Jake is building a toy box for his son that is shaped like a rectangular prism. The toy box has a volume of 72 ft3 and a height of 3 ft. What are the possible dimensions for the area of the base of the toy box?​

For problem 1a, the solution is y(x) = -x + sin(x) - cos(x).

For problem 1b, the solution is y(x) = 3/4 - (1/4)e³ˣ + eˣ.

For problem 2, the solution is y(x) = (2x² + x⁴)/(x⁴ - 2x² + 4).



1a:
Step 1: Find the complementary function by solving the homogeneous equation y'' + y' = 0.
Step 2: Use variation of parameters to find a particular solution.
Step 3: Combine complementary function and particular solution.
Step 4: Apply initial conditions to find constants.

1b:
Step 1: Form a characteristic equation and solve for the roots.
Step 2: Write the general solution using the roots.
Step 3: Apply initial conditions to find constants.

2:
Step 1: Rewrite the given equation in the form of dy/y² -1 = dx/x² - 1.
Step 2: Integrate both sides.
Step 3: Simplify and rearrange to find y(x).
Step 4: Apply initial condition y(2) = 2 to find the constant.

To know more about homogeneous equation click on below link:

https://brainly.com/question/30767168#

#SPJ11

Answer:

b

Step-by-step explanation:

The balance in Mr. Jenkin's account in dollars and cents at the end of 5 years is $1893.75.

What is the simple interest?

Simple interest, often known as the yearly interest rate, is an annual payment based on a percentage of borrowed or saved money. Simple Interest (S.I.) is a way for figuring out how much interest will accrue on a specific principal sum of money at a certain rate of interest.

Here, we have

Given: Mr. Smith deposited $1,500 into an account that earns 5.25% simple interest annually. He made no additional deposits or withdrawals.

P = $1,500 I = ?, r = 5.25% , t = 5 years

Simple interest:

I = Prt

I = (1,500)(0.0525)(5)

I = 393.75

Total amount = 1,500 + 393.75 = $1893.75

Hence, the balance in Mr. Jenkin's account in dollars and cents at the end of 5 years is $1893.75.

To learn more about the simple interest from the given link

https://brainly.com/question/2294792

#SPJ1

Marco walked 24 km further by taking a detour than if he had been able to walk directly from P to R.

Distance is a numerical measurement of how far apart two points are in physical space. It is typically measured in units such as meters, kilometers, miles, and light-years. Distance is an important concept in mathematics, physics, and other sciences. It is used to measure the length of a path, the speed of an object, and the distance between two objects in the universe. Distance is also used to measure the time it takes for a signal or wave to travel from one point to another.

To calculate the distance Marco walked by taking a detour, we need to subtract the distance from P to T (12 km) from the total distance from P to R.

Distance from P to R = Total Distance - Distance from P to T
Distance from P to R = x - 12km

Since we do not know the total distance from P to R, we must use the Pythagorean Theorem to solve for x.

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two sides (legs) is equal to the square of the hypotenuse.

a2 + b2 = c2

In this case, the hypotenuse (c) is the total distance from P to R, while a and b are the distances from P to T and T to R, respectively.

x2 + 122 = (x - 12)2

Simplifying the equation yields:

x2 - 24x + 144 = 0

By using the quadratic formula (ax2 + bx + c = 0), we can solve for x.

For this equation, a = 1, b = -24 and c = 144.

x = [(-b) ± √(b2 - 4ac)]/2a

x = [(24) ± √(-24)2 - 4(1)(144)]/2(1)

x = [(24) ± √(-576)]/2

x = [(24) ± 24√3]/2

Finally, we can calculate the distance Marco walked, going from P to T to R, as follows:

Distance from P to R = (24 + 24√3)/2 - 12
Distance from P to R = 36 - 12
Distance from P to R = 24 km

Therefore, Marco walked 24 km further by taking a detour than if he had been able to walk directly from P to R.

To know more about distance click-
http://brainly.com/question/23848540
#SPJ1

The limit of the given expression is -12.

