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When dealing with a uniform distribution, calculating the mean and standard deviation helps us understand the spread and central tendency of data points. Here, the volume of shampoo is uniformly distributed between 374 and 380 milliliters. This means that every volume between these limits is equally likely.
The mean is essentially the average volume expected in a container. The formula for the mean of a uniform distribution is given by \( \mu = \frac{a + b}{2} \) where \( a \) and \( b \) are the lower and upper limits respectively. Substituting the values, \( a = 374 \) and \( b = 380 \), we find the mean volume is 377 mL.
The standard deviation provides insight into the variation from the mean, calculated using \( \sigma = \frac{b - a}{\sqrt{12}} \). For our values, standard deviation comes out to approximately 1.73 mL. Lower standard deviation means that most values are close to the mean, indicating consistent filling of the containers.

Calculating probabilities within a uniform distribution is straightforward due to its constant probability density between two limits. Consider the probability of a container being filled with less than 375 mL.
For a uniform distribution, the probability that the volume is less than a specific value \( x \) is calculated using \( P(X < x) = \frac{x - a}{b - a} \). For this task \( x = 375 \), giving \( P(X < 375) = \frac{375 - 374}{380 - 374} \). This results in a probability of approximately 0.167, meaning that around 16.7% of the containers will be filled with less than 375 mL.
Using these probability calculations helps manufacturers to forecast and plan for scenarios where production may fall short of target volumes.

Exceedance volume is a term used to define a specific value that is surpassed by a set percentage of observations. In our case, we need to find the volume that is exceeded by 95% of containers or is equaled/exceeded by the remaining 5%.
To find this value, solve for \( x \) using the formula: \( P(X < x) = 0.95 \). Substituting in our uniform distribution: \( \frac{x - 374}{6} = 0.95 \). Solving gives us \( x = 379.7 \text{ mL} \).
This tells us that 95% of all shampoo container fillings will exceed this value, indicating a very high threshold of consistency in the product's volume for most containers.

Understanding the cost implications of filling a container with more than a specified amount can be crucial for a producer. Here, any shampoo volume exceeding 375 mL incurs additional costs at \$0.002 per milliliter.
The calculation of mean extra cost involves finding the mean of volumes exceeding 375 mL. Using the mean of the uniform values greater than 375 mL, we compute \( E(X | X > 375) = \frac{380 + 375}{2} = 377.5 \text{ mL} \).
The extra volume above 375 mL, on average, is 2.5 mL leading to a mean extra cost per container of \(0.002 \times 2.5 = 0.005\) dollars. Knowing this assists in refining production processes to manage costs effectively while maintaining product quality.