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nevsk [136]
3 years ago
8

An IAB study on the state of original digital video showed that original digital video is becoming increasingly popular. Origina

l digital video is defined as professionally produced video intended only for ad-supported online distribution and viewing. According to IAB data, 30% of American adults 18 or older watch original digital videos each month.
1. Suppose that you take a sample of 100 U.S. adults, what is the probability that fewer than 25 in your sample will watch original digital videos?
Mathematics
1 answer:
never [62]3 years ago
5 0

Answer:

11.51% probability that fewer than 25 in your sample will watch original digital videos

Step-by-step explanation:

I am going to use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

\sqrt{V(X)} = \sqrt{np(1-p)}

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that \mu = E(X), \sigma = \sqrt{V(X)}.

In this problem, we have that:

n = 100, p = 0.30

So

\mu = E(X) = np = 100*0.3 = 30

\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.3*0.7} = 4.58

What is the probability that fewer than 25 in your sample will watch original digital videos?

Using continuity correction, this is P(X < 25 - 0.5) = P(X < 24.5), which is the pvalue of Z when X = 24.5. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{24.5 - 30}{4.58}

Z = -1.2

Z = -1.2 has a pvalue of 0.1151

11.51% probability that fewer than 25 in your sample will watch original digital videos

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2 years ago
Find the quotient for 99diveded by 9,801
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Step-by-step explanation:

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2 years ago
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Solve for substitution 8x+2y=13 and 4x+y=11
damaskus [11]
Solving for X:
1) y=-8x+6.5   y=-4x+11
2) -8x+6.5=-4x+11
3) -8x=-4x+11-6.5        (Subtract 6.5 from both sides)
4) -8x+4x=11-6.5         (Add 4x to both sides)
5)-4x=4.5
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Solving for Y:
y=-8(-1.125)+6.5
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7 0
3 years ago
. Use Lagrange multipliers to find the maximum and minimum values of the function, f, subject to the given constraint, g. (Place
zzz [600]

Answer:

Minimum value of f(x, y, z) = (1/3)

Step-by-step explanation:

f(x, y, z) = x⁴ + y⁴ + z⁴

We're to maximize and minimize this function subject to the constraint that

g(x, y, z) = x² + y² + z² = 1

The constraint can be rewritten as

x² + y² + z² - 1 = 0

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

where λ = Lagrange factor, which can be a function of x, y and z

L(x,y,z) = x⁴ + y⁴ + z⁴ - λ(x² + y² + z² - 1)

We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.

(∂L/∂x) = 4x³ - λx = 0

λ = 4x² (eqn 1)

(∂L/∂y) = 4y³ - λy = 0

λ = 4y² (eqn 2)

(∂L/∂z) = 4z³ - λz = 0

λ = 4z² (eqn 3)

(∂L/∂λ) = x² + y² + z² - 1 = 0 (eqn 4)

We can then equate the values of λ from the first 3 partial derivatives and solve for the values of x, y and z

4x² = 4y²

4x² - 4y² = 0

(2x - 2y)(2x + 2y) = 0

x = y or x = -y

Also,

4x² = 4z²

4x² - 4z² = 0

(2x - 2z) (2x + 2z) = 0

x = z or x = -z

when x = y, x = z

when x = -y, x = -z

Hence, at the point where the box has maximum and minimal area,

x = y = z

And

x = -y = -z

Putting these into the constraint equation or the solution of the fourth partial derivative,

x² + y² + z² = 1

x = y = z

x² + x² + x² = 1

3x² = 1

x = √(1/3)

x = y = z = √(1/3)

when x = -y = -z

x² + y² + z² = 1

x² + x² + x² = 1

3x² = 1

x = √(1/3)

y = z = -√(1/3)

Inserting these into the function f(x,y,z)

f(x, y, z) = x⁴ + y⁴ + z⁴

We know that the two types of answers for x, y and z both resulting the same quantity

√(1/3)

f(x, y, z) = x⁴ + y⁴ + z⁴

f(x, y, z) = (√(1/3)⁴ + (√(1/3)⁴ + (√(1/3)⁴

f(x, y, z) = 3 × (1/9) = (1/3).

We know this point is a minimum point because when the values of x, y and z at turning points are inserted into the second derivatives, all the answers are positive! Indicating that this points obtained are

S = (1/3)

Hope this Helps!!!

6 0
2 years ago
6x - 2y = 5
laiz [17]

Answer:

The given system has NO SOLUTION.

Step-by-step explanation:

Here, the given system of equation is:

6 x -  2 y = 5     .......... (1)

3 x  - y = 10          .... (2)

Multiply equation 2 with (-2), we get:

3 x  - y = 10     ( x -2)

⇒  - 6 x + 2 y = - 20

Now, ADD this to equation (1) , we get:

6 x - 2 y  - 6 x + 2 y  = 5 - 20

or, 0 = - 15

WHICH IS NOT POSSIBLE as 0 ≠ -15

Hence, the given system has NO SOLUTION.

7 0
2 years ago
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