I NEED HELPP ... 26 points!
Quotient: The result of dividing two numbers
Explanation: Just some simple dividing and rounding
Quotient - 102.756098
Rounding - 102.76
Answer: 102.76
Someone please help!!! will give brainliest!!!
Round your answer to the nearest hundredths, if necessary.
Find the surface area of the figure
Answer:161.56
Step-by-step explanation:
8 x5=40
8 x 7.07=56.56
1/2 x 5 x 5 x 2= 25
8 x 5=40
Add that all together
HELP PLEASE I NEED HELP !
Answer:
G
Step-by-step explanation:
out of a total of 280 spinners as the overall.
3/40 were defective
280 * 3/40 = 21
Answer:
G
Step-by-step explanation:
For every 40 spinners 3 are defective
Divide amount made by 40 for numbers of groups of 40
280 ÷ 40 = 7 , then
7 × 3 = 21 ← likely defective spinners → G
the circumfrence is 72 cm what is the length of the minor arc
Answer:
Should be 9 centimeters.
Step-by-step explanation:
Tell which value of the variable is the solution of the equation 30 = 6w W = 3, 5, 6, 8??
Answer: w=5
Step-by-step explanation: Hope this help
66666666 help me plz plz plz
Answer:
XY would also be 7 centimeters which is answer D.
Step-by-step explanation:
This is a parallelogram, meaning that the adjacent sides are congruent. As well, the triangles making up the figure are congruent, so it makes sense that XY would also equal 7 centimeters.
PLEASE HELP !!!! find the focus (parabolas)
(y-2)^2=4(x+3)
Answer:
C. ( -2 , 2 )Step-by-step explanation:
Focus of parabola [tex](y-2)^2 = 4(x+3)[/tex] is (-2 , 2) .
Correct option is C .
Given, Equation of parabola [tex](y-2)^2 = 4(x+3)[/tex]
Focus of parabola :
Standard equation of parabola : (y - k)² = 4a(x - h)
Axis of parabola : y = k
Vertex of parabola : (h, k)
Focus of parabola : (h + a, k)
Compare the equation of parabola with standard equation.
(y - k)² = 4a(x - h)
[tex](y-2)^2 = 4(x+3)[/tex]
k = 2
a = 1
h = -3
So focus of parabola: (h + a, k).
-3 + 1 , 2
Focus of parabola = -2 , 2
Hence the correct option is C .
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Solve the system of equations using the substitution method. Show your work and be sure to include the solution to the system.
let f be a function with derivative given by f x ¢( ) = 3 x + 1. what is the length of the graph of y f = ( )x from x = 0 to x = 1.5 ?
If "f" is function with derivative as f'(x) = √(x³ + 1), then length of graph of y = f(x) from x = 0 to x = 1.5 is (b) 2.497.
To find the length of the graph of y = f(x) from x = 0 to x = 1.5, we use the arc-length formula for a function y = f(x):
Length = ∫ᵇₐ√(1 + [f'(x)]²) dx,
Given the derivative : f'(x) = √(x³ + 1), we substitute it into the arc-length formula:
Length = [tex]\int\limits^{1.5}_{0}[/tex] √(1 + (√(x³ + 1))²) dx,
Simplifying the expression inside the square root:
We get,
Length = [tex]\int\limits^{1.5}_{0}[/tex] √(1 + x³ + 1) dx
= [tex]\int\limits^{1.5}_{0}[/tex]√(x³ + 2) dx
= 2.497.
Therefore, the correct option is (b).
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The given question is incomplete, the complete question is
Let f be a function with derivative given by f'(x) = √(x³ + 1). What is the length of the graph of y = f(x) from x = 0 to x = 1.5?
(a) 4.266
(b) 2.497
(c) 2.278
(d) 1.976
PR and os are diameters of circle T. What is the
measure of SR?
50°
* 80°
• 100°
120
Answer:
100degrees
Step-by-step explanation:
Find the diagram attached
From the diagram
<PQT =<TRS = 40
Since the triangle STR is isosceles, here,
<TSR =<TRS = 40
Also the sum of angle in a triangle is 189, hence:
Arc SR+<TSR +<TRS = 18₩
ArcSR +40+40=180
ArcSR +80=180
ArcSR = 180-80
ArcSR = 100degrees
Hence the measure of SR is 100degrees
The measure of arc SR is 100°. The correct option is the third option - 100°
Calculating the measure of an Arc
From the question, we are to determine the measure of arc SR.
