Customers experiencing technical difficulty with their...

Customers experiencing technical difficulty with their Internet cable service may call an 800 number for technical support. It takes the technician between 30 seconds and 10 minutes to resolve the problem. The distribution of this support time follows the uniform distribution. a. What are the values for $a$ and $b$ in minutes? b. What is the mean time to resolve the problem? What is the standard deviation of the time? c. What percent of the problems take more than 5 minutes to resolve? d. Suppose we wish to find the middle $50 \%$ of the problem-solving times. What are the end points of these two times?

Customers experiencing technical difficulty with their Internet cable service may call an 800 number for technical support. It takes the technician between 30 seconds and 10 minutes to resolve the problem. The distribution of this support time follows the uniform distribution.
a. What are the values for $a$ and $b$ in minutes?
b. What is the mean time to resolve the problem? What is the standard deviation of the time?
c. What percent of the problems take more than 5 minutes to resolve?
d. Suppose we wish to find the middle $50 \%$ of the problem-solving times. What are the end points of these two times?

Key Concepts

Uniform Distribution

The uniform distribution is a type of continuous probability distribution in which all outcomes within a fixed interval are equally likely to occur. This means that the probability density function is constant over the interval between its lower bound (a) and upper bound (b), and zero elsewhere.

Parameters of the Uniform Distribution (a and b)

In the context of a uniform distribution, the parameters a and b represent the minimum and maximum possible values, respectively, that the random variable can take. These parameters define the support of the distribution, the interval over which the probability is uniformly distributed, and are used to calculate key statistical properties such as probabilities, the mean, and the variance.

Mean and Standard Deviation of a Uniform Distribution

The mean (or expected value) of a uniform distribution is calculated as (a + b)/2, reflecting the symmetric nature of the distribution about its center. The standard deviation, which measures the spread or dispersion of the distribution, is found using the formula (b - a)/?12. These formulas rely directly on the parameters a and b and indicate that as the range increases, both the mean and the standard deviation are affected proportionally.

Percentile Calculation in Continuous Distributions

Percentiles in a continuous distribution like the uniform distribution are determined by finding the value below which a given percentage of the data falls. Because the probability density is constant in a uniform distribution, calculating percentiles involves a straightforward linear transformation of the interval endpoints, enabling the determination of probabilities such as the proportion of times the variable exceeds a specific threshold.

Middle 50% Range (Interquartile Range)

The middle 50% range, often determined by the 25th and 75th percentiles (also known as the lower and upper quartiles), is a measure of the central spread of the data. In a uniform distribution, these quantiles are found by taking 25% and 75% of the distance from the lower bound to the upper bound and adding this to the lower bound. This interval provides insight into the variability of the central half of the data, effectively capturing the interquartile range.

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