As we have shown that the expected value of ô2 is equal to σ², which means that ô2 is an unbiased estimator for σ².
Assuming that we have a random sample of size n drawn from a normal population with a mean of µ and a variance of σ², we can estimate the population variance using the given formula σ² = 1/2k Σi=1 k (Y2i-Y2i-1)², where n = 2k.
Now, the question asks us to show that this estimator, denoted by ô2, is an unbiased estimator for σ².
To show that ô2 is an unbiased estimator, we need to calculate its expected value and show that it is equal to σ². The expected value of ô2 can be calculated as follows:
E(ô2) = E(1/2k Σi=1 k (Y2i-Y2i-1)²) = 1/2k Σi=1 k E((Y2i-Y2i-1)²)
Now, since Y1, Y2, ..., Yn are drawn from a normal population with mean µ and variance σ², we know that the difference Y2i-Y2i-1 follows a normal distribution with mean 0 and variance 2σ². Therefore, the expected value of (Y2i-Y2i-1)² is given by:
E((Y2i-Y2i-1)²) = Var(Y2i-Y2i-1) + [E(Y2i-Y2i-1)]² = 2σ² + 0² = 2σ²
Substituting this into the expression for E(ô2), we get:
E(ô2) = 1/2k Σi=1 k E((Y2i-Y2i-1)²) = 1/2k Σi=1 k 2σ² = σ²
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Find the length of the spiraling polar curve r = 2e30 From 0 to 2x The length is
The length of the spiraling polar curve r = 2[tex]e^{3\theta[/tex] from 0 to 2π is (2/3)√[10]([tex]e^{6\pi[/tex] - 1).
The length of a polar curve can be found using the formula:
L = [tex]\int\limits^a_b[/tex]√[r² + (dr/dθ)²] dθ
where r is the polar function, and a and b are the starting and ending angles of the curve, respectively.
In this case, the polar function is r = 2[tex]e^{3\theta[/tex], and we want to find the length from 0 to 2π. First, we need to find the derivative of r with respect to θ:
dr/dθ = 6[tex]e^{3\theta[/tex]
Plugging in these values into the formula, we get:
L = [tex]\int\limits^{2 \pi}_0[/tex] √[4[tex]e^{6\theta[/tex] + 36[tex]e^{6\theta[/tex]] dθ
L = [tex]\int\limits^{2 \pi}_0[/tex] 2[tex]e^{3\theta[/tex] √[10] dθ
L = 2√[10] [tex]\int\limits^{2 \pi}_0[/tex] [tex]e^{3\theta[/tex] dθ
Using integration by substitution, we can solve this integral:
L = 2√[10] [[tex]e^{3\theta[/tex]/3] (0 to 2π)
L = 2√[10] [([tex]e^{6\pi[/tex] - 1)/3]
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Find the indefinite integral: Sx^-1/6dx
The indefinite integral of [tex]\int\limits x^{(-1/6)} dx[/tex] is [tex]6x^{(5/6) }+ C[/tex] where C is a constant of integration, under the given condition that no limits are provided since its a indefinite integral.
So here we have to proceed by performing a set of calculations that fall under the criteria provided by the principles of indefinite integral
The given indefinite integral is [tex]\int\limits x^{(-1/6)} dx[/tex]
Evaluating the form
[tex]\int\limits x^{(-1/6)} dx[/tex]
[tex]= \int\limits 1/x^{(1/6)} dx[/tex]
[tex]= 6x^{(5/6)} + C[/tex] here C is constant concerning the integration.
Indefinite integral refers to the form expression Which has no limits that projects the family of function that differentiate by a constant.
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the surface defined by the equation 2 = 9(2,y) at the point (1, 2, 1). 2. [3 marks] The function f(x,y) 4.2 + 2.r- 10.2y + xy 10.xy + xy - 4.ry2 has four critical points: 3 (21,91) = (-6, -2), (12,42)
For the first part of your question, the equation 2 = 9(2,y) defines a surface in three-dimensional space. This surface is a plane that is perpendicular to the y-axis and intersects the y-axis at the point (0, 2/9, 0). At the point (1, 2, 1), the value of y is 2, so the point lies on the surface. For the second part of your question, the function f(x,y) has four critical points.
For the function f(x,y) = 4.2 + 2x - 10.2y + xy + 10xy + xy - 4xy^2, let's find the critical points. Critical points are the points where the partial derivatives with respect to x and y are both zero.
Step 1: Find the partial derivatives:
fx(x,y) = ∂f/∂x = 2 + y + 10y - 4y^2
fy(x,y) = ∂f/∂y = -10.2 + x + 10x + x - 8xy
Step 2: Set the partial derivatives to zero and solve the system of equations:
2 + y + 10y - 4y^2 = 0
-10.2 + x + 10x + x - 8xy = 0
Step 3: Solve the system of equations for x and y to find the critical points. This may require techniques such as substitution or elimination.