The measure of arc SR = <STR
Now, we will determine the measure of <STR
In the diagram, T is the center of the circle.
∴ TP and TQ are radii.
Then, we can conclude that ΔPQT is an isosceles triangle.
Recall: Base angles of an isosceles triangle are equal.
∴ <PQT = <TPQ = 40°
Now, consider ΔPQT
<QTP + <TPQ + <PQT = 180° (Sum of angles in a triangle)
<QTP + 40° + 40° = 180°
<QTP + 80° = 180°
<QTP = 180° - 80°
<QTP = 100°
Also, in the diagram, <QTP and <STR are vertically opposite angles
NOTE: Vertically opposite angles are equal
That is, <QTP = <STR
∴ <STR = 100°
Hence, the measure of arc SR is 100°. The correct option is the third option 100°
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Here is the complete and correct question:
Line PR and Line QS are diameters of circle T. What is the measure of Arc SR?
50°
80°
100°
120°
Please find the attached image
A survey was conducted that asked 1014 people how many books they had read in the past year. Results indicated that x = 12.7 books and s= 16.6 books. Construct a 90% confidence interval for the mean number of books people read. Interpret the interval Click the icon to view the table of critical t-values. Construct a 90% confidence interval for the mean number of books people read and interpret the result. Select the correct choice below and fill in the answer boxes to complete your choice (Use ascending order. Round to two decimal places as needed) OA. There is 90% confidence that the population mean number of books read is between __ and __. if repeated samples are taken, 90% of them will have a sample mean between __ and __. There is a 90% probability that the true mean number of books read is between __ and __ .
There is 90% confidence that the population mean number of books read is between 9.85 and 15.55. If repeated samples are taken, 90% of them will have a sample mean between 9.85 and 15.55. There is a 90% probability that the true mean number of books read is between 9.85 and 15.55.
What is the 90% confidence interval for the mean number of books read?The survey results indicate that the mean number of books read in the past year is estimated to be 12.7, with a standard deviation of 16.6. To construct a 90% confidence interval, we can use the t-distribution and the sample size of 1014. Using the critical t-values from the table, we calculate the margin of error by multiplying the standard error (s / √n) with the t-value. Adding and subtracting the margin of error from the sample mean gives us the lower and upper bounds of the confidence interval.
The confidence interval for the mean number of books read is calculated as 12.7 ± (t-value * 16.6 / [tex]\sqrt{1014}[/tex]), which simplifies to 12.7 ± 2.58. Therefore, the confidence interval is (9.85, 15.55).
In interpretation, this means that we can be 90% confident that the true mean number of books read in the population falls between 9.85 and 15.55. If we were to repeat the survey and take different samples, 90% of those samples would produce a mean number of books read within the range of 9.85 to 15.55. The confidence interval provides a range of values within which we can reasonably estimate the true population mean.
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y=5x
y=-3x+24
Solve by substitution
Answer:
x=12
Step-by-step explanation:
5x = -3x+24
2x = 24
x = 12
Select 2A316 in base 10.
A number cube has sides numbered 1 through 6. The probability of rolling a 2 is 1/6. What is the probability of not rolling a 2?
a. 1/6
b. 5/6
c. 1/5
d. 1/4
Probability refers to the measure of the likelihood that a particular event will occur. It is represented as a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.
The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6.
Here's why: When we roll a number cube with sides numbered 1 through 6, there are six possible outcomes, each with an equal probability of 1/6:1, 2, 3, 4, 5, 6.The probability of rolling a 2 is 1/6, which means there is only one way to roll a 2 out of the six possible outcomes. The probability of not rolling a 2 is the probability of rolling any of the other five possible outcomes. Each of these outcomes has an equal probability of 1/6. Therefore, the probability of not rolling a 2 is:1 - (1/6) = 5/6. Answer: b. 5/6.
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Given that the number cube has sides numbered 1 through 6. The probability of rolling a 2 is 1/6. The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6.
The probability of rolling any of the numbers 1, 3, 4, 5, or 6 is also 1/6 each.
