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.
![P(X \geq x) = 1 - \frac{x - a}{b-a}](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%20x%29%20%3D%201%20-%20%5Cfrac%7Bx%20-%20a%7D%7Bb-a%7D)
The probability that we find a value X between c and d is given by the following formula.
![P(c \leq X \leq d) = \frac{d - c}{b-a}](https://tex.z-dn.net/?f=P%28c%20%5Cleq%20X%20%5Cleq%20d%29%20%3D%20%5Cfrac%7Bd%20-%20c%7D%7Bb-a%7D)
The mean of the uniform distribution is:
![M = \frac{a + b}{2}](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7Ba%20%2B%20b%7D%7B2%7D)
The standard deviation of the uniform distribution is:
![S = \sqrt{\frac{(b-a)^{2}}{12}}](https://tex.z-dn.net/?f=S%20%3D%20%5Csqrt%7B%5Cfrac%7B%28b-a%29%5E%7B2%7D%7D%7B12%7D%7D)
Anywhere between 6.5 and 7.5 ounces.
This means that ![a = 6.5, b = 7.5](https://tex.z-dn.net/?f=a%20%3D%206.5%2C%20b%20%3D%207.5)
(a) Find the mean and standard deviation of x.
Mean
![M = \frac{6.5 + 7.5}{2} = 7](https://tex.z-dn.net/?f=M%20%3D%20%5Cfrac%7B6.5%20%2B%207.5%7D%7B2%7D%20%3D%207)
Standard deviation
![S = \sqrt{\frac{{7.5-6.5)^{2}}{12}} = 0.29](https://tex.z-dn.net/?f=S%20%3D%20%5Csqrt%7B%5Cfrac%7B%7B7.5-6.5%29%5E%7B2%7D%7D%7B12%7D%7D%20%3D%200.29)
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.
![P(X \geq 7) = 1 - \frac{7 - 6.5}{7.5 - 6.5} = 0.5](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%207%29%20%3D%201%20-%20%5Cfrac%7B7%20-%206.5%7D%7B7.5%20-%206.5%7D%20%3D%200.5)
50% probability that x is at least 7 ounces.
(c) Find the probability that x is between 6.5 and 7.25 ounces.
![P(6.5 \leq X \leq 7.25) = \frac{7.25 - 6.5}{7.5 - 6.5} = 0.75](https://tex.z-dn.net/?f=P%286.5%20%5Cleq%20X%20%5Cleq%207.25%29%20%3D%20%5Cfrac%7B7.25%20-%206.5%7D%7B7.5%20-%206.5%7D%20%3D%200.75)
75% probability that x is between 6.5 and 7.25 ounces.