For the second part of your question, the function f(x,y) has four critical points.which are points where the gradient of the function is equal to zero. These points are (2,1), (-6,-2), (12,42), and (21,91). At these points, the partial derivatives of the function with respect to x and y are both zero. The critical points can be used to determine the maximum, minimum, or saddle point of the function.
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a fair die is rolled and a fair coin is flipped. what is the probability that either the die will come up 2 or 3, or the coin will land heads up?
The probability that either the die will come up 2 or 3, or the coin will land heads up is 5/6.
To find the probability of either event happening, we can add the probabilities of each individual event happening and then subtract the probability of both events happening together (since that would be counted twice).
The probability of the die coming up 2 or 3 is 2/6, or 1/3, since there are two out of six equally likely outcomes that meet this condition.
The probability of the coin landing heads up is 1/2, since there are two equally likely outcomes (heads or tails).
To find the probability of both events happening together, we can multiply the probabilities of each event: (1/3) * (1/2) = 1/6.
So, the probability of either the die coming up 2 or 3, or the coin landing heads up is:
(1/3) + (1/2) - (1/6) = 5/6
Therefore, the probability is 5/6.
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The country of Sudan has an estimated annual growth rate of 2 percent. At this rate of growth, approximately how many years will it take for the population of Sudan to double?30 years35 years50 years80 years140 years
Using the rule of 70, we can estimate that it will take approximately 35 years for the population of Sudan to double with an estimated annual growth rate of 2 percent,since 70/2 = 35.
What is rate?Rate refers to the measure of change in one quantity with respect to another over a given time period. It can be expressed as a ratio or percentage and is commonly used in finance, economics, science, and mathematics.
What is percent?Percent, denoted by the symbol "%", means "per hundred" and is used to express a fraction or ratio in relation to 100. It is commonly used in business, finance, and statistics to represent changes, growth rates, and other relative measures.
According to the given information:
If a country's population is growing at a constant rate, the time it takes to double its population can be estimated using the rule of 70. The rule of 70 states that you can estimate the number of years it will take for a population to double by dividing 70 by the annual growth rate as a percentage.
In the case of Sudan, with an estimated annual growth rate of 2 percent, it would take approximately 35 years for the population to double. This is calculated by dividing 70 by 2, which equals 35. Therefore, if the growth rate remains constant, Sudan's population is expected to double in approximately 35 years
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Q2. Let X1, ... , Xn be an independent random sample from the p.d.f. given by f(x; θ) = θ/x^2, 0<θ< x <[infinity]. (i) Find the MLE of θ; (5 marks) (ii) Find one MME of θ and determine whether it is consistent. (5 marks)
the MLE of θ is θ = max(xi)
the MME is consistent.
To find the maximum likelihood estimator (MLE) of θ, we first write the likelihood function as:
[tex]L(\theta|x_1, ..., x_n) = \theta^n / (\pi xi^2)[/tex]
Taking the logarithm of this function, we have:
[tex]l(\theta|x_1, ..., x_n) = n log(\theta) - 2 \Sigma log(xi)[/tex]
To find the maximum, we differentiate with respect to θ and set the derivative equal to zero:
[tex]dl(\theta|x_1, ..., x_n) / d\theta = n/\theta = 0[/tex]
Solving for θ, we obtain the MLE:
[tex]\theta = max(xi)[/tex]
(ii) To find a method of moments estimator (MME) of θ, we first find the population mean:
[tex]E(X) =\int \theta/x f(x; theta) dx[/tex]
[tex]= \theta \int 1/x^2 dx[/tex]
= θ
We set the sample mean equal to the population mean:
[tex]1/n \Sigma xi = \theta[/tex]
Solving for θ, we obtain the MME:
[tex]\theta = (1/n) \Sigma xi[/tex]
To determine whether this estimator is consistent, we use the weak law of large numbers, which states that as n approaches infinity, the sample mean converges in probability to the population mean. Since the MME is based on the sample mean, if the sample mean converges in probability to the population mean, then the MME is consistent.
In this case, since E(X) = θ and the sample mean is an unbiased estimator of E(X), we have:
[tex]\lim_{n \to \infty} P(|(1/n)\Sigma xi - \theta| < \epsilon) = 1[/tex]
for any [tex]\epsilon > 0[/tex].
The MME is consistent.
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Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 43.8 months and a standard deviation of 8.3 months. in USE SALT (a) If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of its batteries will the company expect to replace? (Round your answer to two decimal places.) % (b) If Quick Start does not want to make refunds for more than 8% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)? months
To ensure no more than 8% of batteries are refunded, Quick Start should guarantee the batteries for 31 months.
