Suppose the sample space S = {-4, -3, -2, -1,0,1,2,3,4,5,6,7,8,9,10). Let A = {x|x is odd integer and 1 sx s5} B = {x|x is even integer and 1 s * s 8} C = {x|x E S. x 50) D = {x|x ES, x 29) = = By using probability axiom of postulate probability, explain why P(S) is not permissible.

The 95% confidence interval for the population mean height of women is 64.26 inches to 66.14 inches.

To construct the 95% confidence interval for the population mean height of women, we'll use the sample mean, sample size, and standard deviation you provided. The sample mean is 65.2 inches, the sample size is 25, and the standard deviation is 2.4 inches.

For a 95% confidence interval, we'll use a Z-score of 1.96. The formula for the confidence interval is:

CI = sample mean ± (Z-score × (standard deviation / √sample size))

CI = 65.2 ± (1.96 × (2.4 / √25))

CI = 65.2 ± (1.96 × (2.4 / 5))

CI = 65.2 ± (1.96 × 0.48)

CI = 65.2 ± 0.94

The 95% confidence interval for the population mean height of women is 64.26 inches to 66.14 inches.

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The midline of the graph is y = -3

The midline of a graph is a horizontal line that divides the graph into two equal parts. It is used in various fields such as mathematics, physics, and economics to represent the average or equilibrium value of a function or data set.

midline = (highest point + lowest point) / 2

midline = (2 + -8) / 2

midline = (2 - 8) / 2

midline = -6 / 2

midline = -3

Therefore, the midline of the attached graph is at y = -3.

If the graph is not symmetrical, the midline may not accurately represent the average or equilibrium value.

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The correct answer is D) 50, 55, 59. These three data values belong in the same class because they fall within the 50-60 range. In this class, the lower limit is 50, and the upper limit is 60. The given data values, 50, 55, and 59, all fit within these limits, making them part of the same class.

1. First, identify the classes: 40-50, 50-60, 60-70, 70-80, 80-90, and 90-100.
2. Now, categorize each data value into the appropriate class:
- 41 and 42 belong to the 40-50 class.
- 50, 55, and 59 belong to the 50-60 class.
- 60 belongs to the 60-70 class.
3. Finally, determine which three values belong in the same class. The only group of three values in the same class is 50, 59, and 60, which belong to the 50-60 class.

Therefore, the correct answer is D) 50, 55, 59. These three data values belong in the same class because they fall within the 50-60 range. In this class, the lower limit is 50, and the upper limit is 60. The given data values, 50, 55, and 59, all fit within these limits, making them part of the same class.

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The population will be 5618 students in 6 years.

We have,

To write an exponential growth model for the future population, we can use the formula:

P(x) = P(0) x (1 + r)^x

where P(0) is the initial population, r is the growth rate, and x is the number of years.

Given that the initial population is 2500 and the growth rate is 13.4%, we have:

P(x) = 2500 (1 + 0.134)^x

Simplifying:

P(x) = 2500 x 1.134^x

To find the population after 6 years, we can substitute x = 6 into the formula:

P(6) = 2500 x 1.134^6

P(6) = 2500 x 2.247

P(6) = 5617.5

Rounding to the nearest student, the population will be 5618 students in 6 years.

Thus,
The population will be 5618 students in 6 years.

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The ratio that the odds used for determining the payoff is 23 : 1 (option d).

The odds used for determining the payoff in this game of chance can be found by dividing the total amount that will be paid out (including the original bet) by the amount of the bet. In this case, the total amount that will be paid out is $115, and the amount of the bet is $5. Therefore, the odds can be calculated as follows:

Odds = Total amount paid out : Amount of bet

Odds = $115 : $5

To simplify this ratio, we can divide both sides by $5:

Odds = $23 : $1

This means that for every $1 bet, the payout will be $23 if the event occurs.

Therefore, the answer to this question is option (d) 23:1.

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The equation of the tangent line is 6.033333333.

We have,

We have f(x) = √x, so f'(x) = 1/2√x

At x=36, the slope of the tangent line is f'(36) = 1/2√36 = 1/12.

The equation of the tangent line is y - f(36) = f'(36) (x-36), or

y - 6 =  1/12 x (x - 36)

which simplifies to y = x/12 + 3.

To approximate √36.4 using the tangent line,

We plug in x = 36.4 to get y = 1/12 x (36.4) + 3 = 6.03333333....

Rounding to 9 significant figures, we get

= 6.033333333.

Thus,

The equation of the tangent line is 6.033333333.

