

Answers

Answer: We know that the area of the base of a cylinder is given by the formula:

A = πr²

where A is the area of the base, π is the mathematical constant pi (approximately equal to 3.14), and r is the radius of the base.

To find the perimeter of the base, we need to know the circumference of the circle that forms the base of the cylinder. The formula for the circumference is:

C = 2πr

where C is the circumference and r is the radius.

To find the radius of the base, we can rearrange the formula for the area:

A = πr²

r² = A/π

r = √(A/π)

Substituting the given value for the area of the base, we get:

r = √(38.5/π) ≈ 3.5 m

Now, we can use the formula for the circumference to find the perimeter of the base:

C = 2πr = 2π(3.5) ≈ 22 m

Therefore, the perimeter of the base is approximately 22 meters.

Step-by-step explanation:

PLEASE HELP ASAP!!!

Question in photo

Answers

Answer:

-5 is your answer

Step-by-step explanation:

What do you mean leading coefficient of the following polynomial.

Look at the x^n and see who has the biggest values.

-5x^7 is the leading coefficient.

simplify: 1÷2+2 3÷4-3÷8​

Answers

The expression 1÷2+2 3÷4-3÷8​ when simplified is 9/4

How to simplify the expression

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

1÷2+2 3÷4-3÷8​

Express the terms properly as fractions

So, we have the following

1/2 + 2 3/4 - 3/8

Express each term as proper/improper fraction

So, we have the following representation

1/2 + 11/4 - 3/8

Express the denominator as the same

So, we have

4/8 + 22/8 - 3/8

Evaluate the like terms

18/8

So, we have

9/4

Hence, the solution is 9/4

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Banks and Credit Unions. (choose all that apply)


A. . Are very similar. They offer the same products and services. What makes them different is the way in which they are legally organized.


B. . Have similar interest rates. However, credit unions often have higher savings rates (better) and lower loan rates (better) than banks.


C. . Are different because credit unions cannot issue ATM and credit cards, making it inconvenient to access your funds.


D. . Are very different. Most people cannot join a credit union

Answers

A. Differently organized; B. Different rates; C. Inconvenient access; D. Limited membership.

To calculate the interest rate on a loan, you will need to figure out the amount of interest you will need to pay over the life of the loan. First, you will need to determine the loan amount, the interest rate, and the loan term. Then, you will need to calculate the total interest by multiplying the loan amount by the interest rate and the loan term. Finally, you will need to divide the total interest by the loan amount and then multiply that by 100 to get the annual percentage rate (APR). This will give you the interest rate you will need to pay over the life of the loan.

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If the exponential model f(x)=8(9)x is written with the base e, it will take the form A0ekx. What is A0 and what is k?

Answers

Answer:

429

Step-by-step explanation:

someone help me asap

Answers

Answer:

Step-by-step explanation:

angle

What is the length of the hypotenuse?
8ft 6ft c=?

Answers

Answer:10ft

Step-by-step explanation:

we must use pythagorean theorem

[tex]which \ is \ a^{2} +b^{2} =c^{2} \\6^{2} + 8^{2} =36+64=100=10^{2}\\c=10--- > hypotenuse[/tex]

Round all numbers to the nearest whole number. Find the missing measurements of the triangle

Answers

The missing measurements of the triangle given include:

Angle B = 63 degreesAB = 26 .4 units AC = 23. 6 units

How to find the missing measurements ?

The value of angle B can be found thanks to the knowledge that the interior angles of a triangle add up to 180 degrees and we have two of the angle measurements :

= 180 - angles measurements

= 180 - 27 - 90

= 63 degrees

The value of AB can be found by using the Sin theta operation such that:

Sin theta = Opposite / hypotenuse

Sin ( 27 degrees ) = 12 / hypotenuse which is AB

hypotenuse = 12 / ( Sin ( 27 degrees ))

= 26 .4 units

The value of AC measurement is:

Tan ( 27 degrees ) = 12 / AC

AC = 12 / Tan ( 27 degrees )

AC = 23. 6 units

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Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4

Answers

We can say that after answering the offered question The terms in the equation given equation can be streamlined and rearranged to produce the following equation: 23p – 101 = 64p – 40

What is equation?