(a) To determine the percentage of batteries that fail within 36 months, we will use the Z-score formula:
Z = (X - μ) / σ
Where X is the target value (36 months), μ is the mean (43.8 months), and σ is the standard deviation (8.3 months).
Z = (36 - 43.8) / 8.3 ≈ -0.94
Using a standard normal distribution table, we find that the percentage of batteries failing within 36 months is approximately 17.36%. So, Quick Start can expect to replace about 17.36% of its batteries under the 36-month guarantee.
(b) If Quick Start does not want to make refunds for more than 8% of its batteries, we need to find the corresponding Z-score for 8%. Using a standard normal distribution table, we find a Z-score of approximately -1.41.
Now, we need to find the corresponding number of months (X) for this Z-score:
X = μ + Z × σ
X = 43.8 + (-1.41) × 8.3 ≈ 31.3 months
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Please express the following two functions in power series andstate the IOC:х 2 – 3 In 2 + x х x + 2 2.x2 – X – 1 х 1 2.
The power series are:
1. x^2 - 3x + 3/2(x-1)^2 - 1/2(x-1)^3 + ... and IOC is (0,2).
2. 2(x-1/2) - 2ln(1-(x-1/2)) + C and IOC is (1/2,3/2).
1. f(x) = x^2 - 3ln(2) + x - 2/(x+2)
Using the power series expansions for ln(1+x), 1/(1+x), and 1/(1-x), we can write:
f(x) = x^2 - 3ln(2) + x - 2/(x+2)
= x^2 - 3ln(1+(x-1)) + x - 2/(x+2)
= x^2 - 3((x-1) - (x-1)^2/2 + (x-1)^3/3 - ...) + x - 2/(x+2)
= x^2 - 3x + 3/2(x-1)^2 - 1/2(x-1)^3 + ...
The series converges for |x-1| < 1, so the interval of convergence is (0,2).
2. g(x) = 2x^2 - x - 1
We can factor g(x) as:
g(x) = 2(x-1/2)(x+1)
Using the power series expansions for 1/(1-x) and ln(1+x), we can write:
g(x) = 2(x-1/2)(x+1)
= 2(x-1/2) - 2(x-1/2)^2 + 2(x-1/2)^3 - ...
= 2(x-1/2) - 2ln(1-(x-1/2)) + C
where C is a constant. The series converges for |x-1/2| < 1, so the interval of convergence is (1/2,3/2).
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Question 4: Assume that the building costs are normally distributed with mean = 2,800K and standard deviation a = 200K. and that this is known from the beginning of Year 1. Determine the expected value of the investment and decide whether the project is worth going ahead. Investigate if the decision is still valid if x = 2,800K but o varies by +10% from its stated value. (Hint: Use the result stated in Question 3 above. 20 marks)
The expected value of investment is 400K.
Let's assume that the expected revenue from the project is 3,200K, which is the midpoint of the given range.
Then, the expected value of the investment can be calculated as follows:
Expected value of investment = Expected revenue - Expected cost
Expected revenue = 3,200K
Expected cost = Expected value of building costs
Expected value of building costs = Mean of building costs = 2,800K
Therefore, the expected value of investment = 3,200K - 2,800K = 400K
Since the expected value of the investment is positive, the project is worth going ahead.
Now, let's investigate if the decision is still valid if the standard deviation varies by +10% from its stated value.
New standard deviation = 220K (10% increase from 200K)
Using the formula from Question 3, we can calculate the probability that the total cost of the project will exceed 2,800K with the new standard deviation:
z = (2,800K - 2,800K) / 220K = 0
P(z > 0) = 0.5
Therefore, the probability that the total cost of the project will exceed 2,800K is still 0.5.
Using the same calculations as above, the expected value of investment with the new standard deviation can be calculated as follows:
Expected value of investment = Expected revenue - Expected cost
Expected revenue = 3,200K
Expected cost = Expected value of building costs
Expected value of building costs = Mean of building costs = 2,800K
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Please help me with this question,it's really simple.
What is the probability of landing on heads on the coin, and a number less than 7 on the spinner?
Answer choices:
A.)3/8
B.)1/4
C.)1/16
D.)2/3
The picture will help you out
The probability of landing on heads on the coin is 1/2. The probability of landing on a number less than 7 on the spinner is 6/8 or 3/4. Since these two events are independent, the probability of both events happening is the product of their individual probabilities:
[tex](1/2) \times (3/4) = 3/8[/tex]
Note:- I'm sorry to bother you but can you please mark me BRAINLEIST if this ans is helpfull
If f(x) is a continuous function such that f(x)≥0,∀x∈[2,10] and ∫ 48 f(x)dx=0, then the value of f(6), is
A function is considered continuous at a point if its limit exists at that point and is equal to the function's value at that point.
a function is continuous at a point if it has no gaps, jumps, or holes in its graph at that point. Since the integral of f(x) from 2 to 10 is zero, and f(x) is continuous and non-negative on this interval, it follows that f(x) must be identically zero on [2, 10].