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The magnitude of the magnetic field inside the solenoid is 0.094 T.

A solenoid is a long coil of wire wrapped around a cylinder or other object. It is used to generate a nearly uniform magnetic field within the cylinder.

To determine the magnetic field inside a solenoid, the following equation can be used:

B = μ0nI

Where B is the magnetic field inside the solenoid, μ0 is the permeability of free space (4π × 10-7 T m/A),n is the number of turns per unit length, and I is the current in the solenoid.

Substitute the given values into the formula and solve for the magnetic field:

μ_0nI= (4π × 10^-7 T m/A) × (2100 turns/m) × (0.140 A)= 0.094 T[/tex]

Therefore, the magnitude of the magnetic field inside the solenoid is 0.094 T.

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complete question:

a solenoid of length 2.50 m and radius 1.80 cm carries a current of 0.140 a. determine the magnitude of the magnetic field inside if the solenoid consists of 2100 turns of wire.

The expression sinâ¹(â3/2) is undefined. The value of cosâ¹(2/2) = 0 radians.

In the expression sinâ¹(â3/2), Since the sine function is only defined for angles between -π/2 and π/2, we cannot find an angle with a sine of -â3/2. Therefore, the expression is undefined.

In the expression cosâ¹(2/2), since the cosine of an angle is equal to the adjacent side over the hypotenuse in a right triangle, we can draw a right triangle with adjacent side 2 and hypotenuse 2. Using the Pythagorean theorem, we find that the opposite side has length 0.

Therefore, we have a right triangle with adjacent side 2, opposite side 0, and hypotenuse 2. This means that the angle we are looking for is a zero-degree angle, or 0 radians. Therefore, cosâ¹(2/2) = 0 radians.

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The required F- test statistic is 5.1788 (approximately up to 4 decimal places) to test the claim that the population variances are equal for the standard deviation of the first sample is, [tex]s_{1}[/tex] = 2.8404 and that of second sample is [tex]s_{2}[/tex] = 6.4639.

An F-test is defined as a statistical test in which the test statistic has an F-distribution under the null hypothesis. F- test is performed  in hypothesis testing to check if the variances of (any) two populations or (any) two samples are equal to each other or not.

It is said that both the distributions are normal.

We have, the standard deviation of the first sample is, [tex]s_{1}[/tex] = 2.8404 ,

and the standard deviation of the second sample is, [tex]s_{2}[/tex] = 6.4639

The value of F- statistic, that is, F- ratio is the ratio of the larger sample variance to smaller sample variance.

Therefore,

F- ratio = [tex]\frac{s_{1}^2}{s_{2}^2}[/tex] = (6.4639)² / (2.8404)² = 41.78200321 / 8.06787216

= 5.17881324609

= 5.1788 (approximately up to 4 decimal places)

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The best interpretation of the calculated 90% confidence paired-t interval (0.887, 7.493) is:

We are 90% confident that the average difference in the cholesterol levels of the patients sampled is between 0.887 and 7.493.

Option 2 is the correct answer.

The best interpretation of the given 90% confidence paired-t interval is option 2) We are 90% confident that the average difference in the cholesterol levels of the patients sampled is between 0.887 and 7.493.

This interval gives us an estimate of the range of plausible values for the true population mean difference in cholesterol levels before and after administering the drug.

The interval indicates that we are 90% confident that the true mean difference in cholesterol levels falls between 0.887 and 7.493 mg/dL.

Option 2 is the correct answer.

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On solving the provided query we have As a result, we may infer from the information provided that 39 pupils in Phil's school's seventh grade would select football as their favourite sport.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

39 is equal to 0.2N * (number of students who choose soccer/total number of students polled).

When we simplify this equation, we obtain:

The percentage of students who choose football was 39 * (the total number of students surveyed/0.2N).

By realising that the entire number of respondents to the survey is equal to 20% of all seventh-graders, we can further reduce this equation:

Students who choose football: 39 * (0.2N/0.2N) = 39

As a result, we may infer from the information provided that 39 pupils in Phil's school's seventh grade would select football as their favourite sport.

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Answer:

[tex]x + 2 = - 3x[/tex]

[tex] - 4x = 2[/tex]

[tex]x = - \frac{1}{2} [/tex]

[tex] - 3( - \frac{1}{2} ) = \frac{3}{2} = 1 \frac{1}{2} [/tex]

So the lines intersect at (-1/2, 1 1/2), or

(-.5, 1.5).

The population for this study is all hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit.

