Customers experiencing technical difficulty with their...

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 30 seconds to 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 percent of the problem-solving times. What are the end points of these two times?

Customers experiencing technical difficulty with their Internet cable hookup may call an 800 number for technical support. It takes the technician between 30 seconds to 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 percent of the problem-solving times. What are the end points of these two times?

Key Concepts

Quantiles and Percentile Calculation

Quantiles and percentiles in a uniform distribution are determined using the linear cumulative distribution function (CDF). Since the outcomes are evenly distributed over the interval, calculating the value for a certain percentile or the endpoints for a central percentage of data (for example, the middle 50%) involves solving simple linear equations based on the desired probability thresholds.

Standard Deviation of a Uniform Distribution

The standard deviation of a uniform distribution quantifies the spread or dispersion of the outcomes around the mean. It is given by the formula (b - a) / ?12, which provides a measure of how much the values typically deviate from the central mean.

Mean of a Uniform Distribution

The mean or expected value of a uniform distribution over the interval [a, b] is calculated as (a + b) / 2. This value represents the central point of the distribution where, on average, the outcomes are centered.

Parameters of a Uniform Distribution

The uniform distribution is defined by its two parameters, usually denoted by a and b, which represent the lower and upper bounds of the interval, respectively. These parameters are critical as they set the range over which the random variable is spread in an equally likely manner.

Uniform Distribution

A uniform distribution is a type of probability distribution in which all outcomes are equally likely within a specified interval. This means that the probability of an event occurring in any subinterval of the total range is proportional to the length of that subinterval, making the distribution constant across the interval.

Transcript

00:01 So we have a uniform distribution, and we have our low value for the amount of time it takes to be half a minute, 30 seconds, half a minute.

00:12 And our upper part is 10 minutes.

00:15 And so if this is going to be a uniform distribution, we know the area underneath that has to be equal to 1.

00:21 So this is 9 .5, 9 .5, which is 19 halves.

00:28 And so this height of this distribution would have to be 219.

00:33 So if we want to find any areas, that's helpful to know what that frequency value is.

00:39 So on part a, we want to know what is a, that is half a minute, and what is b.

00:43 That is going to be 10 minutes because i don't want to mix 30 seconds with 10 minutes.

00:50 We either go all in seconds or we go all in minutes.

00:53 Part b, we want to find what the mean is, and the mean of that distribution is going to be to add point.

01:00 5 and 10 and then divided by 2.

01:03 And so 10 .5 divided by 2, we know it's going to be 5 .25 minutes.

01:10 So that's where that middle is, is 5 .25 minutes.

01:15 That would be the cutoff point for 50 % here, 50 % there.

01:19 And our standard deviation is going to be the difference between these two times.

01:23 So 9 .5 squared divided by 12.

01:27 Now that's the variance.

01:28 We square root it and we get the standard deviation.

01:30 So the square root of 0 .9, excuse me, 0 .95, square root of 9 .5 squared divided by 12...

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