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Box plots and the standard normal distribution. What
Chapter 4, Problem 116E(choose chapter or problem)
Problem 116E
Box plots and the standard normal distribution. What relationship exists between the standard normal distribution and the box-plot methodology (Section 2.8) for describing distributions of data using quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal.
a. Calculate the values of the standard normal random variable z, call them zL and zU, that correspond to the hinges of the box plot—that is, the lower and upper quartiles, QL and QU—of the probability distribution.
b. Calculate the z values that correspond to the inner fences of the box plot for a normal probability distribution.
c. Calculate the z values that correspond to the outerfences of the box plot for a normal probability distribution.
d. What is the probability that an observation lies beyond the inner fences of a normal probability distribution? The outer fences?
e. Can you better understand why the inner and outer fences of a box plot are used to detect outliers in a distribution? Explain.
Questions & Answers
QUESTION:
Problem 116E
Box plots and the standard normal distribution. What relationship exists between the standard normal distribution and the box-plot methodology (Section 2.8) for describing distributions of data using quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal.
a. Calculate the values of the standard normal random variable z, call them zL and zU, that correspond to the hinges of the box plot—that is, the lower and upper quartiles, QL and QU—of the probability distribution.
b. Calculate the z values that correspond to the inner fences of the box plot for a normal probability distribution.
c. Calculate the z values that correspond to the outerfences of the box plot for a normal probability distribution.
d. What is the probability that an observation lies beyond the inner fences of a normal probability distribution? The outer fences?
e. Can you better understand why the inner and outer fences of a box plot are used to detect outliers in a distribution? Explain.
ANSWER:
Solution
Step 1 of 5
a) We have to find the probabilities for lower quartile and upper quartile
Let z is a standard normal random variable
QL =zL is the value which divides the standard normal distribution data into 2 parts 25% of the data on the left and 75% of the data on the right
Then
Look at the standard normal tables for the value 0.25
We will get at -0.67
QU =zU is the value which divides the standard normal distribution data into 2 parts 75% of the data on the left and 25% of the data on the right
Then
Look at the standard normal tables for the value 0.75
We will get at 0.67