The major shortcomings in the obtained data are the small sample size (only 100 out of 400 managers responded) and the potential for response bias (managers who are more satisfied or dissatisfied with their computer systems may be more likely to respond to the survey).

Confidence interval for a population mean with a known standard deviation:

[tex]CI = \bar x \± z\times (\sigma/\sqrt n)[/tex]

[tex]\bar x[/tex] is the sample mean, [tex]\sigma[/tex] is the population standard deviation (assumed to be 1.75), n is the sample size (100), and z is the z-score corresponding to the desired level of confidence (95% corresponds to a z-score of 1.96).

Substituting the given values, we have:

[tex]CI = 5.396 \± 1.96\times (1.75/\sqrt 100)[/tex]

[tex]CI = 5.396 \± 0.34[/tex]

[tex]CI = (5.056, 5.736)[/tex]

The 95% confident that the true population mean for satisfaction with ease of use of computer systems for hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit is between 5.056 and 5.736.

To repeat this survey many times and construct a 95% confidence interval based on each sample, about 95% of the intervals would contain the true population mean for satisfaction with ease of use of computer systems.

The same formula as in part B, but with a z-score of 2.58 (corresponding to 99% confidence), we have:

[tex]CI = 4.398 \± 2.58\times (1.75/\sqrt 100)[/tex]

[tex]CI = 4.398 \± 0.45[/tex]

[tex]CI = (3.948, 4.848)[/tex]

The 99% confident that the true population mean for satisfaction with level of computer training for hotel managers in hotels of size 200 to 500 rooms in Chicago and Detroit is between 3.948 and 4.848.

To repeat this survey many times and construct a 99% confidence interval based on each sample, about 99% of the intervals would contain the true population mean for satisfaction with level of computer training.

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Based on the breach of these implied terms, Eric may have a valid claim to return the car and seek a refund. However, Peter's offer to replace the windscreen and compensate for the mileage discrepancy may be considered as an attempt to remedy the situation. It is ultimately up to Eric to decide whether to accept Peter's proposed solutions or pursue legal action to return the car and obtain a refund.

Based on the implied terms under the Sales of Goods Ordinance (Cap 26) ("SOGO"), Eric has the right to return the car to Peter and get a refund. One of the implied terms is that the goods sold must be of satisfactory quality, which includes being free from defects and safe to use. The scratch on the windscreen makes the car unsafe to drive and therefore, not of satisfactory quality.

Additionally, the inspection report revealing that the mileage meter had been adjusted also breaches the implied term that the goods sold must match their description. The sign board on the windscreen stated that the car had a mileage of 12,200km, but the actual mileage is 13,420km. This is a material difference that would have influenced Eric's decision to purchase the car.

Therefore, Eric can reject the car and get a refund under the implied term that the goods sold must be of satisfactory quality and match their description. The compensation offered by Peter for the understatement of the mileage reading does not address the issue of the scratch on the windscreen, which is a breach of an implied term.
Under the Sales of Goods Ordinance (Cap 26) ("SOGO"), there are implied terms concerning the quality and fitness of goods being sold. In Eric's case, two key implied terms are relevant: the satisfactory quality of the goods and the accuracy of the description provided.

1. Satisfactory quality (Section 16 of SOGO): Goods sold should be of satisfactory quality, which includes appearance, freedom from minor defects, and safety. The scratch on the windscreen makes the car unsafe and illegal to drive, which means the car is not of satisfactory quality. Peter's offer to replace the windscreen may address this issue, but it does not automatically negate Eric's right to reject the car.

2. Accuracy of description (Section 15 of SOGO): When the seller provides a description of the goods, the goods should correspond to the description. In this case, the mileage stated on the signboard (12,200km) differs from the actual mileage (13,420km). This misrepresentation could entitle Eric to reject the car.

Based on the breach of these implied terms, Eric may have a valid claim to return the car and seek a refund. However, Peter's offer to replace the windscreen and compensate for the mileage discrepancy may be considered as an attempt to remedy the situation. It is ultimately up to Eric to decide whether to accept Peter's proposed solutions or pursue legal action to return the car and obtain a refund.

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The vertex form of the equation f f(x) is: B. f(x) = −(x + 1)² + 25.

In this exercise, you are required to determine the vertex form of a quadratic function h(x) that is written in standard form. Mathematically, the vertex form of a quadratic equation is given by this formula:

f(x) = a(x - h)² + k

Where:

Since the quadratic function f(x) has roots of 4 and -6, and it passes through the point (1, 21), the leading coefficient can be determined as follows;

f(x) = a(x + 6)(x - 4)

f(x) = a(x² + 2x - 24).