An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.

Using the equality properties, we can rewrite the preceding equation as follows:

2.3p – 10.1 = 6.5p – 4 – 0.01p

2.3p – 10.1 = 6.49p – 4

We can alter the equations using the equality characteristics to look for equations that have the same solution. The answers to the next two equations are the same as those for the previous one:

2.3p – 10.1 = 6.4p – 4

The following equation can be created by adding 0.09p to both sides of the given equation:

2.3p - 10.1 + 0.09p = 6.5p - 4 - 0.01p + 0.09p

2.39p – 10.1 = 6.4p – 4

23p – 101 = 65p – 40 – p

The terms in the given equation can be streamlined and rearranged to produce the following equation:

23p – 101 = 64p – 40

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NO EXPLANATION JUST ANSWER!

Answers

The surface area of a cube with a volume of 4913 cubic yards is 972 square yards.

What is surface area?

Surface area is the measurement of the total area of a 3D object. It is the sum of all the individual faces of a 3D object, including the area of each side of the object. It is a way to measure the total exposed area of an object, including any protrusions or indentations.

The surface area of a cube is the sum of the areas of the six faces of the cube. The formula for the surface area of a cube is 6 × s2, where s is the length of the side of the cube.

In this case, the volume of the cube is 4913 cubic yards, so the length of the side of the cube is ∛4913 = 17. Therefore, the surface area of the cube is 6 × 172 = 972 square yards.

To find the surface area of a cube, the first step is to determine the length of the side of the cube. This is done by finding the cube root of the volume (in this case, 4913). Once the length of the side is known, the surface area of the cube is simply 6 times the length of the side squared.

In conclusion, the surface area of a cube with a volume of 4913 cubic yards is 972 square yards.

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For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions. 10. f(x)=x−14​ 11. f(x)=5x+22​ 12. f(x)=x2−9x​ 13. f(x)=x2+5x−36x​ 14. f(x)=x3−273+x​ 15. f(x)=x3−16x3x−4​ 16. f(x)=x3+9x2+14xx2−1​ 17. f(x)=x2−25x+5​ 18. f(x)=x−6x−4​ 19. f(x)=3x−14−2x​ For the following exercises, find the x - and y-intercepts for the functions. 20. f(x)=x2+4x+5​ 21. f(x)=x2−xx​ 22. f(x)=x2+11x+30x2+8x+7​ 23. f(x)=x2−10x+24x2+x+6​ 24. f(x)=3x2−1294−2x2​

Answers

10. Domain: all real numbers; Vertical Asymptotes: x=14; Horizontal Asymptotes: none
11. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=5x+22
12. Domain: all real numbers; Vertical Asymptotes: x=0 and x=9; Horizontal Asymptotes: none
13. Domain: all real numbers; Vertical Asymptotes: x=-5; Horizontal Asymptotes: y=0
14. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x3
15. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x3-16x3
16. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=x2-1
17. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=0
18. Domain: all real numbers; Vertical Asymptotes: x=-6; Horizontal Asymptotes: y=0
19. Domain: all real numbers; Vertical Asymptotes: none; Horizontal Asymptotes: y=3x-14
20. x-intercepts: -4 and 1; y-intercepts: 5
21. x-intercepts: 0 and 1; y-intercepts: 0
22. x-intercepts: -8 and -3; y-intercepts: 7
23. x-intercepts: -1 and 6; y-intercepts: 0
24. x-intercepts: -2 and 0; y-intercepts: -1294

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15. The expression √x is equivalent to 14i√3. What is the value of x?
a.-588
b. -588i
c.588
d. 588i

16. An expression is shown below.
x^5 + 6x^3 +8x
Which value of x makes the expression equal to 0?
a. -2i²
b. -2i
c. 4i
d. 4i²

Answers

(15) The value of x is -588 (option A).

(16) The value of x makes the expression equal to 0 is -2i. (option B).

What is the value of x?