The sum of the probabilities of all possible outcomes is 1.
The probability of an event happening is defined as the number of ways the event can occur, divided by the total number of possible outcomes.
The total number of possible outcomes is 6 (the numbers 1 through 6).
Thus, if the probability of rolling a 2 is 1/6, then the probability of not rolling a 2 is 1 - 1/6 = 5/6.
Therefore, the correct option is b. 5/6.
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Find the critical points, relative extrema, and saddle points. Make a sketch indicating the level sets. (a) f(x, y) = x - x2 - y2 (b) f(x, y) = (x + 1)(y – 2). (c) f(x, y) = sin(xy). (d) f(x, y) = xy(x - 1).
The critical points function relative extrema and saddle points.
(a) f(x, y) = x - x2 - y2 =f(x, y) = 0: x - x² - y²= 0
(b) f(x, y) = (x + 1)(y – 2)=: x + 1 = 0 and y - 2 = 0.
(c) f(x, y) = sin(xy)= cos(xy),
(d) f(x, y) = xy(x - 1)= (0, 0) and (1, y)
(a) For the function f(x, y) = x - x² - y²
To find the critical points, to find where the gradient is zero or undefined. The gradient of f(x, y) is given by (∂f/∂x, ∂f/∂y):
∂f/∂x = 1 - 2x
∂f/∂y = -2y
Setting both partial derivatives to zero,
1 - 2x = 0 -> x = 1/2
-2y = 0 -> y = 0
The only critical point is (1/2, 0).
To determine the nature of the critical point, examine the second-order partial derivatives:
∂²f/∂x² = -2
∂²f/∂y² = -2
∂²f/∂x∂y = 0
The determinant of the Hessian matrix is Δ = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² = (-2)(-2) - (0)² = 4.
Since Δ > 0 and ∂²f/∂x² = -2 < 0, the critical point (1/2, 0) is a local maximum.
To sketch the level sets, set f(x, y) to different constant values and plot the corresponding curves. For example:
f(x, y) = -1: x - x² - y² = -1
This equation represents a circle with radius 1 centered at (1/2, 0).
f(x, y) = 0: x - x² - y² = 0
This equation represents a parabolic shape that opens downward.
(b) For the function f(x, y) = (x + 1)(y - 2):
To find the critical points, we set both partial derivatives to zero:
∂f/∂x = y - 2 = 0 -> y = 2
∂f/∂y = x + 1 = 0 -> x = -1
The only critical point is (-1, 2).
To determine the nature of the critical point, we can examine the second-order partial derivatives:
∂²f/∂x² = 0
∂²f/∂y² = 0
∂²f/∂x∂y = 1
Since the second-order partial derivatives are all zero, we cannot determine the nature of the critical point based on them. We need further analysis.
To sketch the level sets, set f(x, y) to different constant values and plot the corresponding curves. For example:
f(x, y) = 0: (x + 1)(y - 2) = 0
This equation represents two lines: x + 1 = 0 and y - 2 = 0.
(c) For the function f(x, y) = sin(xy):
To find the critical points, both partial derivatives to zero:
∂f/∂x = ycos(xy) = 0 -> y = 0 or cos(xy) = 0
∂f/∂y = xcos(xy) = 0 -> x = 0 or cos(xy) = 0
From y = 0 or x = 0, the critical points (0, 0).
When cos(xy) = 0, xy = (2n + 1)π/2 for n being an integer. In this case, infinitely many critical points.
To determine the nature of the critical points, we can examine the second-order partial derivatives:
∂²f/∂x² = -y²sin(xy)
∂²f/∂y² = -x²sin(xy)
∂²f/∂x∂y = (1 - xy)cos(xy)
Since the second-order partial derivatives involve the trigonometric functions sin(xy) and cos(xy), it is challenging to determine the nature of the critical points without further analysis.
To sketch the level set f(x, y) to different constant values and plot the corresponding curves.
(d) For the function f(x, y) = xy(x - 1):
To find the critical points, both partial derivatives to zero:
∂f/∂x = y(x - 1) + xy = 0 -> y(x - 1 + x) = 0 -> y(2x - 1) = 0
∂f/∂y = x(x - 1) = 0
From y(2x - 1) = 0, y = 0 or 2x - 1 = 0. This gives us the critical points (0, 0) and (1/2, y) for any y.
