All real numbers can be written as what?

Let X represent the amount of gasoline (gallons) purchased by a randomly selected customer at a gas station. Suppose that the me

Alexus [3.1K]

Answer:

a) 18.94% probability that the sample mean amount purchased is at least 12 gallons

b) 81.06% probability that the total amount of gasoline purchased is at most 600 gallons.

c) The approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers is 621.5 gallons.

Step-by-step explanation:

To solve this question, we use the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are 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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For sums, we can apply the theorem, with mean $\mu$ and standard deviation $s = \sqrt{n}*\sigma$

In this problem, we have that:

$\mu = 11.5, \sigma = 4$

a. In a sample of 50 randomly selected customers, what is the approximate probability that the sample mean amount purchased is at least 12 gallons?

Here we have $n = 50, s = \frac{4}{\sqrt{50}} = 0.5657$

This probability is 1 subtracted by the pvalue of Z when X = 12.

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

By the Central Limit theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{12 - 11.5}{0.5657}$

$Z = 0.88$

$Z = 0.88$ has a pvalue of 0.8106.

1 - 0.8106 = 0.1894

18.94% probability that the sample mean amount purchased is at least 12 gallons

b. In a sample of 50 randomly selected customers, what is the approximate probability that the total amount of gasoline purchased is at most 600 gallons.

For sums, so $mu = 50*11.5 = 575, s = \sqrt{50}*4 = 28.28$

This probability is the pvalue of Z when X = 600. So

$Z = \frac{X - \mu}{s}$

$Z = \frac{600 - 575}{28.28}$

$Z = 0.88$

$Z = 0.88$ has a pvalue of 0.8106.

81.06% probability that the total amount of gasoline purchased is at most 600 gallons.

c. What is the approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers.

This is X when Z has a pvalue of 0.95. So it is X when Z = 1.645.

$Z = \frac{X - \mu}{s}$

$1.645 = \frac{X- 575}{28.28}$

$X - 575 = 28.28*1.645$

$X = 621.5$

The approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers is 621.5 gallons.

5 0

3 years ago

Which function is represented by the graph?

Masja [62]

Answer:

<h2>Option B</h2>

please see the attached picture

hope it helps..

Good luck on your assignment

3 0

3 years ago

I am redesigning a rectangular prism in geometry where I have to minimize the surface area while maintaining the same volume. I

Romashka [77]

Strategy:

You will need to make use of the formulas for volume and area of the shapes you have chosen.

... volume of a sphere = (4/3)πr³

... area of a sphere = 4πr²

You will have to use the volume equation to find the radius of the sphere with the desired volume. Then use the area formula to verify the area is smaller than for your rectangular prism. (A sphere has the smallest surface area for a given volume of any figure.)

___

For your pyramid, it's a bit more complicated. You can use the formula for volume to find a relationship between the side length and the height. Then you can solve for height and put that expression into the equation for surface area. Now, you have an equation for surface area as a function of side length, which you can use to choose side lengths that give surface area in the range you want. You may find a graphing utility to be helpful for this.

... volume of a pyramid = (1/3)b^2h . . . . . b is the edge length of the base; h is the verical height.

... area of a pyramid = b^2 + 2bs, where s is the slant height: s=√((b/2)^ +h^2)

___

The cylinder is solved essentially the same way the pyramid is solved, except the formulas are slightly different.

... volume of a cylinder = πr²h

... area of a cylinder = 2πr² + 2πrh

Numbers:

If I did it right, a pyramid with a base edge length of 4 or 5 inches (or somewhere between) should give a smaller surface area for the volume you want.

For the cylinder, the radius may be around 2 or 2.1 inches.

5 0

3 years ago

What is the current density of the solution a/m2

Jlenok [28]

Current density of solution a/m2 is given below

Step-by-step explanation:

Current density is the measure of current generated per unit surface area of electrode.

Eg: In microbial fuel cell/fuel cells, if you are using electrode with size of 5cm x 2cm and current produced is 200mA, then current density will be,

CD= Current/(projected area of electrode)

area of rectangle= LxB= 5cm x 2cm =10cm2

CD= 200mA/10cm2

CD= 20mA/cm2

Your Current will be 200mA.

Current density will be 20mA/cm2.

8 0

3 years ago

A telephone pole that is 12 ft. tall has fallen against a house. If the top of the telephone pole touches the house 10 ft. above

Ostrovityanka [42]

Answer:

56.44 degrees

Step-by-step explanation:

sin(measure of the angle)=10/12

sin^-1(sin(measure of the angle))=sin^-1(10/12)

measure of the angle=56.44

4 0

3 years ago

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