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Exercises Probability and the bell curve

Example 7.10 from Statistics, 7.3 Using the Central Limit Theorem

Using the Central Limit Theorem: example 7.10

Find the median, the first quartile, and the third quartile for the sample mean time of diagnosis in the United States.
Find the median, the first quartile, and the third quartile for the sum of sample times of diagnosis in the United States.
Find the probability that a diagnosis occurs on average between 1.75 and 1.85 minutes.
Find the value that is two standard deviations above the sample mean.
Find the IQR for the sum of the sample times.

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.

1
Solution
1. We have μ_x = μ = 2 and σ_x = σn = 0.510 = 0.05. Therefore,
  1. 50^th percentile = μ_x = μ = 2,
  2. 25^th percentile = invNorm(0.25,2,0.05) = 1.97, and
  3. 75^th percentile = invNorm(0.75,2,0.05) = 2.03.
2. We have μ_Σx = n(μ_x) = 100(2) = 200 and σ_μx = n(σ_x) = 10(0.5) = 5. Therefore,
  1. 50^th percentile = μ_Σx = n(μ_x) = 100(2) = 200,
  2. 25^th percentile = invNorm(0.25,200,5) = 196.63, and
  3. 75^th percentile = invNorm(0.75,200,5) = 203.37.
3. P(1.75 < $\bar{x}$ < 1.85) = normalcdf(1.75,1.85,2,0.05) = 0.0013
4. Using the z-score equation, $z = \frac{\bar{x} - μ_{\bar{x}}}{σ_{\bar{x}}}$ , and solving for x, we get x = 2(0.05) + 2 = 2.1.
5. The IQR is 75^th percentile – 25^th percentile = 203.37 – 196.63 = 6.74.

How did it go?

PreviousUsing the clt to find percentiles (part 2)Next Using the Central Limit Theorem: example 7.11

From Statistics (High school edition) by OpenStax (Rice University), section 7.3, 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.