21 = a(1 + 2 - 24)

-21a = 21

a = -1.

For the x-coordinate of the vertex, we have:

x = -b/2a = 2/-2 = -1.

For the y-coordinate of the vertex, we have:

y = -(b² - 4ac)/4a

y = -((-2)² - 4(-1)(24))/-4

y = 25.

Therefore, the vertex form of the quadratic function is given by:

f(x) = a(x - h)² + k

f(x) = −(x + 1)² + 25

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Answer:

Step-by-step explanation:

To solve by completing the square, we need to have the equation in the form:

(x - h)^2 = k

where h and k are constants. To get the equation in this form, we need to move the constant term to the right side and group the x terms together. So let's start by moving 9 to the right side:

x^2 + 4x = 9

Next, we need to add and subtract a constant term that will allow us to complete the square. The term we need to add is (b/2a)^2, where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 1 and b = 4, so (b/2a)^2 = (4/2)^2 = 4. So we add and subtract 4:

x^2 + 4x + 4 - 4 = 9

Now we can group the first three terms and simplify:

(x + 2)^2 - 4 = 9

Add 4 to both sides:

(x + 2)^2 = 13

So the answer is not given in any of the options provided.

The following parts can be answered by the concept from Sets.

(a) No, ɸ is not an element of the power set P(S).

(b) Yes, a is an element of the power set P(S).

(a) The power set P(S) of a set S is the set of all possible subsets of S, including the empty set ɸ and the set S itself. However, ɸ is not an element of P(S) because it is not a subset of S. A subset must have at least one element, but ɸ has no elements, so it cannot be a subset of any set, including S. Therefore, ɸ is not an element of P(S).

(b) The element 'a' is an element of set S, and therefore it can be a subset of S. Since every element of S is also a subset of S, 'a' is also a subset of S. Hence, 'a' is an element of P(S) because P(S) includes all possible subsets of S. Therefore, 'a' is an element of P(S).

Therefore,

(a) ɸ is not an element of P(S).

(b) 'a' is an element of P(S)

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The warranty period should be 37.6 months to ensure that the company doesn't have to replace more than 2% of the total machines sold due to malfunctioning.

To solve this problem, we need to use the concept of deviation again. We know that the standard deviation of the PYQ machines is 8 months. Therefore, we can use the z-score formula to calculate the number of standard deviations from the mean that corresponds to the 2% malfunction rate. The z-score is calculated as:

z = (x - μ) / σ

where x is the value we want to find, μ is the mean, and σ is the standard deviation. In this case, we want to find the warranty period (x) that corresponds to a malfunction rate of 2%. The mean is 54 months, and the standard deviation is 8 months. Therefore, we can rearrange the formula as:

z = (x - 54) / 8

We want to find the z-score that corresponds to the 2% malfunction rate. We can use a standard normal distribution table to find the z-score that corresponds to a cumulative probability of 0.02. The z-score turns out to be -2.05.

Now that we know the z-score, we can solve for x:

-2.05 = (x - 54) / 8

-16.4 = x - 54

x = 37.6

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Answer:

B. Tenths

Step-by-step explanation:

Nine is one spot to the left of the decimal. this means that it is a decimal. It is in the first spot, otherwise known as the tenths place value. This means nine is in the tenths place value.

The earthquake in California with a magnitude of 4.9 had an amplitude 7943.28 times stronger than the baseline amplitude Ao.

The formula relating the Richter scale,  magnitude M to the amplitude A of an earthquake is

M = log10(A/Ao)

Solving for A, we get

A = Ao * 10^(M)

For the earthquake in California with a magnitude of 4.9, we can use this formula to find the ratio of its amplitude to the baseline amplitude Ao

4.9 = log10(A/Ao)

10^4.9 = A/Ao

A/Ao = 10^4.9

A/Ao = 7943.28

This means that the wave amplitude A of the earthquake in California was 7943.28 times stronger than the baseline amplitude Ao.

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--The given question is incomplete, the complete question is given

"An earthquake in california measured 4. 9 on richter scale use formula r=log(a/ao) to determine how many times stronger the wave amplitude a of the earthquake was than ao."--

The indicated derivative of the function is -6(x + 1) -4 (option b).

The second derivative of f(x) is 0.