Therefore, f(6) = 0
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b. Let X be the concentration of ethanol in a chemical solution and Y be the acidity of the solution. Suppose the joint probability density function of these two variables is given by 365,1)=CC36-28-39 ) (C(30 - 2x - 3y) (x) = 3 0 OS*s 4, OS y S4 elsewhere. Evaluate i. the value of the constant C. [4 marks] the marginal probability density functions fx(x) of Xand f(y) of Y. [6 marks]
The marginal PDF of X is: fX(x) = (90 - 6x)/160, for 0 ≤ x ≤ 3, And the marginal PDF of Y is: fY(y) = (120 - 6y)/160, for 0 ≤ y ≤ 4
To find the value of the constant C, we integrate the joint probability density function over the entire range of X and Y, and set the result equal to 1, since the total probability over the entire range of the two variables must be equal to 1:
∫∫ f(x,y) dxdy = 1
∫∫ C(30 - 2x - 3y) dxdy = 1
We can evaluate this double integral by integrating over Y first and then X:
∫∫ C(30 - 2x - 3y) dxdy = C∫[0,4] ∫[0,3-2/3y] (30 - 2x - 3y) dxdy
= C∫[0,4] [30x -[tex]x^2[/tex] - 3xy] evaluated from 0 to 3-2/3y dy
= C∫[0,4] (90 - 36y + 4[tex]y^2[/tex])/3 dy
= C[(30y^2 - 36[tex]y^3/2[/tex] + 4[tex]y^3[/tex]/3)/3] evaluated from 0 to 4
= C(160/3)
Setting this equal to 1, we get:
C(160/3) = 1
C = 3/160
Therefore, the constant C is 3/160.
Now, we can find the marginal probability density functions of X and Y by integrating the joint probability density function over the range of the other variable. For the marginal PDF of X:
fX(x) = ∫ f(x,y) dy
fX(x) = ∫ 3/160 (30 - 2x - 3y) dy, for 0 ≤ x ≤ 3
fX(x) = (90 - 6x)/160, for 0 ≤ x ≤ 3
And for the marginal PDF of Y:
fY(y) = ∫ f(x,y) dx
fY(y) = ∫ 3/160 (30 - 2x - 3y) dx, for 0 ≤ y ≤ 4
fY(y) = (120 - 6y)/160, for 0 ≤ y ≤ 4
Therefore, the marginal PDF of X is:
fX(x) = (90 - 6x)/160, for 0 ≤ x ≤ 3
And the marginal PDF of Y is:
fY(y) = (120 - 6y)/160, for 0 ≤ y ≤ 4
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what is the probability that a random integer from 92 to 734 is divisible by 15? (all integers in the given range are equally likely to be chosen).
There are 43 integers in the range from 92 to 734 that are divisible by 15. The probability that a random integer from 92 to 734 is divisible by 15 is approximately 0.067.
To find the probability that a random integer from 92 to 734 is divisible by 15, we need to first determine the total number of integers in this range. The difference between 734 and 92 is 642, but since we want to include both endpoints, we need to add 1 to this difference. So there are a total of 643 integers in the range from 92 to 734. Next, we need to determine how many of these integers are divisible by 15. To do this, we need to find the smallest multiple of 15 that is greater than or equal to 92, and the largest multiple of 15 that is less than or equal to 734. The smallest multiple of 15 that is greater than or equal to 92 is 105 (which is 7 times 15), and the largest multiple of 15 that is less than or equal to 734 is 720 (which is 48 times 15).
So the integers from 105 to 720 (inclusive) are all divisible by 15. To count the number of integers in this range, we can divide the difference between 720 and 105 by 15, and add 1 to account for the first multiple of 15:
(720 - 105) / 15 + 1 = 43
Therefore, there are 43 integers in the range from 92 to 734 that are divisible by 15.
To find the probability that a random integer from this range is divisible by 15, we can divide the number of integers that are divisible by 15 (43) by the total number of integers in the range (643):
43 / 643 ≈ 0.067
So the probability that a random integer from 92 to 734 is divisible by 15 is approximately 0.067.
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y′′+αy′+βy=t+e^(4t).
Suppose the form of the particular solution to this differential equation as prescribed by the method of undetermined coefficients is
yp(t)=A1t^2+A0t+B0te^(4t).
Determine the constants α and β.
The constants value of α = -4 and β = 0.
