Posterior probabilities are _____.
a. simple probabilities
b. conditional probabilities
c. joint probabilities
d. marginal probabilities

The total surface area of the trapezoidal prism is S = 3,296 inches²

Given data ,

Let the total surface area of the trapezoidal prism is S

Now , the measures of the sides of the prism are

Side a = 10 inches

Side b = 32 inches

Side c = 10 inches

Side d = 20 inches

Length l = 40 inches

Height h = 8 inches

Lateral area of prism L = l ( a + b + c + d )

L = 40 ( 10 + 32 + 10 + 20 )

L = 2,880 inches²

Surface area S = h ( b + d ) + L

On simplifying the equation , we get

S = 2,880 inches² + 8 ( 52 )

S = 3,296 inches²

Hence , the surface area of prism is S = 3,296 inches²

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Minimise cost is given by  50X + 25Y

To determine how many litres of each ingredient (X and Y) should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements, follow these steps:

1. Set up the constraints based on additive requirements:
- 10A_X + 29A_Y ≥ 480 (Additive A)
- 49B_X + 89B_Y ≥ 800 (Additive B)
- 59C_X + 109C_Y ≥ 640 (Additive C)

2. Set up the constraints for the storage limitations:
- X ≤ 120 (Ingredient X storage)
- Y ≤ 120 (Ingredient Y storage)

3. Define the objective function to minimise cost:
- Cost = 50X + 25Y

4. Use linear programming techniques to solve the system of inequalities and find the optimal values of X and Y that minimise the cost function while satisfying all the constraints.

5. The optimal solution for X and Y will indicate the number of litres of each ingredient that should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements and storage limitations.

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The rule for the translation by the vector (0,5), plus a reflection around the line y = 5 is given as follows:

(x, y) -> (x, |y - 5| + 5).

The general coordinates of the point in a coordinate plane is given as follows:

(x,y).

The rule for the translation by the vector (0,5) is obtained as follows:

(x, y) -> (x + 0, y + 5) -> (x, y + 5).

For the reflection about the line y = 5, the x-coordinate remains constant, while the y-coordinate is moved on the opposite direction to y = 5, as follows:

(x, y) -> (x, |y - 5| + 5).

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To find the equation of a line that is perpendicular to another line, we need to use the fact that the slopes of perpendicular lines are negative reciprocals of each other. So the equation of the line passing through (-4,3) and perpendicular to 1x + 3y = 5 is y = 3x + 15.

First, we need to find the slope of the given line. We can rearrange the equation 1x + 3y = 5 to get it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

1x + 3y = 5

3y = -1x + 5

y = (-1/3)x + 5/3

So the slope of the given line is -1/3.

To find the slope of the line we want to write the equation for, we know that it must be the negative reciprocal of -1/3. So:

slope of perpendicular line = -1/(-1/3) = 3

Now we have the slope of the perpendicular line and a point it passes through (-4,3), so we can use point-slope form to write the equation:

y - y1 = m(x - x1)

where m is the slope we just found (3) and (x1, y1) is the given point (-4,3).

y - 3 = 3(x - (-4))

y - 3 = 3(x + 4)

y - 3 = 3x + 12

y = 3x + 15

So the equation of the line passing through (-4,3) and perpendicular to 1x + 3y = 5 is y = 3x + 15.

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The series ∑n-1 to ∞ [tex](-1)^{n-1} )/\frac{\sqrt(n}{n+5} }[/tex] is convergence (C).

The given series is:

∑n-1 to ∞ [tex](-1)^{n-1} )/\frac{\sqrt(n}{n+5} }[/tex]

To determine if the series converges or diverges, we can use the alternating series test. The alternating series test states that if a series has alternating terms that decrease in absolute value and converge to zero, then the series converges.

In this series, the terms alternate in sign and decrease in absolute value, since the denominator (n) increases as n increases. Also, as n approaches infinity, the term [tex](-1)^{n-1}[/tex]oscillates between 1 and -1, but does not converge to a specific value. However, the absolute value of the term 1/n approaches 0 as n approaches infinity.

Therefore, by the alternating series test, the given series converges. The answer is C (convergence).

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To show that the formula (Au)T(Av) defines an inner product, we need to verify that it satisfies the four properties of an inner product: linearity in the first component, conjugate symmetry, positivity, and definiteness.

