Math Problem Statement
Solution
Let's solve the two parts of the problem provided in the image.
Part (a): Draw a density curve
The problem states that the amount of time a customer waits on hold can be modeled by a uniform distribution on the interval seconds.
For a uniform distribution, the probability density function (PDF) is constant over the interval. The formula for the PDF of a uniform distribution is:
Here, and , so the density function becomes:
This means the density curve is a horizontal line from to , with a constant height of . On the graph:
- The x-axis represents the time (in seconds) from 0 to 120.
- The y-axis represents the probability density, which is .
Part (b): Percent of the time customer waits between 100 and 120 seconds
To find this percentage, we calculate the probability that the wait time is between 100 and 120 seconds for the uniform distribution . The probability for an interval in a uniform distribution is given by:
Substituting the values , , , and :
This means that about 16.67% of the time, a customer has to wait between 100 and 120 seconds.
Would you like a graph for the density curve or further clarifications?
Here are 5 related questions to consider:
- How would the density curve change if the maximum wait time increased to 150 seconds?
- What would the density function be if the wait time followed a non-uniform distribution?
- How would you calculate the probability of waiting less than 50 seconds?
- How do you interpret the height of the density curve in a uniform distribution?
- How would the probability change if the interval was between 80 and 100 seconds?
Tip: Uniform distributions are useful for modeling situations where all outcomes in a range are equally likely, such as random wait times within a known time limit.
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Math Problem Analysis
Mathematical Concepts
Probability
Uniform Distribution
Formulas
Probability Density Function (PDF) of Uniform Distribution: f(x) = 1/(b - a) for a ≤ x ≤ b
Probability of an Interval in Uniform Distribution: P(x_1 ≤ X ≤ x_2) = (x_2 - x_1)/(b - a)
Theorems
Properties of Uniform Distribution
Suitable Grade Level
Grades 11-12
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