To find the limit of the given expression lim(n→0) (x³ - 6x + 6 sin x)/x⁵, we can use the Maclaurin series for sin x, which is sin x = x - (x³/3!) + (x⁵/5!) - ... . Since sin 2 = 2(1 - (2³/3!) + (2⁵/5!) - ...), we can rewrite the expression as:

(x³ - 6x + 6(x - (x³/3!) + (x⁵/5!) - ...))/x⁵

Now, we can simplify the expression:

(x³ - 6x + 6x - 6(x³/3!) + 6(x⁵/5!) - ...)/x⁵

Notice that the -6x and 6x terms cancel out:

(x³ - 6(x³/3!) + 6(x⁵/5!) - ...)/x⁵

Divide each term by x⁵:

(x³/x⁵ - 6(x³/3!)/x⁵ + 6(x⁵/5!)/x⁵ - ...)

Which simplifies to:

(1/x² - 6/(3!x²) + 6/(5!) - ...)

As n approaches 0, the limit of this expression is -12.

To know more about Maclaurin series click on below link:

https://brainly.com/question/31383907#

#SPJ11

The correct statement regarding the conditions for inference is given as follows:

No, the 10% condition is not met.

The four conditions for inference are given as follows:

In the context of this problem, we must check the sample size condition, also known as the 10% condition, which states that on each trial there must have been at least 10 successes and 10 failures.

On the second container, there were only 8 beads, hence the 10% condition is not met.

More can be learned about conditions for inference at https://brainly.com/question/18597943

#SPJ1

The logistical growth function, 2400 = 3000 / (1 + 21e^(-0.78t)) and the population of squirrels on campus will reach 2400 approximately 5.36 years after the beginning of 1855.

To find the time t when s(t) = 2400, we can substitute 2400 for s(t) in the logistical growth function and solve for t.

2400 = 3000 / (1 + 21e^(-0.78t))

Multiplying both sides by the denominator:

2400 + 2400*21e^(-0.78t) = 3000

2400*21e^(-0.78t) = 600

Dividing both sides by 2400:

21e^(-0.78t) = 0.25

Taking the natural logarithm of both sides:

ln(21) - 0.78t = ln(0.25)

Solving for t:

t = (ln(21) - ln(0.25)) / 0.78

t ≈ 5.36 years

Therefore, the population of squirrels on campus will reach 2400 approximately 5.36 years after the beginning of 1855.
To find the time t when the squirrel population s(t) is equal to 2400, you can use the given logistical growth function:

s(t) = 3000 / (1 + 21e^(-0.78t))

You want to find t when s(t) = 2400, so substitute s(t) with 2400 and solve for t:

2400 = 3000 / (1 + 21e^(-0.78t))

First, isolate the term with t:

(3000 / 2400) - 1 = 21e^(-0.78t)

(5/4) - 1 = 21e^(-0.78t)

1/4 = 21e^(-0.78t)

Now, divide both sides by 21:

(1/4) / 21 = e^(-0.78t)

1/84 = e^(-0.78t)

Next, take the natural logarithm (ln) of both sides:

ln(1/84) = -0.78t

Finally, solve for t by dividing both sides by -0.78:

t = ln(1/84) / (-0.78)

Using a calculator, you'll find:

t ≈ 3.18

So, the time t when the squirrel population on campus reaches 2400 is approximately 3.18 years after the beginning of 1855.

To learn more about function, click here:

brainly.com/question/12431044

#SPJ11

The scatter plot for Age, on the other hand, appears more scattered and does not show as clear of a correlation. However, it is important to note that further analysis using correlation or regression techniques would be necessary to confirm this initial observation.

Based on the two scatter plots provided for Age and Seniority, it appears that Seniority may be the better predictor of salary. This is because the scatter plot for Seniority shows a clearer positive correlation between the two variables, indicating that as Seniority increases, so does Salary. The scatter plot for Age, on the other hand, appears more scattered and does not show as clear of a correlation. However, it is important to note that further analysis using correlation or regression techniques would be necessary to confirm this initial observation.

learn more about scatter plot

https://brainly.com/question/13984412

#SPJ11

The probability that both marbles are red is 9/23.

To find the probability of both marbles being red, follow these steps:

1. Calculate the total number of marbles in the jar: 19 red + 25 blue = 44 marbles.

2. Determine the probability of picking a red marble on the first draw: 19 red marbles / 44 total marbles = 19/44.

3. After picking one red marble, there are 18 red marbles and 43 total marbles left. Calculate the probability of picking a red marble on the second draw: 18 red marbles / 43 total marbles = 18/43.

4. Multiply the probabilities from steps 2 and 3 to find the overall probability: (19/44) x (18/43) = 342/1892.
5. Simplify the fraction: 342/1892 = 9/23.