First, we need to show that (Au)T(Av) is linear in the first component. Let u, v, and w be vectors in R^n and let a be a scalar. Then we have:

(Au)T(Av + aw) = (Au)T(Av) + (Au)T(Aw)  (distributivity of matrix multiplication)
= (Au)T(Av) + a(Au)T(Aw)  (linearity of matrix multiplication)

Thus, (Au)T(Av + aw) is linear in the first component. Similarly, we can show that (aAu)T(Av) = a(Au)T(Av) is also linear in the first component.

Next, we need to show that (Au)T(Av) satisfies conjugate symmetry. This means that for any u and v in R^n, we have:

(Au)T(Av) = (Av)T(Au)*

Taking the conjugate transpose of both sides, we get:

[(Au)T(Av)]* = (Av)T(Au)

Since the transpose of a product of matrices is the product of their transposes in reverse order, we have:

[(Au)T(Av)]* = (vTAu)* = uTAv

Therefore, we have:

(Au)T(Av) = (Au)T(Av)*

Thus, (Au)T(Av) satisfies conjugate symmetry.

Next, we need to show that (Au)T(Av) is positive for nonzero vectors u. This means that for any nonzero u in R^n, we have:

(Au)T(Au) > 0

Expanding the formula, we have:

(Au)T(Au) = uTA^T(Au)

Since A is nonzero, its transpose A^T is also nonzero. Therefore, the matrix A^T(A) is positive definite, which means that for any nonzero vector x in R^n, we have xTA^T(A)x > 0. Substituting u for x, we get:

uTA^T(A)u > 0

Thus, (Au)T(Au) is positive for nonzero vectors u.

Finally, we need to show that (Au)T(Au) = 0 if and only if u = 0. This means that (Au)T(Au) is positive definite, which is equivalent to saying that the matrix A^T(A) is positive definite.

Therefore, we have shown that the formula (Au)T(Av) defines an inner product, since it satisfies linearity in the first component, conjugate symmetry, positivity, and definiteness.

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Answer:

The answer to your problem is, C. 11

Step-by-step explanation:

The range of a data set in statistics is the difference between the largest and the smallest values. While range does have different meanings within different areas of statistics and mathematics, this is its most basic definition. Using the MY OWN EXAMPLE example:

( I have used this sample in many of my answers )

2, 10, 21, 23, 23, 38, 38

    38 - 2 = 36

The range in this example is 36. Similar to the mean, range can be significantly affected by extremely large or small values. Using the same example as previously:

2, 10, 21, 23, 23, 38, 38, 1027892

The range, in this case, would be 1,027,890 compared to 36 in the previous case. As such, it is important to extensively analyze data sets to ensure that outliers are accounted for.

Thus the answer to your problem is, C. 11

Answer: The box is large enough to ship the coat.

15000cm to the power of 3>10000cm to the power of 3

Step-by-step explanation:

V box=25x30x20

         =750+20

         =15000cm to the power of 3

So the box is large enough to ship the coat

The price at which the video game should be sold to maximize profit is $19.40 (to the nearest cent).

We have,

To find the price at which the video game should be sold to maximize profit, we need to find the x-value that corresponds to the maximum value of y.

The equation that relates profit to selling price is:

y = -5x^2 + 194x - 990

To find the x-value that maximizes profit, we need to find the vertex of the parabolic graph represented by this equation.

The x-coordinate of the vertex is given by:

x = -b/2a

where a is the coefficient of the x^2 term, and b is the coefficient of the x term.

In this case,

a = -5 and b = 194, so:

x = -194/(2 (-5)) = 19.4

Thus,

The price at which the video game should be sold to maximize profit is $19.40 (to the nearest cent).

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The value of x is 116°

Given that a circle, with two congruent chords, FG and DE we need to find the central angle x,

Here we will use the properties of circle,

Since, the chords are equal, so, the measure of the arc intercepted by them will be equal,

arc DE = arc FG = 116°

Also, we know that the central angle is equal to the measure of the arc intercepted by it,

Therefore, x = m arc DE

x = 116°

Hence, the value of x is 116°

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Answer:

To find the surface area of a regular hexagonal pyramid, we need to find the area of the six triangular faces and the area of the hexagonal base, and then add them together.