The value of x is calculated by solving the equation as follows;

(15) We have the equation:

√x = 14i√3

Squaring both sides, we get:

x = (14i√3)^2

x = -588

(16) Factoring the expression, we get:

x(x⁴ + 6x² + 8)

Setting each factor equal to 0, we get:

x = 0 or x⁴ + 6x² + 8 = 0

Let's solve for the second equation:

Let y = x²

Then we have:

y² + 6y + 8 = 0

Factorizing, we get:

(y + 2)(y + 4) = 0

So y = -2 or y = -4

Since y = x², this means:

x² = -2 or x² = -4

Taking the square root of both sides, we get:

x = ±√(-2) or x = ±√(-4)

Simplifying using imaginary numbers, we get:

x = ±i√(2) or x = ±2i

Thus, for the expression √x is equivalent to 14i√3,  the value of x is -588. And for the expression x⁵ + 6x³ + 8x, x = -2i.

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khomo and Peter bought a house for R575 000 as an investment. khomo payed R245 000 and Peter payed the rest . they sold the house 5 years later and made a profit of R 234 500 if they share the profit in the same ratio as their respective investment, how much profit will Peter receive​

Answers

Peter will receive R134,417.50 of the profit.

What is Investment ?

Investment refers to the purchase of goods that are not consumed today but are used in the future to create wealth. In financial terms, investment typically involves buying assets such as stocks, bonds, real estate or other financial instruments with the expectation of earning a profit or income from them.

Khomo paid R245,000 for the house, so his investment ratio is:

245,000 / 575,000 = 0.426

Peter's investment ratio is:

1 - 0.426 = 0.574

If they share the profit in the same ratio as their respective investment, then Peter will receive:

0.574 x 234,500 = 134,417.5

Therefore, Peter will receive R134,417.50 of the profit.

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The prime factorization of 6. Use exponents when appropriate and order the factors from least to greatest

Answers

The prime factorization of 6 is the expression of 6 as a product of its prime factors.

A prime factor is a prime number that is a factor of the given number. To find the prime factorization of 6, we need to express 6 as a product of its prime factors. Since 6 can be evenly divided by 2 and 3, which are both prime numbers, the prime factorization of 6 is 2 x 3. We can also write this as 2^1 x 3^1, using exponents to show that there is only one 2 and one 3 in the prime factorization of 6. Finally, since 2 and 3 are both prime, we order the factors from least to greatest as 2 x 3.

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Publishing scientific papers online is fast, and the papers can be long. Publishing in a paper journal means that the paper will forever in libraries. The British Medical Journal combines the two: it prints short and readable versions, with longer versions available online. The journal asked a random sample of 104 of its recent authors several questions. One question was "Should the journal continue using this system?" In the sample, 72 said "yes." Do the data give good evidence that more than 67% of authors support continuing this system? Interpret the p-value in the context of the problem. If an error has been committed, explain which type of error could it be.

Answers

There is no significant evidence that more than two-thirds (67%) of authors support continuing this system.

Define null hypothesis?

This hypothesis is either rejected or accepted, depending on whether the population or sample under investigation is viable.

In other words, the null hypothesis is an assertion that the sample observations are the result of chance. It is asserted that surveyors who are interested in the data have made this allegation. It has the prefix H0.

Let p be the proportion of authors who support continuing the system.

Then hypotheses are:

H0: p=0.67

Ha: p>0.67

For calculating the test statistic:

z = p(s)-p/√ [p0(1-p)/N]

where, p(s) = 72/104 ≈ 0.692, p is 0.67 and N is the sample size =104

Then Z ≈ 0.477

The test statistic's p-value is 0.317.

Since 0.317>0.05 and we're assuming a significance level of 0.05, we are unable to reject the null hypothesis.

p-value 0.317 is the likelihood that the sample is chosen from the distribution assumed under null hypothesis, that is where the fraction of authors favouring the new publication scheme is at most 0.67.

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2 + 3x = 12 - 5x

Why I am having so much trouble with this?