Differential Equation:A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x) Here “x” is an independent variable and “y” is a dependent variable. For example, dy/dx = 5x.
The function is :
[tex]y"+\alpha y'+\beta y=t+e^(^4^t^)[/tex]
the form of the particular solution to this differential equation as prescribed by the method of undetermined coefficients is
[tex]yp(t)=A1t^2+A0t+B0te^(^4^t^).[/tex]
=> [tex]y ' = 2A1t + A0 + B0 [e^(^4^t^) +4 te^(^4^t^) ][/tex]
[tex]y ' = 2A1t + A0 + B0 e^(^4^t^) +4B0 te^(^4^t^)[/tex]
=> [tex]y '' = 2A1 + 4B0e^(^4^t^) + 4B0 [ e^(^4^t^) + 4te^(^4^t^)[/tex]
[tex]y '' = 2A1 + 4B0e^(^4^t^) + 4B0e^(^4^t^) + 16B0te^(^4^t^)[/tex]
Now substitute the values of y ' and y '' in the differential equation:
[tex]y"+\alpha y'+\beta y=t+e^(^4^t^)[/tex]
[tex]2A1 + 4B0e^(^4^t^) + 4B0e^(^4^t^) + 16B0te^(^4^t^) + \alpha {2A1t + A0 + B0e^(^4^t^) + 4B0 te^(^4^t^) } + \beta {A1 t^2 + A0 t + B0 t e6(^4^t^)} = t + e^(^4^t^)[/tex]
Next, we equate coefficients
1) Constant terms of the left side = constant terms of the right side:
[tex]2A1+ 2\alpha A0 = 0[/tex] ..... eq (1)
2) Coefficients of [tex]e^(^4^t^)[/tex] on both sides
8B0 + αB0 = 1 => B0 (8 + α) = 1 .... eq (2)
3) Coefficients on t
2αA1 + βA0 = 1 .... eq (3)
4) Coefficients on [tex]t^2[/tex]
βA1 = 0 ....eq (4)
A1 ≠ 0 => β =0
5) terms on [tex]te^(^4^t^)[/tex]
16B0 + 4αB0 + βB0 = 0 => B0 (16 + 4α + β) = 0 ... eq (5)
B0 ≠ 0 => 16 + 4α + β = 0
Use the value of β = 0 found previously
16 + 4α = 0 => α = - 16 / 4 = - 4.
α = - 4 and β = 0
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2. If a marble is selected at random from Adrian's Bag of Marbles,
Which expression can be used to determine the probability the
Marble selected will NOT be red?
If a marble is selected at random from Adrian's Bag of Marbles, then the probability that marble selected from Adrian's bag will not be red is 0.7.
The "Probability" of an "event-A" occurring is defined as the ratio of the number of favorable outcomes for event A to the total number of possible outcomes in a given sample space. It is denoted as P(A).
To find the probability that marble selected will not be red,
we need to find "total-number" of marbles in Adrian's bag and the number of marbles that are not red.
We know that,
⇒ Number of red marbles = 3,
⇒ Number of blue marbles = 7,
So, Total marbles in bag = Number of red marbles + Number of blue marbles,
⇒ 3 + 7 = 10,
The Number of marbles that are not red = Number of blue marbles = 7,
So, probability that marble selected will not be red is :
⇒ Probability (not red) = (Number of marbles that are not red)/(Total number of marbles),
⇒ 7/10,
⇒ 0.7
Therefore, the required probability is 0.7.
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The given question is incomplete, the complete question is
Adrian's Bag of marbles contain 3 Red and 7 Blue Marbles, If a marble is selected at random from Adrian's Bag of Marbles, then What is the probability the Marble selected will NOT be red?
In a study conducted in 2004, it was found that the share of online advertisement worldwide, as a percentage of the total ad market, was expected to grow at the rate of R(t) = −0.021t^2 + 0.3004t + 0.06 0 ≤ t ≤ 6 percent per year at time t (in years), with t = 0 corresponding to the beginning of 2000. The online ad market at the beginning of 2000 was 1.7% of the total ad market.
(a) What is the projected online ad market share at any time t? S(t) =
(b) What is the projected online ad market share (as a percentage) at the beginning of 2005? (Round your answer to two decimal places.) %
The projected online ad market share (as a percentage) at the beginning of 2005 is 26.7%.