The area of each triangular face is given by the formula:

(1/2) x base x height

In this case, the base of each triangle is the side length of the hexagon (8), and the height is the slant height of the pyramid (16). Therefore, the area of each triangular face is:

(1/2) x 8 x 16 = 64

The hexagonal base can be divided into six equilateral triangles, each with side length 8. The area of each equilateral triangle is:

(1/4) x sqrt(3) x side length^2

Plugging in the values, we get:

(1/4) x sqrt(3) x 8^2 = 16sqrt(3)

To find the total surface area, we add the area of the six triangular faces and the area of the hexagonal base:

6 x 64 + 16sqrt(3) = 384 + 16sqrt(3)

Rounding to the nearest tenth, the surface area of the regular hexagonal pyramid is:

398.6 square units (rounded to one decimal place)

For the first question, the rope is 64 inches.

For the second question, the average gallons of water in the buckets is 4.725 gallons.

For the third question, the most commonly occurring number is called the mode.

To convert feet to inches, you multiply by 12. So 5 feet x 12 = 60 inches. Then you subtract the 4 inches that were cut, giving you 60 - 4 = 56 inches.

To find the average, you add up all the amounts of water and then divide by the number of buckets. So 3.2 + 4.6 + 0.3 + 9.8 = 18.9. Then you divide 18.9 by 4 to get 4.725 gallons.

For the third question, the most commonly occurring number is called the mode. So in this list, the mode is 6 because it appears most frequently.

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the probability that the sample mean amount spent on non-medical mask is greater than 22 dollars but less than 25 dollars is approximately 0.0668.

We can use the central limit theorem to approximate the sampling distribution of the sample mean. The mean of the sampling distribution of the sample mean is equal to the population mean, which is 28 dollars, and the standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size, which is 8/sqrt(36) = 4/3 dollars.

Now we need to find the probability that the sample mean is greater than 22 dollars but less than 25 dollars. Let X be the sample mean amount spent on non-medical mask. Then we need to find P(22 < X < 25).

We can standardize X as follows:

Z = (X - μ) / (σ / sqrt(n))

where μ = 28, σ = 8, and n = 36.

Substituting the values, we get:

Z = (X - 28) / (8/√36)

Z = (X - 28) / (4/3)

So we need to find P((22 - 28)/(4/3) < Z < (25 - 28)/(4/3)), which simplifies to P(-4.5 < Z < -1.5).

Using a standard normal table or calculator, we find:

P(Z < -1.5) ≈ 0.0668

P(Z < -4.5) ≈ 0.00003

Therefore, P(-4.5 < Z < -1.5) ≈ 0.0668 - 0.00003 ≈ 0.0668.

So the probability that the sample mean amount spent on non-medical mask is greater than 22 dollars but less than 25 dollars is approximately 0.0668.

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The probability that a data value is between 60 and 36 is 95.44%.

We have,

Mean = 34

Standard deviation = 2

So, P( 30 < x < 36)

= P (30 - 34/2) - P(36-34/2)

= P(-2) - P(2)

=  0.9772498 -0.0227501

= 0.9544

= 95.44%

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The average total cost when Q=1000 is approximately 1003.

To find the average total cost when Q=1000 and given the total cost function TC = 100 + aQ + bQ² with a = 3 and b = 1, follow these steps:

1. Plug in the values of a, b, and Q into the total cost function:
TC = 100 + 3(1000) + 1(1000)²

2. Calculate the total cost:
TC = 100 + 3000 + 1000000 = 1003100

3. Calculate the average total cost by dividing the total cost by the quantity (Q):
Average Total Cost (ATC) = TC / Q = 1003100 / 1000 = 1003.1

4. Round the average total cost to the nearest integer:
ATC ≈ 1003

So, the average total cost when Q=1000 is approximately 1003.

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a) The linear function giving the cost after x months is given as follows: C(x) = 88 - 8x.

b) The cost of the shoes after 8 months is given as follows: $24.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

Each month, the balance decays by $8.00, hence the slope m is given as follows:

m = -8.

Hence:

y = -8x + b.

When x = 1, y = 80, hence the intercept b is given as follows:

80 = -8 + b

b = 88.

Hence the function is:

C(x) = 88 - 8x.

The cost after 8 months is given as follows:

C(8) = 88 - 8(8) = 88 - 64 = $24.