Answers

Answer:x=5/4

Step-by-step explanation:

you can take numbers with same variable(i mean x or y or other) in the same side:

2+3x=12-5x

2-12=-5x-3x

-10=-8x

x=10/8=5/4

Answer:

x = 5/ 4

Step-by-step explanation:

2 + 3x =12- 5x

3x + 5x = 12 - 2

8x = 10

x = 10/8

x = 5/4

Fill in the gaps below:
a. from A to B
b. from B to C
c. from A to C

Answers

Answer:

a. from A to B, a rotation by 90 degrees anti-clockwise about (0,0).

b. from B to C, a translation by vector (-3/4, -1/4).

c. from A to C, a rotation by 60 degrees, anti-clockwise about (0,0).

Hope this helps, I'm sorry if it doesn't. If you need more help, ask me! :]

A rectangular portrait is 1 yard wide and 2 yards high. It costs $10.17 per yard to put a gold frame around the portrait. How much will the frame cost?

Answers

Answer:

The perimeter of the portrait is $2\times(1+2)=6$ yards.

To find the cost of the frame, we need to calculate the length of the frame needed, which is the same as the perimeter of the portrait. So, the length of the frame needed is 6 yards.

The cost of the frame will be the cost per yard multiplied by the length of the frame:

$6~\text{yards} \times $10.17/\text{yard} = $61.02$

Therefore, the frame will cost $61.02.

Which system of equations is represented by the graph? Graph of a quadratic function intersecting a linear function at point 2 comma 0. y = x2 + 3x + 2 x + y = 2 y = x2 − 3x + 2 x − y = 2 y = x2 − 3x + 2 x − y = 3 y = x2 − 3x + 2 x + y = 3

Answers

After answering the presented question, we can conclude that The point of intersection of these two equations is (2, 0), which is the spot on the graph where the parabola and line intersect.

What is equation?

An equation is a mathematical statement that validates the equivalent of two expressions linked by the equal symbol '='. For example, 2x - 5 = 13. 2x-5 and 13 are two phrases. The character '=' is used to connect the two expressions. An equation is a mathematical formula that has two algebras on either side of an assignment operator (=). It illustrates the equivalency relationship between the left and middle formulas. In any formula, L.H.S. = R.H.S. (left side = right side).

The graph represents the following equation system:

[tex]y = x^2 + 3x + 2 x + y = 2[/tex]

This is due to the fact that the graph of a quadratic function intersecting a linear function at point (2, 0) is represented by a parabola and a straight line intersecting at that point (2, 0).

The equation [tex]y = x^2 + 3x + 2[/tex] represents an upward-opening parabola that goes through the point (-2, 0).

The equation x + y = 2 denotes a straight line that connects the locations (0, 2) and (2, 0).

The point of intersection of these two equations is (2, 0), which is the spot on the graph where the parabola and line intersect.

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ABCD is a trapezium.
EBC is a straight line.
F is the point on AB so that DFE is a straight line.
Angle BEF = 45°
Angle FBC = 100°
Work out the value of angle ADF.
Give a reason on the same line of each
stage of your working.
+
F
Answer: angle ADF =
E
45°
100°
B
Diagram not drawn to scale
C
Total marks: 4

Answers

Answer:

35 Degrees

Step-by-step explanation:

Given in figure.

Happy to help :)

The solution is, the value of the angle ADF = 35° .

Here, we have,

from the given diagram, we get,

ABCD is a trapezium.

EBC is a straight line.

F is the point on AB so that DFE is a straight line.

Angle BEF = 45°

Angle FBC = 100°

now, we have,

angle FBE = 180 - 100

                 = 80 degrees

now, from triangle BEF , we get,

Angle EFB = 55 degrees

again, as AFB is vertically opposite angle,

so, angle AFB = Angle EFB = 55 degrees

finally from triangle ADF , we get,

Angle ADF = 180 - 90 - 55

                  = 90 - 55

                  = 35 degrees

so, the value of the angle ADF = 35° .

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Pleaseeeee hurryyyyy

Julia had a bag filled with gumballs. There were 1 watermelon, 2 lemon-lime, and 3 grape gumballs. What is the correct sample space for the gumballs in her bag?

