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Kamila [148]
3 years ago
15

A population proportion is . A sample of size will be taken and the sample proportion will be used to estimate the population pr

oportion. Use z-table. Round your answers to four decimal places. Do not round intermediate calculations. a. What is the probability that the sample proportion will be within of the population proportion? b. What is the probability that the sample proportion will be within of the population proportion?
Mathematics
1 answer:
padilas [110]3 years ago
6 0

Complete Question:

A population proportion is 0.4. A sample of size 200 will be taken and the sample proportion p will be used to estimate the population proportion. Use z- table Round your answers to four decimal places. Do not round intermediate calculations. a. What is the probability that the sample proportion will be within ±0.03 of the population proportion? b. What is the probability that the sample proportion will be within ±0.08 of the population proportion?

Answer:

A) 0.61351

Step-by-step explanation:

Sample proportion = 0.4

Sample population = 200

A.) proprobaility that sample proportion 'p' is within ±0.03 of population proportion

Statistically:

P(0.4-0.03<p<0.4+0.03)

P[((0.4-0.03)-0.4)/√((0.4)(.6))/200 < z < ((0.4+0.03)-0.4)/√((0.4)(.6))/200

P[-0.03/0.0346410 < z < 0.03/0.0346410

P(−0.866025 < z < 0.866025)

P(z < - 0.8660) - P(z < 0.8660)

0.80675 - 0.19325

= 0.61351

B) proprobaility that sample proportion 'p' is within ±0.08 of population proportion

Statistically:

P(0.4-0.08<p<0.4+0.08)

P[((0.4-0.08)-0.4)/√((0.4)(.6))/200 < z < ((0.4+0.08)-0.4)/√((0.4)(.6))/200

P[-0.08/0.0346410 < z < 0.08/0.0346410

P(−2.3094 < z < 2.3094)

P(z < -2.3094 ) - P(z < 2.3094)

0.98954 - 0.010461

= 0.97908

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Answer:

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

Step-by-step explanation:

f(x, y, z) = x⁴ + y⁴ + z⁴

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4x² = 4z²

4x² - 4z² = 0

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

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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

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when x = -y = -z

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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)

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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)

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