To know more about probability click on below link:

https://brainly.com/question/30034780#

#SPJ11

The population of Toledo, Ohio for any year t after 2000, assuming that the population continues to grow at a constant rate of 4.9% per year.

We can model the population of Toledo, Ohio as an exponential function of time, since it is increasing at a constant percentage rate per year. Let P(t) be the population of Toledo t years after the year 2000.

We know that in the year 2000, the population was approximately 530,000. So, we have:

P(0) = 530,000

We are also given that the population is increasing at a rate of 4.9% per year. This means that the population is growing by a factor of 1 + 0.049 = 1.049 per year.

Therefore, we can write the exponential function as:

P(t) = 530,000 * (1.049)^t

where t is the number of years since 2000.

This function gives us the population of Toledo, Ohio for any year t after 2000, assuming that the population continues to grow at a constant rate of 4.9% per year.

To learn more about exponential visit:

https://brainly.com/question/2193820

#SPJ11

The domain is p ≠ -2 or in interval notation, (-∞, -2) U (-2, ∞).

The function given is [tex]y = 35/(p+2)^(^2^/^5^)[/tex]. This function is continuous for all values of p except when the denominator is zero. The denominator becomes zero when p = -2. So, no, this function is not continuous for all values of p.

Since the function is continuous for all values of p except p = -2, and 24 is not equal to -2, yes, this function is continuous at p = 24.

For p ≥ 0, the function is continuous, as the only discontinuity occurs at p = -2, which is not in the range p ≥ 0. So, yes, this function is continuous for all p ≥ 0.

The domain for this application is all real numbers except for the point of discontinuity, which is p = -2. Therefore, the domain is p ≠ -2 or in interval notation, (-∞, -2) U (-2, ∞).

Learn more about Domain

brainly.com/question/28135761

#SPJ11

The following can be answered by the concept of Probability.

a. The probability of rolling each number is 1/6.

b. The sum of the probabilities of all possible outcomes should equal 1.

(a) To compute P(Dc), which represents the probability of rolling a 1, 4, 5, or 6 on a fair six-sided die, we'll determine the probability of each outcome and add them together. Since there are 6 equally likely outcomes on the die, the probability of rolling each number is 1/6.

P(Dc) = P(rolling a 1) + P(rolling a 4) + P(rolling a 5) + P(rolling a 6) = (1/6) + (1/6) + (1/6) + (1/6) = 4/6 = 2/3.

(b) To compute P(D) + P(Dc), we need to first determine P(D), which is the complementary event of P(Dc). Since there are only 6 possible outcomes on a die, the complementary event includes rolling a 2 or a 3. The probability of each outcome is still 1/6.

P(D) = P(rolling a 2) + P(rolling a 3) = (1/6) + (1/6) = 2/6 = 1/3.

Now, we can add P(D) and P(Dc) together:

P(D) + P(Dc) = (1/3) + (2/3) = 3/3 = 1.

This makes sense, as the sum of the probabilities of all possible outcomes should equal 1.

To learn more about Probability here:

brainly.com/question/30034780#

#SPJ11

The slope of the tangent to the curve r = -6 + 4cos(θ) at θ = π/2 is equal to 0.

To find the slope of the tangent to the curve at θ = π/2, we need to first find the polar coordinates (r, θ) at θ = π/2.

Substituting θ = π/2 in the equation of the curve, we get:

r = -6 + 4cos(π/2)

r = -6 + 0

r = -6

So the polar coordinates at θ = π/2 are (-6, π/2).

To find the slope of the tangent, we need to find the derivative of the polar equation with respect to θ:

dr/dθ = -4sin(θ)

dθ/dt = 1

Now, we can find the slope of the tangent by using the formula:

dy/dx = (dy/dθ) / (dx/dθ) = (r sinθ + dr/dθ cosθ) / (r cosθ - dr/dθ sinθ)

Substituting the values we found earlier, we get:

dy/dx = (r sinθ + dr/dθ cosθ) / (r cosθ - dr/dθ sinθ)

At θ = π/2, this becomes:

dy/dx = [(r sin(π/2) + dr/dθ cos(π/2)) / (r cos(π/2) - dr/dθ sin(π/2))] = [(6)(0) / (-6)] = 0

To learn more about slope click on,

https://brainly.com/question/31056856

#SPJ4

a. These outcomes are mutually exclusive or disjoint.

b. The probability of rolling a 1, 4, or 5 on a fair die is 1/2 or 50%.

(a) The outcomes 1, 4, and 5 are disjoint because they cannot occur at the same time. For example, if we roll a die and it shows 1, then it cannot also show 4 or 5 at the same time. Similarly, if it shows 4, it cannot also show 1 or 5, and if it shows 5, it cannot also show 1 or 4. Therefore, these outcomes are mutually exclusive or disjoint.

(b) The Addition Rule for disjoint outcomes states that the probability of either one of two or more disjoint events occurring is the sum of their individual probabilities. In this case, we want to find the probability of rolling a 1 or 4 or 5. Since these outcomes are disjoint, we can simply add their individual probabilities to find the total probability:

P(1 or 4 or 5) = P(1) + P(4) + P(5)

Assuming we have a fair die, the probability of rolling each of these outcomes is 1/6:

P(1 or 4 or 5) = 1/6 + 1/6 + 1/6 = 3/6

Simplifying the fraction, we get:

P(1 or 4 or 5) = 1/2

Therefore, the probability of rolling a 1, 4, or 5 on a fair die is 1/2 or 50%.

To learn more about Probability here:

brainly.com/question/30034780#

#SPJ11

The relationship between the unit circle and the periodicity of the sine and cosine functions affects the graphs of these functions.

What is the trigonometric function?

the trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.

The unit circle is a circle centered at the origin with a radius of 1, which is used to define the values of sine and cosine functions.

As we move around the unit circle in a counterclockwise direction starting from the point (1, 0) on the x-axis, the angle formed by the radius and the positive x-axis increases.

The sine and cosine of each angle can be found by calculating the y- and x-coordinates of the point on the unit circle that corresponds to that angle.

The sine and cosine functions are periodic functions, which means that they repeat their values after a certain interval of the input.

The period of both functions is 2π, which means that the value of the function repeats itself after an angle of 2π (or 360 degrees).

This periodicity is related to the unit circle because as we move around the circle, the values of sine and cosine repeat themselves at each interval of 2π.

The relationship between the unit circle and the periodicity of the sine and cosine functions affects the graphs of these functions. The sine and cosine graphs have a repeating wave-like pattern, where each period is a complete cycle of the function.

The x-axis of the graph represents the angle in radians, and the y-axis represents the value of the function.

The maximum and minimum values of the sine and cosine functions are 1 and -1, which correspond to the points (1, 0) and (-1, 0) on the unit circle.

The x-intercepts of the sine function occur at every multiple of π, and the x-intercepts of the cosine function occur at every multiple of π/2.

Hence, The relationship between the unit circle and the periodicity of the sine and cosine functions affects the graphs of these functions.

To learn more about the trigonometric function visit:

https://brainly.com/question/25618616

#SPJ1

The area under the curve to the right of 64 is approximately 0.2514.

To find the area under the curve to the right of 64 for a normal distribution with a mean (μ) of 60 and a standard deviation (σ) of 6, follow these steps:

Step 1: Convert the raw score (64) to a z-score. z = (X - μ) / σ z = (64 - 60) / 6 z = 4 / 6 z ≈ 0.67

Step 2: Use a standard normal distribution table or a calculator to find the area to the left of the z-score. For z ≈ 0.67, the area to the left is approximately 0.7486.

Step 3: Find the area to the right of the z-score.

Since the total area under the curve is 1, subtract the area to the left from 1 to find the area to the right. Area to the right = 1 - 0.7486 Area to the right ≈ 0.2514

So, the area under the curve to the right of 64 is approximately 0.2514.

Learn more about z-score,

https://brainly.com/question/25638875

#SPJ11

Answer:

(x + 6, y + 6)

Step-by-step explanation:

Points (6,-8) → (12,-2)

We see the y increase by 6 and the x increase by 6, so the translation is

(x + 6, y + 6)

The probability there is no event given the lot is full is 0.09.

To compute the probability of no event given the lot is full (P(no event | lot is full)), we will use the complementary rule, as the sum of probabilities for all events should equal 1.

The complementary rule states: P(A') = 1 - P(A), where A' is the complement of event A.

In this case, P(sporting event) = 0.56, and P(academic event) = 0.35.

First, we need to find the total probability of both events occurring when the lot is full: P(sporting event) + P(academic event) = 0.56 + 0.35 = 0.91.

Now we can apply the complementary rule to find the probability of no event given the lot is full: P(no event | lot is full) = 1 - P(events) = 1 - 0.91 = 0.09.

