The manager of a local soft drink bottling company believes that a new beverage- dispensing machine, which is designed to dispense 7 ounces, in fact dispenses anywhere between 6.5 and 7.5 ounces.
If the amount dispensed is represented by a uniformly distributed random variable x:
(a) Find the mean and standard deviation of x.
(b) Find the probability that x is at least 7 ounces.
(c) Find the probability that x is between 6.5 and 7.25 ounces.

The mean is 7 ounces and the standard deviation is of 0.29 ounces.

50% probability that x is at least 7 ounces.

75% probability that x is between 6.5 and 7.25 ounces.

Given that,

New beverage- dispensing machine, which is designed to dispense 7 ounces, in fact dispenses anywhere between 6.5 and 7.5 ounces.

We have to determine,

The amount dispensed is represented by a uniformly distributed random variable x.

According to the question,

The continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions.

In a uniform distribution the probability of each event is exactly the same.

For this situation, A lower limit of the distribution denoted as a that an upper limit denoted as b that ,

The probability that value X at least x is given by the following formula.

The probability that a value X between c and d is given by the following formula.

The mean of uniform distribution is given by,

And The standard deviation of uniform distribution is given by,

Anywhere between 6.5 and 7.5 ounces .

Then, a = 6.5 and b = 7.5

Mean and standard deviation of x is given by,

The value of mean is 7.

Standard deviation is given by,

The standard deviation is of 0.29 ounces.

The probability that x is at least 7 ounces.

50% probability that x is at least 7 ounces.

75% probability that x is between 6.5 and 7.25 ounces.

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a) The mean is 7 ounces and the standard deviation is of 0.29 ounces.

b) 50% probability that x is at least 7 ounces.

c) 75% probability that x is between 6.5 and 7.25 ounces.

Step-by-step explanation:

An uniform probability is a case of probability in which each outcome is equally as likely.

For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.

The probability that we find a value X at least x is given by the following formula.

The probability that we find a value X between c and d is given by the following formula.

The mean of the uniform distribution is:

The standard deviation of the uniform distribution is:

Anywhere between 6.5 and 7.5 ounces.

This means that

(a) Find the mean and standard deviation of x.

Mean

Standard deviation

The mean is 7 ounces and the standard deviation is of 0.29 ounces.

(b) Find the probability that x is at least 7 ounces.

50% probability that x is at least 7 ounces.

(c) Find the probability that x is between 6.5 and 7.25 ounces.

75% probability that x is between 6.5 and 7.25 ounces.

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step-by-step explanation:

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