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Answer:
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.
[tex]P(X \geq x) = 1 - \frac{x - a}{b-a}[/tex]
The probability that we find a value X between c and d is given by the following formula.
[tex]P(c \leq X \leq d) = \frac{d - c}{b-a}[/tex]
The mean of the uniform distribution is:
[tex]M = \frac{a + b}{2}[/tex]
The standard deviation of the uniform distribution is:
[tex]S = \sqrt{\frac{(b-a)^{2}}{12}}[/tex]
Anywhere between 6.5 and 7.5 ounces.
This means that [tex]a = 6.5, b = 7.5[/tex]
(a) Find the mean and standard deviation of x.
Mean
[tex]M = \frac{6.5 + 7.5}{2} = 7[/tex]
Standard deviation
[tex]S = \sqrt{\frac{{7.5-6.5)^{2}}{12}} = 0.29[/tex]
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.
[tex]P(X \geq 7) = 1 - \frac{7 - 6.5}{7.5 - 6.5} = 0.5[/tex]
50% probability that x is at least 7 ounces.
(c) Find the probability that x is between 6.5 and 7.25 ounces.
[tex]P(6.5 \leq X \leq 7.25) = \frac{7.25 - 6.5}{7.5 - 6.5} = 0.75[/tex]
75% 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.
[tex]P(X\geq x) = 1-\frac{x-a}{b-a} \\[/tex]
The probability that a value X between c and d is given by the following formula.
[tex]P(c\leq x\leq d) = \frac{d-c}{b-a}[/tex]
The mean of uniform distribution is given by,
[tex]M = \frac{a+b}{2}[/tex]
And The standard deviation of uniform distribution is given by,
[tex]s = \sqrt{\frac{(b-a)^{2} }{12} }[/tex]
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,
[tex]M = \frac{6.5+7.5}{2} \\\\M = \frac{14}{2} \\\\M = 7[/tex]
The value of mean is 7.
Standard deviation is given by,
[tex]s = \sqrt{\frac{(7.5-6.5)^{2} }{12} }\\\\s = \sqrt{\frac{(1.5)^{2}} {12}[/tex]
[tex]s = \sqrt{ \frac{2.25}{12} }\\\\s = \sqrt{0.18} \\\\s = 0.424[/tex]
The standard deviation is of 0.29 ounces.
The probability that x is at least 7 ounces.
[tex]P(x\geq 1) = 1 - \frac{7-6.5}{7.5-6.5} \\\\p(x\geq 1) = 1- \frac{0.5}{1.5} \\\\P(x\geq 1) = \frac{1.5-0.5}{1.5} \\\\ P(x\geq 1) = 0.5[/tex]
50% probability that x is at least 7 ounces.
- The probability that x is between 6.5 and 7.25 ounces,
[tex]P(6.5\leq X\leq 7.5) = \frac{7.25-6.5}{7.5-6.5} = \frac{0.75}{1} = 0.75[/tex]
75% probability that x is between 6.5 and 7.25 ounces.
For more information about Probability click the link given below.
https://brainly.com/question/15802566