From x(x - 1) = 0, x = 0 or x = 1. These values correspond to the critical points (0, 0) and (1, y) for any y.
To determine the nature of the critical points, examine the second-order partial derivatives:
∂²f/∂x² = 2y
∂²f/∂y² = 0
∂²f/∂x∂y = 2x - 1
For the critical point (0, 0), the second-order partial derivatives are ∂²f/∂x² = 0, ∂²f/∂y² = 0, and ∂²f/∂x∂y = -1. Based on the second partial derivative test, this critical point is a saddle point.
For the critical points (1, y) and (1/2, y) where y can be any value, the second-order partial derivatives are ∂²f/∂x² = 2y, ∂²f/∂y² = 0, and ∂²f/∂x∂y = 1. The nature of these critical points depends on the value of y.
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how do i find the area of this triangle
Step-by-step explanation:
You can apply cosinus theory for finding are
Area=cos40°3.4(ft)*2.7(ft)/2 like thia
2, 3, 1, 6, 4, 5, 3, 2, 3, 4 is the set
Answer:
mean: 3.3
median: 3
mode: 3
range: 5
Q1 = 2
Q3 = 4
IQR = 2
Step-by-step explanation:
Worth five points! it doesnt tell me if what answer is right but if i get 75% or up i will mark the first person who answered with an actual answer brainliest and i don't lie about brainliest!! Please no nonsense answers I just want help :(
Triangle DEF is an isosceles triangle with DE = EF, and mE =92°
What is mD?
A. 45
B. 44°
C. 88°
D. 90°
Answer:
Step-by-step explanation:
Answer:
44 is the awnser
Step-by-step explanation:
becuase if you were to look at the first persons work it is correct showing him solving the equasion and witch it is not 44
find the shaded region of the figure below
Answer:
-x³ + 3x² - 14x + 12
Step-by-step explanation:
Area of outer rectangle = (x² + 3x - 4) * (2x - 3)
= (x² + 3x - 4) * 2x + (x² + 3x - 4) * (-3)
=x²*2x + 3x *2x - 4*2x + x² *(-3) + 3x *(-3) - 4*(-3)
=2x³ + 6x² - 8x - 3x² - 9x + 12
= 2x³ + 6x² - 3x² - 8x - 9x + 12 {Combine like terms}
= 2x³ + 3x² - 17x + 12
Area of inner rectangle = (x² - 1)* 3x
= x² *3x - 1*3x
= 3x³ - 3x
Area of shaded region = area of outer rectangle - area of inner rectangle
= 2x³ + 3x² - 17x + 12 - (3x³ - 3x)
= 2x³ + 3x² - 17x + 12 -3x³ + 3x
= 2x³ - 3x³ + 3x² - 17x + 3x + 12
= -x³ + 3x² - 14x + 12
Use the Fundamental Theorem to calculate the following integral exactly: 7/6 Jo de cos? 0
The exact value of the integral [tex]\int\limits^0_{\pi/6}[/tex] 2/cos²θ dθ is √3/3 using the Fundamental Theorem of Calculus.
To calculate the integral ∫[0 to π/6] 2/cos²θ dθ using the Fundamental Theorem of Calculus, we need to find the antiderivative of the integrand and evaluate it at the upper and lower limits of integration.
The antiderivative of 2/cos²θ is tan(θ), so applying the Fundamental Theorem of Calculus:
[tex]\int\limits^0_{\pi/6}[/tex] 2/cos²θ dθ = [tan(θ)] evaluated from θ = 0 to θ = π/6
Substituting the upper and lower limits of integration:
= tan(π/6) - tan(0)
Since tan(0) = 0, we have:
= tan(π/6)
Using the value of tan(π/6) = √3/3, the exact value of the integral is √3/3.
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The question is -
Use the Fundamental Theorem to calculate the following integral exactly:
[tex]\int\limits^0_{\pi/6}[/tex] 2/cos²∅ d∅ = _______
A rectangular swimming pool is 6 it deep. One side of the pool is 3.5 times longer than the other. The amount of water needed to fill the swimming pool is
1344 cubic feet. Find the dimensions of the pool.
Answer:
8 feet by 28 feet by 6 feet
Step-by-step explanation:
So volume is length times width times height
It tells us that the volume is 1344 cubic feet (the water used to fill it)
And it also tells us that the height/depth (which are the same thing in this case) is 6ft
All we need now are length and width
We know that one of the sides is 3.5 times the other one. So we can just say length is x and width is 3.5x
So plugging that in, the equation becomes
[tex]3.5x*x*6=1344[/tex]
3.5 x times x is just 3.5x squared so
[tex]3.5x^2*6=1344[/tex]
divide both sides by 6
[tex]3.5x^2=244[/tex]
divide by 3.5
[tex]x^2 =64[/tex]
[tex]x=\sqrt{64}[/tex]
x = 8
So that means the one side is 8 feet long and the other side is 3.5 times that, which is 28 feet long.
So the dimensions of the pool are 8 feet by 28 feet by 6 feet
EQUAÇO
1. x + 5 - 25=x + 3x - 4
2. 1 - 2x = 3 - 2(x + 1)
Can someone please help me answer this question asap thank you
Which point is not on the line
The overhead reach distances of adult females are normally distributed with mean of 202.5 cm and standard deviation of 8.3 cm Find the probability that an individual dislance is greater Ihan 211.80 cm
The probability that an individual distance is greater than 211.80 cm is 0.1292.
The problem statement is:
The overhead reach distances of adult females are normally distributed with mean of 202.5 cm and standard deviation of 8.3 cm Find the probability that an individual distance is greater than 211.80 cm.
We need to find the z-score first as follows:$\begin{aligned}z&=\frac{x-\mu}{\sigma} \\z&=\frac{211.80-202.5}{8.3} \\z&=1.122\end{aligned}$
Using the standard normal table,
The standard normal distribution table is a compilation of areas from the standard normal distribution, more commonly known as a bell curve, which provides the area of the region located under the bell curve and to the left of a given z-score to represent probabilities of occurrence in a given population.
The probability is given by:P(Z > 1.122) = 0.1292
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The given information is Population Mean (µ) is 202.5 cm
Standard deviation (σ) is 8.3 cm
The distance we have to find is x = 211.80 cm.
Hence, the probability that an individual distance is greater than 211.80 cm is 0.1314.
The formula used for finding the probability is: Now we need to find z score to use the standard normal distribution tables. The formula for finding z score is:
z = (x - µ) / σ
Substitute the values in the above formula, we get
z = (211.80 - 202.5) / 8.3
z = 1.12
Now use this z value to look up in the standard normal distribution tables to find the probability.
P(z > 1.12) = 1 - P(z < 1.12)
From standard normal distribution tables, the probability of P(z < 1.12) is 0.8686.
Therefore, P(z > 1.12) = 1 - P(z < 1.12)
= 1 - 0.8686
= 0.1314
Hence, the probability that an individual distance is greater than 211.80 cm is 0.1314.
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What is the volume of the pyramid in
cubic centimeters?
Answer:
3328 cubic centimeters
Step-by-step explanation:
volume of pyramid equation:
V=(lwh)/3
V = (12·26·32) / 3
V =3328
Answer:
The answer is
[tex]9984 cm {}^{3} [/tex]
Step-by-step explanation:
The way i solved this was by using the formula to volume. I also am doing this but for me it is a bit easier. The simple formula is Width x Length x Height. Since i already have the numbers, it is easier to plug in the numbers
What is the value of Point C on the number line below?
A) 0.208
B) 0.28
C) 0.302
D) 0.32
Answer:
0.28
Step-by-step explanation:
All you need to do is count.
0.20, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.30
C
Point C sits on the point 0.28.
Seats in a theater are curved from the front row to the back. The front row has 10 chairs, the second has 16 and the third has 22, and so on.
A. Write a recursive rule for this series
B. Write an explicit rule for this series
C. Using the explicit formula, find the number of chairs in row 5
D. The auditorium can hold 17 rows of chairs. Write a sigma notation for this series, and then use either series formula to calculate how many chairs can fit in the auditorium
Answer:
The first term is 10.
The second term is 16
The third term is 22.
We can see that the first term plus 6, is:
10 + 6 = 16
Then the first term plus 6 is equal to the second term.
And the second term plus 6 is:
16 + 6 = 22
Then the second term plus 6 is equal to the third term.
A) As we already found, the recursive rule is:
Aₙ = Aₙ₋₁ + 6
B) The explicit rule is:
Aₙ = A₁ + (n - 1)*6
Such that A1 is the first term, in this case A₁ = 10
Then:
Aₙ = 10 + (n - 1)*6
C)
Now we want to find A₅, then:
A₅ = 10 + (5 - 1)*6 = 34
There are 34 chairs in row 5.
D)
Here we have 17 rows, then we can have 17 terms, this means that the total number of chairs will be:
C = A₀ + A₁ + ... + A₁₆
This summation can be written as:
∑ 10 + (n - 1)*6 such that n goes from 0 to 16.
The formula for the sum of the first N terms of a sum like this is:
S(N) = (N)*(A₁ + Aₙ)/2
Then the sum of the 17 rows gives:
S(17) = 17*(10 + (10 + (17 - 1)*6)/2 = 986 chairs.
There are total 986 chairs in the considered auditorium and there are 34 chairs in the fifth row.
The recursive rule for this series is: [tex]T_n = T_{n-1} + 6[/tex]The explicit rule for this series is: [tex]T_n = 6n + 4[/tex]What is recursive rule?A rule defined such that its definition includes itself.
Example: [tex]F(x) = F(x-1) + c[/tex] is one such recursive rule.
For this case, we're provided that:
Seats in rows are 10 in front, 16 in second, 22 in third, and so on.
10 , 16 , 22 , .....
16 - 10 = 6
22 - 16 = 6
...
So consecutive difference is 6
If we take [tex]T_i[/tex] as ith term of the series then:
[tex]T_2 - T_1 = 6\\T_3 - T_2 = 6\\T_4 - T_3 = 6 \\T_5 - T_4 = 6\\\cdots\\T_{n} - T_{n-1} = 6[/tex]
Thus, the recursive rule for the given series is [tex]T_{n} - T_{n-1} = 6[/tex] or [tex]T_n = T_{n-1} + 6[/tex]
From this recursive rule, we can deduce the explicit formula as:
[tex]T_n = T_{n-1} + 6\\T_n = T_{n-2} + 6 + 6\\\cdots\\T_n = T_{n-k} + k \times 6\\T_n = T_1 + 6(n-1)\\T_n = 10 + 6(n-1) \: \rm (as \: T_1 = 10)\\[/tex]
Thus, the explicit rule for this series is [tex]T_n = 10 + 6(n-1)[/tex]
For 5th row, putting n = 5 gives us:
[tex]T_n = 10 + 6(n-1) = 6n + 4\\T_5 = 6(5) + 4 = 34[/tex]
If the auditorium has 17 rows, then total chairs are:
[tex]T = T_1 + T_2 + \cdots + T_{17} = \sum_{n=1}^{17} T_n\\\\T = \sum_{n=1}^{17} (10 + 6(n-1))\\\\T = \sum_{n=1}^{17} (6n + 4)\\\\T = 6\sum_{n=1}^{17} n + \sum_{n=1}^{17}4 = 6\sum_{n=1}^{17} n + 4 \times 17\\\\T = 6\left( \dfrac{17(18)}{2}\right) + 68 = 918 + 68\\\\T = 986[/tex]
(it is because [tex]\sum_{k=1}^n k = 1 + 2 + \cdots + n = \dfrac{n(n+1)}{2}[/tex] )
Thus, there are total 986 chairs in the considered auditorium. There are 34 chairs in the fifth row. The recursive rule for this series is: [tex]T_n = T_{n-1} + 6[/tex] The explicit rule for this series is: [tex]T_n = 6n + 4[/tex].
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The unprecedented shift to remote learning during the Covid-19 pandemic offered a chance to learn about student experiences and needs and possible future trends in unit design. An educator set out to understand the impact of remote learning and assumed that 46% of students would report their studies in the new situation (online) is the same as in the face-to-face context.
In a random sample of 40 university students, 20 rated their overall learning in the virtual format as on par with the face-to-face learning.
Research Question: Has the proportion of students reporting an equal preference for online and face-to-face learning changed due to the Covid-19 pandemic?