(a) The projected online ad market share at any time t can be found by integrating the rate function R(t) with respect to t:
S(t) = ∫(−0.021t^2 + 0.3004t + 0.06) dt
S(t) = −0.007t^3 + 0.1502t^2 + 0.06t + C
where C is the constant of integration. We can find the value of C by using the fact that the online ad market share at the beginning of 2000 was 1.7%:
S(0) = −0.007(0)^3 + 0.1502(0)^2 + 0.06(0) + C = 0.017
C = 0.017
So the projected online ad market share at any time t is:
S(t) = −0.007t^3 + 0.1502t^2 + 0.06t + 0.017
(b) The beginning of 2005 corresponds to t = 5, so we can use the function S(t) to find the projected online ad market share at that time:
S(5) = −0.007(5)^3 + 0.1502(5)^2 + 0.06(5) + 0.017 = 0.267
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Pls help me with this question
The value of angle x is 40^o
What are supplementary angles?A given set of angles are said to be supplementary if and only if on addition of the measures of the angles, it forms 180 degrees. This is the measure of an angle on a straight line.
In the given diagram, to find the value of x;
Triangle ABD is an isosceles, thus base angles are equal. So that;
ABD ≅ ADB = 25^o
Thus,
<ABD + x + 115 = 180 (definition of supplementary)
25 + x + 115 = 180
140 + x = 180
x = 180 - 140
= 40
x = 40^o
Therefore, the value of x is 40^o.
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An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions are __________.
conditions
approximate
given
violated
met
The complete sentence is,
An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions are ''Violated''.
Given that;
To find An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions.
Now, We know that;
Robust inference is inference that is insensitive to (smaller or larger) deviations from the assumptions under which it is derived.
Hence, We get;
An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions are ''Violated''.
Hence, Option 4 is true.
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31. How many 7-digit even numbers can be formed using all the digits 0, 1, 2, 2, 3, 5, and 5? (T: 2)(A: /3)
There are 1458, 7-digit even numbers that can be formed using all the digits 0, 1, 2, 2, 3, 5, and 5.
To form a 7-digit even number, the last digit must be even. We have two choices for the last digit: 2 or 5. Once we have chosen the last digit, we have 6 digits left to fill the first six positions. We have two 2's, two 5's, one 0, one 1, and one 3 to choose from.
To count the number of 7-digit even numbers, we will use the multiplication principle. There are 2 choices for the last digit and 3 choices for each of the first six digits (since we cannot use the same digit twice). Therefore, the total number of 7-digit even numbers that can be formed is:
2 x 3 x 3 x 3 x 3 x 3 x 3 = 2 x 3^6 = 1458
So there are 1458 7-digit even numbers that can be formed using all the digits 0, 1, 2, 2, 3, 5, and 5.
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pls pls help whoever gets it right gets marked brainliest
Answer: -2/3
Step-by-step explanation:
Question 1 (10 marks) A salad shop is selling fruit cups. Each fruit cup consists of two types of fruit, strawberries and blue berries. The weight of strawberries in a fruit cup is normally distribute with mean 160 grams and standard deviation 10 grams. The weight of blue berries in a fruit cup is normally distributed with mean μ grams and standard deviation σ grams. The weight of strawberries and blue berries are independent, and it is known that the weight of a fruit cup with average of 300 grams and standard deviation of 15 grams.
(a) Find the values of μ and σ.
(b) The weights of the middle 96.6% of fruit cups are between (300 – K, 300 + K) grams. Find the value
of K.
(c) The weights of the middle 96.6% of fruit cups are between (L1, L2) grams. Find the values of L1 and
L2.
a. σ = 5√5
b. The value of K is 30.75 grams.
c. The values of L1 and L2 are 268.25 grams and 331.75 grams, respectively.
(a) We know that the mean weight of a fruit cup is 300 grams, which is
the sum of the mean weights of strawberries and blueberries. Thus, we
have:
160 + μ = 300
Solving for μ, we get:
μ = 140
Next, we can use the formula for the variance of the sum of independent
random variables to find the variance of the weight of a fruit cup:
Var(weight) = Var(strawberries) + Var(blueberries)
The variance of strawberries is given as 10^2 = 100, and the variance of
blueberries is σ^2. We know that the variance of the weight of a fruit cup
is[tex]15^2[/tex]= 225. Thus, we have:
100 + σ^2 = 225
Solving for σ, we get:
σ = 5√5
(b) The middle 96.6% of the fruit cups corresponds to the interval between the 2.17th and 97.83rd percentiles of the distribution of fruit cup weights. We can use the standard normal distribution to find the z-scores corresponding to these percentiles:
z1 = invNorm(0.0217) ≈ -2.05
z2 = invNorm(0.9783) ≈ 2.05
Using the formula for the standard error of the mean, we can find the value of K:
K = z2 × (15 / √n)
We know that the mean weight of a fruit cup is 300 grams, so n = 1. Plugging in the values, we get:
K = 2.05 × (15 / √1) = 30.75
(c) We can use the mean and standard deviation values found in part (a) to find the z-scores corresponding to the 2.17th and 97.83rd percentiles:
z1 = invNorm(0.0217) ≈ -2.05
z2 = invNorm(0.9783) ≈ 2.05
Using the z-scores and the formula for the standard error of the mean, we can find the values of L1 and L2:
L1 = 300 + z1 × (15 / √1) = 268.25
L2 = 300 + z2 × (15 / √1) = 331.75
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if all possible samples of a specific size are selected from a population and then the means for each sample are computed,what is this distribution of means called? explain
The distribution of means obtained by taking all possible samples of a specific size from a population and computing the means for each sample is called the sampling distribution of the sample mean.
