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5.7 Practice 2: exponential distribution

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In this module the student will explore the properties of data with an exponential distribution.

Student learning outcomes

  • The student will analyze data following the exponential distribution.

Given

Carbon-14 is a radioactive element with a half-life of about 5730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121 . We start with 1 gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

Describe the data

What is being measured here?

Are the data discrete or continuous?

Continuous

In words, define the Random Variable X size 12{X} {} .

X size 12{X} {} = Time (years) to decay carbon-14

What is the decay rate ( m size 12{m} {} )?

m size 12{m} {} = 0.000121

The distribution for X size 12{X} {} is:

X size 12{X} {} ~ Exp(0.000121)

Probability

Find the amount (percent of 1 gram) of carbon-14 lasting less than 5730 years. This means, find P ( x < 5730 ) size 12{P \( X<"5730" \) } {} .

  • Sketch the graph. Shade the area of interest.
    Blank graph with vertical and horizontal axes.
  • Find the probability. P ( x < 5730 ) size 12{P \( X<"5730" \) } {} =
  • P ( x < 5730 ) size 12{P \( X<"5730" \) } {} = 0.5001

Find the percentage of carbon-14 lasting longer than 10,000 years.

  • Sketch the graph. Shade the area of interest.
    Blank graph with horizontal and vertical axes.
  • Find the probability. P ( x > 10000 ) size 12{P \( X<"5730" \) } {} =
  • P ( x > 10000 ) size 12{P \( X<"5730" \) } {} = 0.2982

Thirty percent (30%) of carbon-14 will decay within how many years?

  • Sketch the graph. Shade the area of interest.
    Blank graph with vertical and horizontal axes.
  • Find the value k size 12{k} {} such that P ( x < k ) = 0 . 30 size 12{P \( X<k \) =0 "." "30"} {} .
  • k size 12{k} {} = 2947.73
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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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