Let's look at the first problem. We are asked to find the derivative of the function f(x) = 6x⁵ - 3x² - 5x + 1. The derivative of a function is written as f'(x) or dy/dx. To find the derivative of this function, we need to use the power rule and the constant multiple rule.

The power rule states that if we have a term of the form xⁿ, then its derivative is nxⁿ⁻¹. The constant multiple rule states that if we have a constant c multiplied by a function f(x), then the derivative of cf(x) is c times the derivative of f(x).

Using these rules, we can find that the derivative of f(x) is f'(x) = 30x⁴ - 6x - 5. This is the rate at which the function is changing at each point x.

The answer choices given are all in different forms, but we can see that choice B, 480x + 6, matches our answer if we simplify it.

Hence the correct option is (b).

Given function is f(x) = x + 1, and we are asked to find its second derivative f''(x). The first derivative of f(x) is found by differentiating the equation f(x) = x + 1 with respect to x:

f'(x) = d/dx(x + 1) = 1

Now, we can find the second derivative of f(x) by differentiating the equation for f'(x) with respect to x:

f''(x) = d/dx(1) = 0

Therefore, the second derivative of f(x) is 0, which means that the slope of the tangent line to the graph of f(x) is constant at all points.

Therefore, the correct answer is none of the above.

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Derivatives play a crucial role in understanding the position, velocity, and acceleration of a moving object.

Position can be represented as a function of time, and its derivative with respect to time is velocity. Velocity, in turn, can be represented as a function of time, and its derivative with respect to time is acceleration. Therefore, by taking derivatives of position with respect to time, we can obtain the velocity of the moving object, and by taking the derivative of velocity with respect to time, we can obtain the acceleration of the moving object. In summary, derivatives help us to analyze and quantify the motion of a moving object by providing us with information about its position, velocity, and acceleration.

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The value of the given integral is 390.

Given that, we need to use Fundamental Theorem of Calculus to evaluate [tex]\int\limits^{27}_8 {(x+3)} \, dx[/tex]

So,

[tex]\int\limits^{27}_8 {(x+3)} \, dx\\\\= \int\limits^{27}_8 {x} \, dx + \int\limits^{27}_8 {3} dx\\\\[/tex]

= [x²/2]²⁷₈ + 3[x]²⁷₈

= 332.5 + 57

= 389.5

≈ 390

Hence, the value of the given integral is 390.

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The probability that the sample mean x will be greater than 11 is approximately 9.01%.

To find the probability that the sample mean (x) will be greater than 11 given a sample size of 45 drawn from a population with a mean of 10 and a standard deviation of 5, follow these steps:

Step 1: Calculate the standard error (SE) of the sample mean.

The standard error is given by the formula:

SE = population standard deviation / √sample size.

In this case, SE = 5 / √45 ≈ 0.745.

Step 2: Calculate the z-score corresponding to the given value of x.

The z-score is given by the formula:

z = (x - population mean) / SE.

In this case, z = (11 - 10) / 0.745 ≈ 1.34.

Step 3: Use the z-score to find the probability that x will be greater than 11.

The probability can be found using a standard normal (z) table or a calculator.

A z-score of 1.34 corresponds to a probability of approximately 0.0901 (9.01%).

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Answer:

yes, 2 divided by 4 is 1/2.

Step-by-step explanation:

4 only goes into 2 half a time, therefore 2 divided by 4 is 1/2.

Answer: I say yes

Step-by-step explanation: I guess like when dividing 2 into 4 it goes into 4, 2 times, but like if you were to look up the answer, it comes up as 0.5 which us basically considered 1/2 imo.

To minimize the amount of tin used, the height of the can must be 4 inches (option D).

The volume of a cylindrical tin can is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height. To minimize the amount of tin used, we need to minimize the surface area, which is given by the formula A = 2πrh + 2πr².

Given the volume is 16π cubic inches, we have:

16π = πr²h

Now, we can find the relationship between r and h:

h = 16/r²

Now, substitute this into the surface area formula:

A = 2πr(16/r²) + 2πr²
A = 32π/r + 2πr²

To minimize the surface area, we can take the derivative with respect to r and set it to 0:

dA/dr = -32π/r² + 4πr

0 = -32π/r² + 4πr

Solving for r:

r³ = 8
r = 2 (since r > 0)

Now, substituting r back into the relationship between r and h:

h = 16/(2²)
h = 16/4
h = 4

Therefore, the height of the can must be 4 inches (option D) to minimize the amount of tin used.