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Answer:

To find the center of the given ellipse, we need to first put the equation in standard form:

9x^2 + y^2 - 18x - 6y + 9 = 0

We can start by completing the square for both the x and y terms. For the x terms, we can add and subtract (18/2)^2 = 81 to get:

9(x^2 - 2x + 81/9) + y^2 - 6y + 9 = 0

Simplifying inside the parentheses, we get:

9(x - 9/3)^2 + y^2 - 6y + 9 = 0

For the y terms, we can add and subtract (6/2)^2 = 9 to get:

9(x - 3)^2 + (y - 3)^2 = 36

Dividing both sides by 36, we get:

[(x - 3)^2]/4 + [(y - 3)^2]/36 = 1

Comparing this to the standard form of an ellipse:

[(x - h)^2]/a^2 + [(y - k)^2]/b^2 = 1

We can see that the center of the ellipse is at the point (h, k), which in this case is (3, 3). Therefore, the center of the given ellipse is (3, 3).

Step-by-step explanation:

Answer:

center, = 9, 3

radius = 9

Step-by-step explanation:

9x² + y² - 18x - 6y + 9 = 0

equation of a circle is,

x² + y² + 2ax + 2by + c = 0

where center of a circle equals, -a, -b

radius = √a² + b² - c

by comparing the general equation from the given equation,

2ax = - 18x

a = -9

2by = -6y

b = -3

center of a circle -a, -b will be 9,3

radius = √81 + 9 -9

=√81

=9

A researcher conducted a study to investigate if a specific diet can help reduce blood pressure in people with high blood pressure. Participants were randomly assigned to a treatment group (following the diet) or a control group (not following the diet).

The mean blood pressure was calculated for both groups, and a 95% confidence interval for the difference in the means between the treatment and control groups was determined.

The interpretation of this 95% confidence interval is that, in 95 out of 100 similar experiments, the true difference in mean blood pressure between the treatment and control groups would fall within the calculated range. If the confidence interval does not include zero, it suggests that there is a significant difference between the treatment and control groups, meaning the diet may have a positive effect on reducing blood pressure. If the confidence interval includes zero, it indicates that the difference may not be statistically significant, and further research may be needed.

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The word "sleepless" has 8 letters. To find the number of distinguishable permutations, we can use the formula for permutations of a set with no repeated elements, which is n!, where n is the number of elements.

Therefore, the number of permutations for the word "sleepless" can be calculated as 8!, which is equal to 40,320. This means that there are 40,320 different ways we can arrange the letters in the word "sleepless" while keeping all the letters distinct.

Note that if the word had repeated letters, we would have to divide the result by the factorials of the number of times each letter was repeated.

a)

P(X = 2) = 0.1839

b)

P(X = 7) = 0.0573

c)

P(X = 1) = 0.3033

d)

P(X = 3) = 0.1413

We have,

a. To find P(X = 2) when λ = 2.5, we can use the Poisson probability mass function:

P(X = 2) = (e^(-λ)  x λ^X) / X!

Substituting λ = 2.5 and X = 2, we get:

P(X = 2) = (e^(-2.5) x 2.5²) / 2!

P(X = 2) ≈ 0.1839 (rounded to four decimal places)

b.

To find P(X = 7) when λ = 8.0, we can again use the Poisson probability mass function:

P(X = 7) = (e^(-λ) x λ^X) / X!

Substituting λ = 8.0 and X = 7, we get:

P(X = 7) = (e^(-8.0) x 8.0^7) / 7!

P(X = 7) ≈ 0.0573 (rounded to four decimal places)

c.

To find P(X = 1) when λ = 0.5, we use the same formula:

P(X = 1) = (e^(-λ) x λ^X) / X!

Substituting λ = 0.5 and X = 1, we get:

P(X = 1) = (e^(-0.5) * 0.5^1) / 1!

P(X = 1) ≈ 0.3033 (rounded to four decimal places)

d.

To find P(X = 3) when λ = 3.7, we can use the Poisson probability mass function:

P(X = 3) = (e^(-λ) * λ^X) / X!

Substituting λ = 3.7 and X = 3, we get:

P(X = 3) = (e^(-3.7) x 3.7³) / 3!

P(X = 3) ≈ 0.1413 (rounded to four decimal places)

Thus,

P(X = 2) = 0.1839

P(X = 7) = 0.0573

P(X = 1) = 0.3033

P(X = 3) = 0.1413

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The given relation:

Input    output

-5             7

1                4

6              1

7               4

Is a function.

A relation maps elements (inputs) from one set into elements (outputs)of another set, and a relation is called a function if every element of the first set is mapped into only one element of the second set.

Here the first set is:

Input

-5

1

6

7

And the correspondent pairings are:

7

4

1

4

Notice that every one of the inputs appears only once, then this is a function.