So, the probability there is no event given the lot is full is 0.09.

To learn more about probability here:

brainly.com/question/30034780#

#SPJ11

A) The average slope of the function on the interval [−4,8] is -24.

B) The value of c that satisfies the Mean Value Theorem is c = 3.

(A) The average slope of the function on the interval [−4,8] is given by:

f(8)−f(−4) / (8−(−4))

= (2−4(8)2) − (2−4(−4)2) / 12

= (-254) − (34) / 12

= -24

(B) By the Mean Value Theorem, we know that there exists a value c in

the open interval (-4,8) such that:

f′(c) = (f(8)−f(−4)) / (8−(−4))

     = -24

We need to find the value of c that satisfies the above equation. The

derivative of f(x) is given by:

f′(x) = -8x

Setting f′(c) = -24, we get:

-8c = -24

c = 3

Therefore, the value of c that satisfies the Mean Value Theorem is c = 3.

for such more question on average

https://brainly.com/question/20118982

#SPJ11

There are several potential problems that could arise from the situation described. Firstly, the response rate of only 30% may not be representative of the entire population, and thus the results may not accurately reflect the views of all constituents.

Additionally, the question itself may be leading or biased, potentially influencing respondents to answer in a certain way. Furthermore, the survey may not have been distributed randomly, which could further skew the results. Lastly, it's important to note that agreement with the statement "soft on crime" can be interpreted in many different ways, making it difficult to draw clear conclusions from the survey results.

Know more about constituents here:

https://brainly.com/question/14101794

#SPJ11

The Jacobian of  ∂ ( x , y ) / ∂ ( u , v )  is   [tex]\left[\begin{array}{ccc}1/5&1/5\\1/5&1/5\end{array}\right][/tex]

The Jacobian is a matrix of partial derivatives that describes the relationship between two sets of variables. In this case, we have two input variables, u and v, and two output variables, x and y.

To find the Jacobian for our change of variables, we need to compute the four partial derivatives in the matrix above. We start by computing ∂ x / ∂ u:

∂ x / ∂ u = − 1 / 5

To compute ∂ x / ∂ v, we differentiate x with respect to v, treating u as a constant:

∂ x / ∂ v = 1 / 5

Next, we compute ∂ y / ∂ u:

∂ y / ∂ u = 1 / 5

Finally, we compute ∂ y / ∂ v:

∂ y / ∂ v = 1 / 5

Putting it all together, we have:

J =  [tex]\left[\begin{array}{ccc}1/5&1/5\\1/5&1/5\end{array}\right][/tex]

This is the Jacobian matrix for the given change of variables. It tells us how changes in u and v affect changes in x and y. We can also use it to perform other calculations involving these variables, such as integrating over a region in the u-v plane and transforming the result to the x-y plane.

To know more about Jacobian here

https://brainly.com/question/29855578

#SPJ4

Complete Question:

Find the Jacobian ∂ ( x , y ) / ∂ ( u , v ) for the indicated change of variables.

x = − 1 / 5 ( u − v ) , y = 1 / 5 ( u + v )

a) To find the exact length L of the curve 3y² = (4x - 3), 1 < x < 2, where y ≥ 20, we will use the arc length formula: L = ∫[a, b] √(1 + (dy/dx)²) dx. First, we find the derivative dy/dx = (d/dx) (3y²) / (d/dx) (4x - 3). Then, we find the integral over the given interval and evaluate it to get the length L.

b) To evaluate the integral ∫ -∞ to 0 x eˣ dx, we use integration by parts. Let u = x and dv = eˣ dx. Find du and v, and then apply the integration by parts formula: ∫ u dv = uv - ∫ v du. Finally, evaluate the resulting expression.

c) To evaluate the integral ∫ 0 to 3 1/(x-1) dx, perform a substitution. Let u = x-1, so du = dx. The new integral is ∫ 1/u du over the transformed interval. Evaluate the integral and substitute back to obtain the final result.

To know more about arc length formula click on below link:

https://brainly.com/question/30760398#

#SPJ11

The answer of the given question based on the expression is equivalent is , [tex]\frac{27}{8}[/tex] .

In mathematics, an expression is a combination of numbers, variables, and mathematical operations that represents a quantity or a value. Expressions can be simple or complex, and they can include constants, variables, coefficients, and exponents. Expressions can be evaluated or simplified using various techniques, like the order of operations, algebraic manipulation, and factoring. The value of an expression depends on the values of its variables and constants.

