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Exercises Probability and the bell curve

Example 5.8 from Statistics, 5.3 The Exponential Distribution (Optional)

(part 2)

b. Half of all customers are finished within how long? (Find the 50^th percentile).

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
b. Find the 50^th percentile.

Figure 5.24

P(x < k) = 0.50, k = 2.8 minutes (calculator or computer)

Half of all customers are finished within 2.8 minutes.

You can also do the calculation as follows:

$P (x < k) = 0.50 a n d P (x < k) = 1 – e^{- 0.25 k}$

Therefore, 0.50 = 1 − e^−0.25k and e^−0.25k = 1 − 0.50 = 0.5.

Take natural logs: ln(e^–0.25k) = ln(0.50). So, –0.25k = ln(0.50).
Solve for k: $k = \frac{l n (0.50)}{- 0.25} = 2.8$ minutes. The calculator simplifies the calculation for percentile k. See the following two notes.
Note

A formula for the percentile k is $k = \frac{l n (1 - A r e a T o T h e L e f t)}{- m}$ where ln is the natural log.

Using the TI-83, 83+, 84, 84+ Calculator

On the home screen, enter ln(1 – 0.50)/–0.25. Press the (–) for the negative.

How did it go?

Previous (part 1)Next (part 3)

From Statistics (High school edition) by OpenStax (Rice University), section 5.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.