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The total sample size, because each pair of data provides one less degree of freedom than the number of individuals in the pair.

The formula used to calculate the degrees of freedom for a test for dependent groups is:

df = n - 1

where n is the number of pairs of data in the sample.

In a dependent groups (or paired samples) test, the same group of individuals is measured twice, or two groups of individuals are matched in pairs. The difference between the two measurements or the two pairs of measurements is the data used in the test.

The degrees of freedom is a measure of the amount of information available to estimate a population parameter, such as the mean difference between the paired observations in the case of a dependent groups test. In general, a higher degrees of freedom value means more information and thus more precision in the estimate.

In the case of a dependent groups test, the degrees of freedom is determined by the number of pairs of data in the sample, rather than the total sample size, because each pair of data provides one less degree of freedom than the number of individuals in the pair.

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The probability that a randomly selected bank balance will be less than $1,000 is 0.2119, or approximately 21.2%.

To solve this problem, we need to use the normal distribution formula:

z = (x - μ) / σ

where:

x = the value we are interested in (in this case, $1,000)

μ = the mean (average) balance, which is $1,200

σ = the standard deviation, which is $250

z = the z-score, which tells us how many standard deviations the value is from the mean

First, we need to calculate the z-score:

z = (1,000 - 1,200) / 250

z = -0.8

Next, we need to find the probability of a z-score of -0.8 using a standard normal distribution table or calculator. The table or calculator tells us that the probability of a z-score of -0.8 is 0.2119.

Therefore, the probability that a randomly selected bank balance will be less than $1,000 is 0.2119, or approximately 21.2%.

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The probability that it will take less than 63 minutes to complete the test is 0.1016, which corresponds to option c) in your list.

To solve this problem, we first need to standardize the value of 63 minutes using the formula:

z = (x - μ) / σ

where:
x = 63 (the given value)
μ = 77 (the mean)
σ = 11 (the standard deviation)

Plugging in these values, we get:

z = (63 - 77) / 11
z = -1.27

Next, we use a standard normal distribution table (or a calculator) to find the probability that a standard normal variable is less than -1.27. The table gives us a probability of approximately 0.1016.

However, we are not dealing with a standard normal distribution, but rather a normal distribution with a specific mean and standard deviation. To account for this, we need to use the following formula:

P(X < 63) = P(Z < -1.27) = Φ(-1.27)

where Φ is the standard normal cumulative distribution function. Using a standard normal distribution table (or a calculator), we find that Φ(-1.27) is approximately 0.1016.

Therefore, the answer is (c) 0.1016.

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a) Using the Law of Cosines for this triangle, we can write the equation:
d² = (36)² + (3s)² - 2(36)(3s)cos(60°)

b) The second ship must travel at approximately 4.24 miles per hour to be 42 miles apart from the first ship at noon.

(a) To write an equation that relates the speed s and the distance d between the two ships at noon using the Law of Cosines, we first need to determine the distance each ship has traveled by noon. Since they leave at 9 a.m. and we're interested in the distance at noon, they travel for 3 hours.

Ship 1:
Speed: 12 miles per hour
Distance traveled: 12 miles/hour * 3 hours = 36 miles

Ship 2:
Speed: s miles per hour
Distance traveled: s miles/hour * 3 hours = 3s miles

Now, we can form a triangle where Ship 1 travels 36 miles, Ship 2 travels 3s miles, and the distance between them (d) is the third side. The angle between Ship 1 and Ship 2 is 180° - (53° + 67°) = 60°.

Using the Law of Cosines for this triangle, we can write the equation:
d² = (36)² + (3s)² - 2(36)(3s)cos(60°)

(b) To find the speed s that the second ship must travel so that the ships are 42 miles apart at noon, we can plug d = 42 into our equation from part (a) and solve for s.

42² = (36)² + (3s)² - 2(36)(3s)cos(60°)

Solving for s, we get:
s ≈ 4.24 miles per hour (rounded to two decimal places)

So, the second ship must travel at approximately 4.24 miles per hour to be 42 miles apart from the first ship at noon.

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The standard deviation of the population is 8.

We have,

The mean of the sampling distribution is the mean of the population, so we have:

[tex]$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \mu$[/tex]

where [tex]$\bar{x}$[/tex] is the sample mean, n is the sample size, [tex]x_1[/tex] are the individual samples, and [tex]$\mu$[/tex] is the population mean.