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(a)

(i) To find the pdf of X, we differentiate the cdf:

f_x(x) = d/dx (1 - e^(-bx)) = b e^(-bx), x >= 0

0, x < 0

(ii) To find the pdf of Y = sqrt(X), we use the transformation method:

f_y(y) = f_x(x) / |dy/dx|, where x = y^2

dy/dx = 1 / (2 sqrt(x)) = 1 / (2y)

f_y(y) = f_x(y^2) / (2y) = b e^(-b y^2) / (2y), y >= 0

(b)

(i) We want to find g(U) such that Y = g(U) has the desired pdf:

f_y(y) = f_u(g(y)) |dg/dy|

Since U is a standard uniform random variable, its pdf is f_u(u) = 1, 0 <= u <= 1.

We want to find g(U) such that Y = g(U) has the pdf:

f_y(y) = 3y^2, 0 < y < 1

0, otherwise

From the above equation, we can see that:

f_u(u) = 1 = f_y(g(u)) |dg/du|

So, we need to find a function g(U) that satisfies:

f_y(y) = 3y^2 = |dg/dy|, 0 < y < 1

We can solve this differential equation by integrating both sides:

∫_0^y 3u^2 du = ∫_0^g(u) |dg/dy| dy

y^3 = g(u)

So, the required function is:

g(u) = u^(1/3)

(ii) If the student has a random variable X in the interval (2,5), then we need to modify the function g(U) to map the interval [0,1] of the standard uniform random variable U to the interval [2,5] of X.

Let X have the pdf f_x(x). Then, the cdf of X is given by:

F_x(x) = ∫_2^x f_x(t) dt

We can use this cdf to transform U to X:

F_x(g(u)) = u, 0 <= u <= 1

Solving for g(u), we get:

g(u) = F_x^(-1)(u) = ∫_2^x f_x(t) dt)^(-1)

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The teacher may use this method to mimic randomly selecting a bay or girt in the future, changing the probability model as more choices are made.

The teacher might use red and blue marbles to create a uniform probability model for simulating randomly selecting a bay or girt from the class. Here's how it's done:

First, the teacher must establish how many pupils are bays and how many are girts in the class.

Assume there are 20 students in the class, with 10 of them being bays and the other ten being girts. To represent each student in the class, the teacher would want a total of 20 marbles. The marbles would be half red (representing the bays) and half blue (representing the girts).

The teacher would then place all 20 marbles in a container, such as a bag or a jar, and start over by Mixing them thoroughly.

To simulate picking a bay or girt at random, the teacher would close their eyes and pick a marble from the container without looking. They'd then keep track of whether the stone was red or blue.

The teacher would go through this process several times, being sure to put each marble back into the container before selecting it again. This ensures that each marble has an equal opportunity to be chosen.

The teacher would keep track of the color of the stone chosen each time until they had collected enough data to create a probability model.

The chance of choosing a bay or girt can be estimated by dividing the number of times a red marble (representing a bay) was chosen by the total number of times.

of selections made. For example, if the teacher chose 50 marbles in total, 25 of which were red (representing bays), the probability of choosing a bay is 0.5, or 50%.

The teacher may use this method to mimic randomly selecting a bay or girt in the future, changing the probability model as more choices are made.

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The value of sin 2Ф is  (√3 )/ 2 ,  cos 2Ф is 1/2 and tan 2Ф is √3 by application of Trigonometric formulas of Sine, Cosine and Tangent.

We have, sin Ф= 1/2

Applying Trigonometric formulas of Sine, Cosine and Tangent, we get

We know,

sin² Ф + cos² Ф = 1

⇒ (1/2)² + cos² Ф = 1

⇒ cos² Ф = 1 - (1/4) = 3/4

Rooting both sides we get,

⇒ cos Ф = (√3 )/ 2

Therefore, sin 2Ф = 2{sin Ф} {cos Ф}

⇒ sin 2Ф = 2 (1/2){(√3 )/ 2}

= (√3 )/ 2

Therefore, cos 2Ф = 2{ cos² Ф} - 1

⇒ cos 2Ф = 2 ( 3/4) - 1

= 1/2

Therefore, tan 2Ф = [tex]\frac{sin (theta)}{cos(theta)}[/tex] = [tex]\frac{1/2}{(\sqrt{3} )/ 2}[/tex] = 1/ (√3 )

So, tan 2Ф = [tex]\frac{2 tan (theta)}{ 1 - tan^{2} (theta) }[/tex] = [tex]\frac{2(1/\sqrt{3}) }{1 - (1/\sqrt{3})^{2} }[/tex] = √3

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