A. Sample space = watermelon, lemon-lime, lemon-lime, grape, grape, grape
B. Sample space = lemon-lime, watermelon, grape
C. Sample space = 1, 2, 3, 4, 5, 6
D. Sample space = 1, 2, 3

Answers

Answer:

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

dudjeisosidheis9sjdhxisjdgskkjehud

Helphelphelphelphelphelp

Answers

Answer: -1

Step-by-step explanation:

An easy way to figure out the slop of a linear graph is the formula:

(y1-y2)/(x1-x2)

In order to use this, find two spots on the line that run exactly through a point, in this example (0,20) and (20,0).

Plug in the xs and ys into the formula, which gets you  

(20-0) / (0-20)

When simplified, the result is -1 and that is your slope

To find the distance from point C to the line AB, you must find the length of the segment from C_______ to AB.


A. Widthwise

B. Perpendicular

C. Parallel

D. Lengthwise

Answers

the distance from point C to line AB, you must find the length of the segment from C perpendicular to AB

A line segment in geometry has two different points on it that define its boundaries. A line segment is sometimes referred to as a section of a line that links two places. The difference between a line and a line segment is that a line has no endpoints and can go on forever in either direction. A ray has only one endpoint and an endlessly long another end, as opposed to a line segment that has two ends.

When the angle of inclination between two line segments is exactly equal, those two segments are said to be perpendicular.

The plus symbol is formed when two perpendicular line segments cross one other.

To find the distance from point C to line AB, you must find the length of the segment from C_______ to AB

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Oliver earns $25.50 per hour and saves 25% of his income. Oliver wants to save at least $160
a. Write this situation as an inequality.

b. What is the minimum number of hours Oliver needs to work?

Answers

Answer: a. 25.50(h)(25%) >= 160  

b. 26

Step-by-step explanation:

a. Let the number of hours Oliver works be h hours.

25.50(h)(25%) >= 160

b. 25.50(h)(25%) >= 160

h >= 25.1 (cor. to 3 sig. fig)

The minimum number of hours Oliver needs to work is 26 hours.

Use the graph to determine

(a) open intervals on which the
function is increasing, if any.

(b) open intervals on which the function is decreasing if any

(c) open intervals on which the function is constant if any

Answers

using graph f(x) is concave down on (-1,0), (1,∞).

What is derivative?

A function's varied rate of change with respect to an independent variable is referred to as a derivative. When there is a variable quantity and the rate of change is irregular, the derivative is most frequently utilized. The derivative is used to assess how sensitive a dependent variable is to an independent variable (independent variable). In mathematics, a quantity's instantaneous rate of change with respect to another is referred to as its derivative. Investigating the fluctuating nature of an amount is beneficial.

f(x) is concave up where f'(x) is increasing and concave down where f'(x) is decreasing

So, using graph f(x) is concave down on (-1,0), (1,∞).

Hence, using graph f(x) is concave down on (-1,0), (1,∞).

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Demonstrate and explain how to evaluate the derivative for each of the following definite integrals using the fundamental theorem of calculus. ?A)d/xd∫ 4x(2(6cos(t)+7) 4)dtB)d/xd∫ x3(4sin(t 3−3))dt

Answers

The derivative for each of the definite integrals using the fundamental theorem of calculus is

a) d/dx ∫ 4x(2(6cos(t)+7)⁴)dt is 4(2(6cos(b)+7)⁴) - 4(2(6cos(a)+7)⁴)

b)  d/dx ∫ x³(4sin(t³−3))dt is 12(b²sin(b³-3) - a²sin(a³-3))

The fundamental theorem of calculus tells us that the derivative of the definite integral of a function f(x) with respect to x is equal to the function evaluated at the upper limit of integration minus the function evaluated at the lower limit of integration. In other words, if we have an integral of the form ∫f(x)dx evaluated from a to b, then

d/dx ∫f(x)dx = f(b) - f(a)

Let's apply this to the first integral, A).

A) d/dx ∫ 4x(2(6cos(t)+7)⁴)dt

We begin by recognizing that the function inside the integral is a function of t, not x. However, we want to take the derivative with respect to x. This means that we need to use the chain rule to differentiate the integrand with respect to x.