The sampling distribution of the sample mean is a theoretical probability distribution that describes the possible values of the sample means that can be obtained from the population. The shape of the sampling distribution of the sample mean is approximately normal if the sample size is large enough (typically, greater than 30) and the population is normally distributed, regardless of the shape of the population distribution. This is known as the Central Limit Theorem.
The sampling distribution of the sample mean is important in statistics because it allows us to make inferences about the population based on the characteristics of the sample. Specifically, we can use the sampling distribution of the sample mean to estimate the population mean and to calculate confidence intervals for the population mean. We can also use the sampling distribution of the sample mean to test hypotheses about the population mean.
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a) Briefly explain the difference between a data point that has been flagged as an outlier and a data point that has high leverage.
b) Can a data point that has been flagged as an outlier also have high leverage? Explain.
c) Are data points that are outliers or that have high leverage necessarily influential?
The influence of a data point depends on its location in the predictor space, its leverage, and its contribution to the fit of the model.
a) An outlier is a data point that deviates significantly from the other observations in a dataset, and it can be either due to measurement errors, natural variability, or genuine extreme observations. On the other hand, high leverage data points are observations that have extreme values on one or more predictor variables, and they have the potential to exert a significant influence on the estimation of regression coefficients.
b) Yes, a data point that is flagged as an outlier can also have high leverage. This occurs when an observation has extreme values on both the response variable and one or more predictor variables.
c) Not necessarily. While outliers and high leverage data points can impact the results of a statistical analysis, they may not necessarily be influential. An influential data point is one that significantly affects the estimated coefficients and can change the results of a statistical analysis if removed. The influence of a data point depends on its location in the predictor space, its leverage, and its contribution to the fit of the model.
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What is the probability that a randomly selected person is exactly 180 cm? Assume you can measure perfectly.
The probability of finding a person with an exact height of 180 cm is close to zero.
The probability of randomly selecting a person who is exactly 180 cm tall is extremely low, since height measurements are typically continuous and not discrete
Height measurements are usually considered as continuous data, meaning they can take on any value within a certain range, and not just specific, discrete values.
In reality, it is highly unlikely to find a person who has an exact height of 180 cm, as height measurements are subject to natural variation and measurement error.
Even with perfect measuring accuracy, the probability of finding a person with an exact height of 180 cm is still extremely low, as human height distribution typically follows a bell-shaped curve or a normal distribution.
The normal distribution of human height is characterized by a range of heights that occur with varying frequencies, and the probability of finding a person with a height that falls exactly at 180 cm is likely to be infinitesimal.
Therefore, based on the principles of continuous data and the natural variation in human height, the probability of randomly selecting a person who is exactly 180 cm tall is essentially negligible.
Therefore, the probability of finding a person who is exactly 180 cm tall is extremely low, close to zero, and can be considered as practically impossible.
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Use LaGrange Multipliers to find the maximum and minimum values of f(x,y)=x22y2−4y subject to the constraint x2+y2=9.
The maximum value is 0 at points (±3, 0) and the minimum value is -12 at points (0, ±3).
To solve this problem using LaGrange Multipliers, let's define the given function and constraint:
f(x, y) = x^2 * y^2 - 4y
g(x, y) = x^2 + y^2 - 9
Now, introduce the LaGrange Multiplier λ (lambda) and find the gradient of both f and g:
∇f(x, y) = (2xy^2, 2x^2y - 4)
∇g(x, y) = (2x, 2y)
The LaGrange Multiplier method states that for optimal points (x, y), ∇f(x, y) = λ∇g(x, y). So, we have the following equations:
2xy^2 = λ(2x) -> (1)
2x^2y - 4 = λ(2y) -> (2)
From equation (1), we get:
xy^2 = λ
Substitute λ in equation (2):
2x^2y - 4 = 2y(xy^2)
Now, solve for x and y using the constraint g(x, y) = 0:
x^2 + y^2 = 9
By solving the system of equations, you will find four critical points: (±3, 0) and (0, ±3). Evaluate f(x, y) at these points:
f(±3, 0) = 0
f(0, ±3) = -12
Thus, the maximum value is 0 at points (±3, 0) and the minimum value is -12 at points (0, ±3).
