Step 1: To tell the null and alternative hypotheses for the test:
Null hypothesis (H0): The mean ACT score of first-born children is equal to or greater than the mean ACT score of second-born children. μ1 ≥ μ2Alternative hypothesis (Ha): The mean ACT score of first-born children is lower than the mean ACT score of second-born children. μ1 < μ2What are the other stepsStep 2: The test statistic for this scenario is the z-score, calculated as:
Step 3 involves comparing the chosen significance level (1% in this instance) with the test statistic to decide whether to reject or accept the null hypothesis. In case the test statistic falls within the critical region, the null hypothesis is rejected, but if it doesn't, the null hypothesis is not rejected.
We can also determine the p-value related to the test statistic and assess its compatibility with the selected level of significance. If the significance level exceeds the p-value, we do not reject the null hypothesis, but if the p-value is lower, then we reject the null hypothesis.
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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
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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the circumfrence is 72 cm what is the length of the minor arc
Answer:
Should be 9 centimeters.
Step-by-step explanation:
Which of the following sets of angle measures can be used to draw an acute isosceles triangle? Select all that apply. 75°, 30°, 75° 80°, 55°, 45° 80°, 80°, 40° 60°, 60°, 60° 50°, 50°, 80° 20°, 140°, 20°
Answer:
1. 80°, 80°, 40°
2. 60°, 60°, 60°
3. 20°, 140°, 20°
4. 50°, 50°, 80°
5. 75°, 30°, 75°
6. 80°, 55°, 45°
6, 2, and 4
Answer:
I think its 6,2, and 4
Step-by-step explanation: hope that helps! °∪°
the radius of a circle is 8 miles. what is the area of a sector bounded by a 144° arc
Answer:
Step-by-step explanation:
The area of a sector and the properties of circles bounded by a 144° arc in a circle with a radius of 8 miles can be calculated using the formula: Area of sector = (θ/360°) * π * r² where θ is the central angle of the sector and r is the radius of the circle.
In this case, the central angle is 144° and the radius is 8 miles. Plugging these values into the formula, we get: Area of sector = (144°/360°) * π * (8 miles)². Simplifying the equation, we have: Area of sector = (0.4) * π * (8 miles)².
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which of the following expressions is equivalent to -10?
a.-7 3
b.-3 - 7
c.3 - 7
d.7 - 3
The expression which is equivalent to -10 is the option b, -3 - 7.
Explanation:
We can use subtraction and addition of integers to get the value of the given expression. We can write the given expression as;
-3 - 7 = -10 (-3 - 7)
The addition of two negative integers will always give a negative integer. When we subtract a larger negative integer from a smaller negative integer, we will get a negative integer.
If we add -3 and -7 we will get -10. This makes the option b the correct answer.
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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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Which point is not on the line
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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From Hardcover Book, Marsden/Tromba, Vector Calculus, 6th ed., Section 2.1., # 40) Using polar coordinates, describe the level curves of the function defined by f (x, y) = - 2xy (22+y2) if (x, y) + (0,0) and f(0,0) = 0.
The level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function. These level curves represent the points (r, θ) where the function f(r, θ) is constant.
To describe the level curves of the function f(x, y) = -2xy / (2^2 + y^2), we can first express the function in terms of polar coordinates. Let's substitute x = r cos(θ) and y = r sin(θ) into the function:
f(r, θ) = -2(r cos(θ))(r sin(θ)) / (r^2 + (r sin(θ))^2)
Simplifying this expression, we get:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2 + r^2 sin^2(θ))
Now, we can further simplify this expression:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2(1 + sin^2(θ)))
f(r, θ) = -2 cos(θ) sin(θ) / (1 + sin^2(θ))
The level curves of this function represent the points (r, θ) in polar coordinates where f(r, θ) is constant. Let's consider a few cases:
1. When f(r, θ) = 0:
This occurs when -2 cos(θ) sin(θ) / (1 + sin^2(θ)) = 0. Since the numerator is zero, we have either cos(θ) = 0 or sin(θ) = 0. These correspond to the lines θ = π/2 + kπ and θ = kπ, where k is an integer.
2. When f(r, θ) > 0:
In this case, the numerator -2 cos(θ) sin(θ) is positive. For the denominator 1 + sin^2(θ) to be positive, sin^2(θ) must be positive. Therefore, the level curves lie in the regions where sin(θ) > 0, which corresponds to the upper half of the unit circle.
3. When f(r, θ) < 0:
Similar to the previous case, the level curves lie in the regions where sin(θ) < 0, which corresponds to the lower half of the unit circle.
In summary, the level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function.
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Select 2A316 in base 10.
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
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
y=5x
y=-3x+24
Solve by substitution
Answer:
x=12
Step-by-step explanation:
5x = -3x+24
2x = 24
x = 12
Solve the system of equations using the substitution method. Show your work and be sure to include the solution to the system.
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
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.
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
EQUAÇO
1. x + 5 - 25=x + 3x - 4
2. 1 - 2x = 3 - 2(x + 1)
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
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.
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
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:
increase 50$ by 15%
Can you say how to do it and answer?
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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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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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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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
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