Using the chain rule, we have

d/dx [4x(2(6cos(t)+7)⁴)] = 4(2(6cos(t)+7)⁴)(d/dx [4x])

= 4(2(6cos(t)+7)⁴)(4)

= 32(2(6cos(t)+7)⁴)

Now, we can apply the fundamental theorem of calculus. Let's say we are evaluating the integral from a to b. Then,

d/dx ∫ 4x(2(6cos(t)+7)⁴)dt = [4b(2(6cos(t)+7)⁴)] - [4a(2(6cos(t)+7)⁴)]

= 4(2(6cos(b)+7)⁴) - 4(2(6cos(a)+7)⁴)

This is the final answer for A).

Now, let's move on to integral B).

B) d/dx ∫ x³(4sin(t³−3))dt

Again, we need to use the chain rule to differentiate the integrand with respect to x.

d/dx [x³(4sin(t³−3))] = (d/dx [x³])(4sin(t³−3))

= 3x²(4sin(t³−3))

Now, we apply the fundamental theorem of calculus. Let's say we are evaluating the integral from a to b. Then,

d/dx ∫ x³(4sin(t³−3))dt = [3b²(4sin(t³−3))] - [3a²(4sin(t³−3))]

= 12(b²sin(b³-3) - a²sin(a³-3))

This is the final answer for B).

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The Water Department checks the city water supply on a regular basis for
contaminants such as trihalomethanes (THMs). The Water Department takes
200 samples and estimates that the concentration of THMs in your drinking
water is 3 ppb (parts per billion), with a standard deviation of 0. 3 ppb.
Assuming the samples were random and unbiased, how much confidence
can you have in this data?​

Answers

We can be 95% confident that the true mean concentration of THMs in your drinking water lies within the range [2.9584 ppb, 3.0416 ppb]. This was calculated using a confidence interval formula for the mean.

The confidence interval for the mean concentration of THMs can be calculated using the formula x ± z * (s/√n), where x is the sample mean, z is the z-score for a given confidence level, s is the sample standard deviation and n is the sample size.

In this case, we have x = 3 ppb, s = 0.3 ppb, and n = 200. To calculate the confidence interval at a certain confidence level (e.g. 95%), we need to find the corresponding z-score. For a 95% confidence level, the z-score is approximately 1.96.

Substituting these values into the formula above gives us:

3 ± 1.96 * (0.3/√200) = [2.9584, 3.0416]

So we can be 95% confident that the true mean concentration of THMs in your drinking water lies within the range [2.9584 ppb, 3.0416 ppb].

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Suppose X and Y are independent and X∼Bernoulli(1/2) and Y∼Bernoulli(1/3). Determine a) the PMF of X+Y b) the CDF of X+Y c) the range of X+Y d) the expectation of X+Y e) the variance of X+Y f) the expectation of X⋅Y

Answers

The PMF of X+Y is given byP(X + Y = k) = P(X = 0,Y = k) + P(X = 1,Y = k-1) + P(X = 1,Y = k) = (1/2)(1/3)^k + (1/2)(1/3)^(k-1) + (1/2)(1/3)^(k-1)for k = 0, 1, 2.

The CDF of X+Y can be determined as below:P(X + Y = 0) = (1/2)(1/3)^0 = 1/2P(X + Y ≤ 1) = P(X = 0,Y = 0) + P(X = 1,Y = 0) + P(X = 0,Y = 1) = 1/2 + (1/2)(1/3) + (1/2)(1/3) = 5/6P(X + Y ≤ 2) = P(X = 0,Y = 0) + P(X = 1,Y = 0) + P(X = 0,Y = 1) + P(X = 1,Y = 1) + P(X = 0,Y = 2) = 1/2 + (1/2)(1/3) + (1/2)(1/3) + (1/2)(1/3)(2/3) + (1/2)(1/3)^2 = 11/18P(X + Y ≤ 3) = 1, as the range of X+Y is 0, 1, 2.

The expectation of X+Y can be determined asE(X + Y) = E(X) + E(Y) = 1/2 + 1/3 = 5/6

The variance of X+Y can be determined asVar(X + Y) = Var(X) + Var(Y) = (1/2)(1/2) + (1/3)(2/3) = 7/18

The expectation of X⋅Y can be determined as below: E(X⋅Y) = P(X = 1,Y = 1) = (1/2)(1/3) = 1/6

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