Problem 58 - Applied Statistics and Probability for Engineers

Question

The volume of a shampoo filled into a container is uniformly distributed between 374 and 380 milliliters. (a) What are the mean and standard deviation of the volume of shampoo? (b) What is the probability that the container is filled with less than the advertised target of 375 milliliters? (c) What is the volume of shampoo that is exceeded by \(95 \%\) of the containers? (d) Every milliliter of shampoo costs the producer \$0.002. Any shampoo more than 375 milliliters in the container is an extra cost to the producer. What is the mean extra cost?

Step-by-Step Solution

Answer

The mean is 377 ml and the standard deviation is 1.732 ml. Probability of less than 375 ml is \( \frac{1}{6} \). Volume exceeded by 95% of containers is 374.3 ml. Mean extra cost is $0.005.

Step 1: Identify the Distribution

The volume of shampoo is uniformly distributed between 374 ml and 380 ml. A uniform distribution means all values are equally likely within this range.

Step 2: Calculate the Mean

The mean of a uniform distribution from a to b is given by \( \frac{a+b}{2} \). Here, \( a = 374 \) and \( b = 380 \). Therefore, the mean is \( \frac{374 + 380}{2} = 377 \) milliliters.

Step 3: Calculate the Standard Deviation

The standard deviation of a uniform distribution from a to b is given by \( \frac{b-a}{\sqrt{12}} \). Therefore, the standard deviation is \( \frac{380 - 374}{\sqrt{12}} \approx 1.732 \) milliliters.

Step 4: Probability of Less than 375 ml

The probability that the container has less than 375 ml is calculated by finding the proportion in the range (374 to 375) out of the total range (374 to 380). This is \( \frac{375 - 374}{380 - 374} = \frac{1}{6} \).

Step 5: Volume Exceeded by 95% of Containers

In a uniform distribution, the k-th percentile is given by \( a + (b-a) \times k \). For the 5th percentile (since 95% exceed), it's \( 374 + (380-374) \times 0.05 = 374.3 \) milliliters.

Step 6: Mean Extra Cost for Volumes Above 375 ml

First, find the expected amount above 375 ml. The expected value in a range (c to b) in uniform distribution (a to b) is \( \frac{c+b}{2} \), so here it is \( \frac{375+380}{2} = 377.5 \). The mean extra volume is \( 377.5 - 375 = 2.5 \) ml. The extra cost is \( 2.5 \times 0.002 = 0.005 \) dollars per container.

Key Concepts

Mean and Standard Deviation

For a uniform distribution, calculating the mean and standard deviation is quite straightforward. Imagine a long, flat plateau where every point is equally likely. That’s essentially what you’re visualizing with uniform distribution. The mean of a uniform distribution gives you the average amount filled in the containers, serving as a central point. It is calculated using the formula: \[ \text{Mean} = \frac{a+b}{2} \] where \(a\) and \(b\) are the endpoints of the distribution. So for our example, the volume of shampoo filled is uniformly distributed between 374 and 380 milliliters. By plugging in these values, we get a mean of 377 milliliters.
The standard deviation, on the other hand, tells us how much the values deviate from the mean - a measure of spread. In the uniform distribution, this is calculated as: \[ \text{Standard Deviation} = \frac{b-a}{\sqrt{12}} \]Plugging our endpoint values into this formula gives us a standard deviation of approximately 1.732 milliliters. A smaller standard deviation in this context indicates that most of the volumes are close to the mean, 377 milliliters.

Probability Calculation

Probability helps us determine the likelihood of a particular event occurring within our uniform distribution. In this case, we want to know the probability that a shampoo container holds less than 375 milliliters.
For a uniform distribution, this probability is the ratio of the length of the segment representing the interest interval (from start value of 374 to 375) to the total length of the distribution range (from 374 to 380).
The calculation thus is: \[ P(X < 375) = \frac{375 - 374}{380 - 374} = \frac{1}{6} \] This means there's a one-sixth chance, or approximately 16.67%, that any given container is filled with less than 375 milliliters of shampoo.

Percentile Calculation

Percentiles are used to understand how a particular value compares with the rest of the data set. To find a value that is exceeded by 95% of the observations, we need to find the 5th percentile in our context (since 95% exceed means we look for the lowest 5%).
The formula for finding the k-th percentile in a uniform distribution is given by:\[ P_k = a + (b-a) \times k \]For our distribution, calculating for the 5th percentile involves:\[ P_{5\%} = 374 + (380 - 374) \times 0.05 = 374.3 \] What this tells us is that 95% of the containers actually contain more than 374.3 milliliters of shampoo.

Cost Analysis

Cost analysis becomes crucial when the producer incurs additional costs for filling beyond a certain volume. In this problem, any shampoo quantity filled over 375 milliliters is considered an extra cost.
For uniform distribution, to calculate the mean extra volume, you find the average of volumes between 375 ml and the upper range. Here, the expected volume is: \[ \text{Expected Volume} = \frac{375+380}{2} = 377.5 \]The extra volume per container is this expected volume minus 375 ml, which is \[ 377.5 - 375 = 2.5 \text{ ml} \]The cost of such extra volume (since the cost per milliliter is \$0.002) is then calculated as:\[ \text{Extra Cost} = 2.5 \times 0.002 = 0.005 \text{ dollars per container} \]This means that on average, each container incurs an extra cost of half a cent due to overfilling.