Instead of focussing on the proportion of university students reporting the same learning experience in online and face-to-face contexts, we shift our attention to the variable X: the number of university students who reported the same learning experience in online and face-to-face contexts.
1A. Assuming the hypothesised value holds, what are the expected numbers of university students who reported the same learning experience in online and face-to-face contexts?
1B. What are the degrees of freedom associated with this hypothesis test?
1C. What is the value of the test statistic associated with this hypothesis test?
The given problem is about hypothesis testing. The sample size is 40, and the proportion of students reporting their studies in the new situation (online) is the same as in the face-to-face context is 46%.
1A. The expected numbers of university students who reported the same learning experience in online and face-to-face contexts are 18.4.
1B. The degrees of freedom associated with this hypothesis test is 39.
1C. The value of the test statistic associated with this hypothesis test is approximately 0.518.
Here, the null hypothesis is H0: p = 0.46 and the alternative hypothesis is Ha: p ≠ 0.46, where p is the proportion of university students reporting the same learning experience in online and face-to-face contexts.
Here, we are interested in testing whether the proportion of students reporting an equal preference for online and face-to-face learning has changed due to the Covid-19 pandemic.
1A. Assuming the hypothesized value holds, the expected numbers of university students who reported the same learning experience in online and face-to-face contexts are 0.46 × 40 = 18.4.
1B. The degrees of freedom associated with this hypothesis test is (n - 1) where n is the sample size.
Here, n = 40.
Hence, the degrees of freedom will be 40 - 1 = 39.
1C. The value of the test statistic associated with this hypothesis test can be calculated as follows:
z = (X - μ) / σ, where X = 20,
μ = np
μ = 18.4, and
σ = √(npq)
σ = √(40 × 0.46 × 0.54)
σ ≈ 3.09.
z = (20 - 18.4) / 3.09
z ≈ 0.518
So, the value of the test statistic associated with this hypothesis test is approximately 0.518.
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Consider a sequence of i.i.d random variables X₁, X2,..., each with a discrete uniform distribution on the set {0, 1,2}. In other words, P(X = 0) = 1/3 = P(X₁ = 1) = P(X = 2), for each k. (a) Compute P(X₁ + X₂ ≤ 1). (b) Determine the mgf of X₁ along with its domain. n (c) Consider a sequence of sample averages, {X}, where X₁ = EX for n € N. Find k=1 the mgf of X, by also stating its domain. Hint. First describe the mgf of X, in terms of the mgf of Xk, and then use the mgf of X.
(a) To compute P(X₁ + X₂ ≤ 1), we can list out all the possible values of X₁ and X₂ that satisfy the inequality: X₁ + X₂ ≤ 10 + 0 = 0, which is impossible, so P(X₁ + X₂ ≤ 1) = P(X₁ = 0, X₂ = 0) + P(X₁ = 1, X₂ = 0) + P(X₁ = 0, X₂ = 1) = (1/3)² + (1/3)² + (1/3)² = 1/3.
(b) The moment generating function (mgf) of X₁ is given by:
M(t) = E(etX₁) = (1/3) et0 + (1/3) et1 + (1/3) et2 = (1/3) + (1/3) et + (1/3) e2t
The domain of M(t) is the set of all values of t for which E(etX₁) exists.
(c) Let X be the sample average of {Xk}, where Xk are i.i.d random variables with the same distribution as X₁.
Then, by the linearity of expectation and the definition of X₁, we have:
E(X) = E( (X₁ + X₂ + ... + Xn)/n ) = (E(X₁) + E(X₂) + ... + E(Xn))/n = (EX₁ + EX₂ + ... + EXn)/n = X₁ = 1
From part (b), we have the mgf of X₁ as M₁(t) = (1/3) + (1/3)et + (1/3)e2t.
Then, the mgf of X is given by the formula: M(t) = E(etX) = et (X₁ + X₂ + ... + Xn)/n) = E(etX₁/n) × E(etX₂/n) × ... × E(etXn/n) = (M₁(t/n)) ⁿ = [(1/3) + (1/3) et/n + (1/3) e2t/n] ⁿ
The domain of M(t) is the set of all values of t for which E(etX) exists.
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