In this case,

We are given the sample size n = 100 and the sample mean [tex]\bar{x}[/tex]

We also know that the sampling distribution comes from a normally distributed population.

The standard error of the mean.

[tex]$SE = \frac{s}{\sqrt{n}}$[/tex]

where s is the sample standard deviation.

The standard error of the mean represents the standard deviation of the sampling distribution.

In this case,

We don't have the sample standard deviation s, but we can estimate it using the sample variance:

[tex]$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$[/tex]

Substituting the values we have:

[tex]$s^2 = \frac{1}{99}\sum_{i=1}^{100}(x_i - 145)^2 = 64$[/tex]

Therefore:

[tex]$s = \sqrt{64} = 8$[/tex]

Substituting this value into the standard error equation:

[tex]$SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{100}} = 0.8$[/tex]

The standard deviation of the population is related to the standard error by the following equation:

[tex]$SE = \frac{\sigma}{\sqrt{n}}$[/tex]

Rearranging this equation, we get:

[tex]$\sigma = SE \times \sqrt{n} = 0.8 \times \sqrt{100} = 8$[/tex]

Therefore,

The standard deviation of the population is 8.

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The amount of ice cream consumed in the United State would be 5,650 millions of pounds in 2028.

Based on the information provided about the mount of ice cream consumed in the United State, a quadratic equation that models the amount of ice cream consumed, in millions of pounds, since 2010 is given by;

y = 12(x - 6)² + 3922

Where:

By substituting the value of y, the number of years can be calculated as follows;

5,650 = 12(x - 6)² + 3922

5,650 - 3922 = 12(x - 6)²

1728 = 12(x - 6)²

144 = (x - 6)²

12 = x - 6

x = 12 + 6

x = 18 years.

Since 2010; 2010 + 18 = 2028.

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We have calculated joint, marginal, conditional probability mass function (pmf).  

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

(a) The joint probability mass function (pmf) of the pair (X, Y) can be found by listing all possible outcomes and their probabilities. There are 2³ = 8 possible outcomes, which are:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

The values of X and Y for each outcome are:

HHH: X=2, Y=2

HHT: X=2, Y=1

HTH: X=1, Y=1

HTT: X=1, Y=0

THH: X=1, Y=2

THT: X=1, Y=1

TTH: X=0, Y=1

TTT: X=0, Y=0

The probability of each outcome can be calculated as (1/2)³ = 1/8, since each coin toss is independent and has a probability of 1/2 of being heads or tails. Therefore, the joint pmf of (X, Y) is:

P(X=0,Y=0) = 1/8

P(X=0,Y=1) = 1/4

P(X=0,Y=2) = 1/8

P(X=1,Y=1) = 1/4

P(X=1,Y=2) = 1/8

P(X=2,Y=1) = 1/4

P(X=2,Y=2) = 1/8

(b) The marginal pmf of X can be found by summing the joint pmf over all possible values of Y:

P(X=0) = P(X=0,Y=0) + P(X=0,Y=1) + P(X=0,Y=2) = 3/8

P(X=1) = P(X=1,Y=1) + P(X=1,Y=2) + P(X=0,Y=1) = 1/2

P(X=2) = P(X=2,Y=1) + P(X=2,Y=2) = 3/8

Similarly, the marginal pmf of Y can be found by summing the joint pmf over all possible values of X:

P(Y=0) = P(X=0,Y=0) + P(X=1,Y=0) = 1/4

P(Y=1) = P(X=0,Y=1) + P(X=1,Y=1) + P(X=2,Y=1) = 1/2

P(Y=2) = P(X=1,Y=2) + P(X=2,Y=2) = 1/4

(c) The conditional pmf of X given Y = 1 is:

P(X=0|Y=1) = P(X=0,Y=1)/P(Y=1) = (1/4)/(1/2) = 1/2

P(X=1|Y=1) = P(X=1,Y=1)/P(Y=1) = (1/4)/(1/2) = 1/2

P(X=2|Y=1) = P(X=2,Y=1)/P(Y=1) = 0

The conditional pmf of X given Y = 2 is:

P(X=0|Y=2) = P(X=0,Y=2)/P(Y=2) = (1/8)/(1/4) = 1/2

P(X=1|Y=2)

Hence, We can conclude that we have calculated joint, marginal, conditional probability mass function (pmf).  

To know more about probability visit :

https://brainly.com/question/13604758

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