The expression "Negative 2 and one-fourth divided by negative two-thirds" we can write as:

[tex]-2\frac{1}{4}[/tex] ÷[tex](-\frac{2}{3} )[/tex]

To simplify this expression, we first need to convert the mixed number [tex]-2\frac{1}{4}[/tex] to an improper fraction:

[tex]-2\frac{1}{4} = -\frac{9}{4}[/tex]

Substituting this value and the fraction ([tex]-\frac{2}{3}[/tex] ) into the expression, we get:

[tex]-\frac{9}{4}[/tex] ÷ [tex](-\frac{2}{3} )[/tex]

To divide fractions, we invert the second fraction and multiply:

[tex]-\frac{9}{4}[/tex] × [tex](-\frac{3}{2} )[/tex]

Simplifying the numerator and denominator, we get:

[tex]\frac{27}{8}[/tex]

Therefore, expression that is equivalent to "Negative 2 and one-fourth divided by negative two-thirds" is [tex]\frac{27}{8}[/tex] .

To know more about Improper fraction visit:

https://brainly.com/question/21449807

#SPJ1

Approximately 24,024 Acrosonic model F loudspeaker systems were sold in the first 4 years after they appeared on the market.

The number of Acrosonic model F loudspeaker systems that appeared on the market can be determined by integrating the rate of sales function f(t) from t=0 to t=4:

∫[0,4] f(t) dt = ∫[0,4] 2,400(3 - 2e^(-t)) dt

Using integration by substitution with u = 3 - 2e^(-t), du/dt = 2e^(-t), and dt = -ln(3/2) du, we can simplify the integral:

∫[0,4] f(t) dt = -2,400ln(3/2) ∫[1,5] u du = -2,400ln(3/2) [(u^2)/2] from 1 to 5
= -2,400ln(3/2) [(5^2)/2 - (1^2)/2]
= -2,400ln(3/2) (12)
≈ -21,098

Since we cannot have a negative number of loudspeaker systems, we round the result to the nearest whole number:

The number of Acrosonic model F loudspeaker systems that appeared on the market is approximately 21,098 systems.

The growth rate of Acrosonic model F loudspeaker systems sales is given by the function f'(t) = 2,400(3 - 2e^(-t)) systems/year, where t represents the number of years the loudspeaker systems have been on the market. To determine the total number of systems sold in the first 4 years, you need to integrate the growth rate function with respect to time (t) from 0 to 4.

∫(2,400(3 - 2e^(-t))) dt from 0 to 4

First, apply the constant multiplier rule:

2,400 ∫(3 - 2e^(-t)) dt from 0 to 4

Now, integrate the function with respect to t:

2,400 [(3t + 2e^(-t)) | from 0 to 4]

Now, substitute the limits of integration:

2,400 [(3(4) + 2e^(-4)) - (3(0) + 2e^(0))]

Simplify the expression:

2,400 [(12 + 2e^(-4)) - 2]

Calculate the final value and round to the nearest whole number:

2,400 (10 + 2e^(-4)) ≈ 24,024

Visit here to learn more about integration:

brainly.com/question/22008756

#SPJ11

We can conclude that Y is a minimal sufficient statistic for p in this Bernoulli model.

In the Bernoulli statistical model where X1, ..., Xn is 0 = Vp, with both & and p taking values in (0,1), the parameter of interest is p. To find a minimal sufficient statistic for θ, we can use the factorization theorem.

Let Y be the number of successes in the sample, i.e., Y = ∑ Xi. Then, the likelihood function can be written as:

L(p; x) = pY (1-p)(n-Y)

Now, let's consider two different samples, x and y. We want to find out whether the ratio of their likelihoods depends on p or not. That is:

L(p; x) / L(p; y) = [pYx (1-p)(n-Yx)] / [pYy (1-p)(n-Yy)]

= p(Yx - Yy) (1-p)(n - Yx - n + Yy)

= p(Yx - Yy) (1-p)(Yy - Yx)

Notice that this ratio only depends on p if Yx - Yy = 0. Otherwise, it depends on both p and Y.

In other words, if we know the value of Y, we have all the information we need to estimate p. This means that any other statistic that depends on the sample but not on Y would be redundant and not necessary for estimating p.

Know more about Bernoulli model here:

https://brainly.com/question/30504672

#SPJ11

When the production manager selects a sample of items that have been produced on her production line and computes the proportion of those items that are defective, the proportion is referred to as a statistic.