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A consumer psychologist wants to determine which fast-food burger is the healthiest. They buy 4 burgers from each of these restaurants: In-n-Out, Jack in the Box, and Whataburger. 4 people measured the grease levels of a burger from each place. The amount of grease was extrapolated from each of these burgers, with lower grease indicating it was healthier. The grease levels of the burgers can be found in the Burgers Grease Levels" excel file on Blackboard. Is this a one-way between or a one-way within groups ANOVA test?
What is the critical fvalue, when alpha is .05?
QUESTION 3 What is the calculated f value?
QUESTION 4 What is the calculated p value? QUESTION 5 What is the partial eta squared?
1) With the given alpha of .05 and the appropriate degrees of freedom (based on the number of groups and the sample size).
2) This can be done in Excel, using the ANOVA function, or with other statistical software.
3) The calculated P value will be given as part of the ANOVA test output.
4 SS error is the sum of squares within groups. These values will also be available in the ANOVA test output.
5) Once you have the data from the Excel file, you can perform these calculations and interpret the results.
Once you have the data from the Excel file, you can perform these calculations and interpret the results.
he critical F value, use an F-distribution table or an online calculator, with the given alpha of .05 and the appropriate degrees of freedom (based on the number of groups and the sample size).
Based on the information provided, this study involves a one-way between-groups ANOVA test. This is because the consumer psychologist is comparing the grease levels of burgers from three different fast-food restaurants (In-n-Out, Jack in the Box, and Whataburger), and the measurements are taken by four different people.
As for the critical F value, calculated F value, calculated P value, and partial eta squared, I am unable to access the "Burgers Grease Levels" Excel file on Blackboard. However, I can provide guidance on how to calculate these values:
1. To find the critical F value, use an F-distribution table or an online calculator, with the given alpha of .05 and the appropriate degrees of freedom (based on the number of groups and the sample size).
2. To calculate the F value, you will need to perform the one-way between-groups ANOVA test on the grease levels data. This can be done in Excel, using the ANOVA function, or with other statistical software.
3. The calculated P value will be given as part of the ANOVA test output.
4. To calculate partial eta squared, use the formula: partial[tex]{\eta}^2=SS_{effect} / (SS_{effect} + SS_{error}),[/tex] where SS_effect is the sum of squares between groups, and SS error is the sum of squares within groups. These values will also be available in the ANOVA test output.
Once you have the data from the Excel file, you can perform these calculations and interpret the results.
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A naturalist leads whale watch trips every morning in March. The number of whales seen has a Poisson distribution with a mean of 4.3. Find the probability that on a randomly selected trip, the number of whales seen is 3.
The probability that on a randomly selected trip, the number of whales seen is 3 is 22.4%.
To find the probability that on a randomly selected whale watch trip in March, the number of whales seen is 3, we'll use the Poisson distribution with a mean of 4.3. The formula for the Poisson probability is:
P(X=k) = (e^(-λ) * (λ^k)) / k!
Where X is the number of whales seen, k is the desired number of whales (in this case, 3), λ is the mean (4.3), and e is the base of the natural logarithm (approximately 2.71828).
Plugging in the values, we get:
P(X=3) = (e^(-4.3) * (4.3^3)) / 3!
P(X=3) = (0.01353 * 79.507) / 6
P(X=3) ≈ 0.22404
So, the probability of seeing 3 whales on a randomly selected trip is approximately 22.4%.
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What is the distance between the coordinates (7.4, 6.8) and (7.4, 2,1)?
1. If 0 = {n € Zin is odd) and B = {ne Z|-4 ≤ n ≤ 6}, calculate the following:(a) O ∩ B(B) B - O(C) O (in Z)
Let's find the solutions for the given sets O and B. The set O is defined as all odd integers n ∈ Z, while set B contains integers n such that -4 ≤ n ≤ 6.
(a) To find O ∩ B, we need to determine the common elements between sets O and B. The odd numbers in the range -4 to 6 are {-3, -1, 1, 3, 5}. Therefore, O ∩ B = {-3, -1, 1, 3, 5}.
(b) To calculate B - O, we need to remove the elements of O from set B. Set B contains {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}. Removing the odd numbers, we get B - O = {-4, -2, 0, 2, 4, 6}.
(c) To find O in Z, we consider all odd integers n in the set of integers Z. Since Z is an infinite set, O in Z is the set of all odd integers, which can be represented as {..., -5, -3, -1, 1, 3, 5, ...}.
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8. Find the probability of each set of independent events.
flipping a tail on a coin and spinning a 5 on a spinner with sections of
equal area numbered 1-5.
a) 1/2
c) 1/5
b) 1/7
d) 1/10
Answer:
Step-by-step explanation:
