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When dealing with a uniform distribution, calculating the mean and standard deviation is straightforward and helps us understand the data's central tendency and spread.

The **mean** of a uniform distribution is like finding the average value of our range. Specifically, for the given problem, the uniform distribution is characterized by having lower bound (minimum time) of 0.5 minutes and an upper bound (maximum time) of 10 minutes.

The formula for finding the mean \( \mu \) of a uniform distribution can be expressed as:

• \( \mu = \frac{a+b}{2} \)

In this case:

• \( \mu = \frac{0.5+10}{2} = 5.25 \) minutes

So, on average, it takes about 5.25 minutes to resolve a customer's problem.

The **standard deviation** informs us about the expected deviation from the mean. In simple terms, it tells us how "spread out" the values are from the mean. Here’s the formula:

• \( \sigma = \frac{b-a}{\sqrt{12}} \)

Using our example values:

• \( \sigma = \frac{10 - 0.5}{\sqrt{12}} \approx 2.735 \) minutes

This result tells us how much variation from the average problem-solving time is usual. A lower standard deviation would indicate times are closer to the mean, while a higher standard deviation indicates more variability.

Calculating probabilities with a uniform distribution can be intuitive due to its rectangular shape. Each outcome within the interval is equally likely.

In part c of the original exercise, we need to determine the percentage of problems that take more than 5 minutes to resolve. Probability calculations for a uniform distribution can be visualized as finding the relative area under the curve.

For values greater than 5 minutes, we find this by the formula:

• \( P(X > x) = \frac{b-x}{b-a} \)

Where \(x\) is 5 minutes. Plugging in the known values:

• \( P(X > 5) = \frac{10 - 5}{10 - 0.5} = \frac{5}{9.5} \approx 0.526 \)

This calculation shows that approximately 52.6% of problem-solving times are over 5 minutes.

Understanding this result helps in making probability-based decisions, such as planning resources for customer support. Knowing more than half of the calls last over 5 minutes could influence how many technicians are needed at any given time.

Uniform distribution gives a simple model for the problem-solving time, making calculations and predictions easier.

In the exercise, we’re interested in finding the middle 50% of problem-solving times—essentially the interquartile range (IQR) specific to a uniform distribution.

To determine this, we find when the cumulative probability is 25% (\(P(X < x_1) = 0.25\)) and 75% (\(P(X < x_2) = 0.75\)). These calculations ensure capturing the central half of the distribution:

• \( x_1 = a + 0.25(b-a) \)
• \( x_1 = 0.5 + 0.25(10 - 0.5) = 2.875 \)
• \( x_2 = a + 0.75(b-a) \)
• \( x_2 = 0.5 + 0.75(10 - 0.5) = 7.625 \)

Therefore, the middle 50% of problem-solving times ranges between 2.875 and 7.625 minutes.

This information is incredibly useful in practice. By understanding this range, businesses can better anticipate and manage typical support service interactions, aiming to maintain service quality and efficiency.