The statement is true.

A statistic can be the sample mean or the sample standard deviation, which is a number computed from sample. Since a sample is random in nature therefore every statistic is a random variable (that is, it differs from sample to sample in such a way that it cannot be predicted with certainty).

Statistics are computed to estimate the corresponding population parameters.

Here the production manager selects a sample of items produced on her production line to compute the proportion of defective items, that is taken from the sample and would later be used to represent the entire bunch of items produced. Thus, the proportion can be referred to as a statistic.

Hence, the statement given is true.

To know more about statistic here

https://brainly.com/question/29093686

#SPJ4

(a) To rank each lottery pair by statewise dominance, we compare the prizes of each lottery for each possible outcome of the die roll. Here are the six rankings:
Lottery 1 >SW Lottery 2
Lottery 1 >SW Lottery 3
Lottery 1 >SW Lottery 4
Lottery 2 ∼SW Lottery 3
Lottery 2 ∼SW Lottery 4
Lottery 3 ∼SW Lottery 4

(b) To rank each lottery pair by first-order stochastic dominance, we compare the cumulative distribution functions (CDFs) of each lottery. The CDF of a lottery gives the probability that the prize is less than or equal to a certain value. Here are the rankings:
Lottery 1 >F OSD Lottery 2 >F OSD Lottery 3 >F OSD Lottery 4
To show why Lottery 1 is first-order stochastically dominant over Lottery 2, consider the following CDFs:
Lottery 1:
Prize ≤ $1: 1/6
Prize ≤ $2: 2/6
Prize ≤ $3: 3/6
Prize ≤ $4: 4/6
Prize ≤ $5: 5/6
Prize ≤ $6: 6/6
Lottery 2:
Prize ≤ $1: 0/6
Prize ≤ $2: 1/6
Prize ≤ $3: 2/6
Prize ≤ $4: 3/6
Prize ≤ $5: 4/6
Prize ≤ $6: 6/6
We can see that for any prize value, the CDF of Lottery 1 is always greater than or equal to the CDF of Lottery 2. This means that the probability of winning a certain prize or less is always greater for Lottery 1 than for Lottery 2, which is the definition of first-order stochastic dominance.
We can similarly compare the CDFs of the other lotteries to arrive at the ranking above.

(c) To rank each lottery pair by second-order stochastic dominance, we compare the CDFs of the lotteries' expected values. The expected value of a lottery is the sum of the prizes multiplied by their probabilities, and the CDF of the expected value gives the probability that the expected value is less than or equal to a certain value. Here are the rankings:
Lottery 1 >SOSD Lottery 2 >SOSD Lottery 4 >SOSD Lottery 3
To show why Lottery 1 is second-order stochastically dominant over Lottery 2, consider the following CDFs of the expected values:
Lottery 1:
Expected value ≤ $1: 1/6
Expected value ≤ $2: 3/6
Expected value ≤ $3: 4/6
Expected value ≤ $4: 5/6
Expected value ≤ $5: 6/6
Expected value ≤ $6: 6/6
Lottery 2:
Expected value ≤ $1: 0/6
Expected value ≤ $2: 1/6
Expected value ≤ $3: 2/6
Expected value ≤ $4: 3/6
Expected value ≤ $5: 4/6
Expected value ≤ $6: 5/6
We can see that for any expected value, the CDF of Lottery 1 is always greater than or equal to the CDF of Lottery 2. This means that the probability of getting an expected value or less is always greater for Lottery 1 than for Lottery 2, which is the definition of second-order stochastic dominance.
We can similarly compare the CDFs of the other lotteries to arrive at the ranking above.

Learn more about it here:

https://brainly.com/question/31581956

#SPJ11

The required sample size if you want to be 95% confident that the sample proportion is within 1% of the population proportion is  3173.

Based on the survey, the population proportion (p) is 660/800 = 0.825. To determine the required sample size (n) with a 95% confidence level and a margin of error (E) of 1% (0.01), we use the following formula:

n = (Z² * p * (1-p)) / E²

Here, Z is the Z-score corresponding to the desired confidence level. For a 95% confidence level, the Z-score is 1.96.

n = (1.96² * 0.825 * (1-0.825)) / 0.01²
n ≈ 3172.23

Know more about margin of error here:

https://brainly.com/question/29101642

#SPJ11