𝐶ℎ. 10.4 𝑁𝑜𝑟𝑚𝑎𝑙 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑠 𝑎𝑛𝑑 𝐵𝑜𝑥𝑝𝑙𝑜𝑡𝑠
µ (Mu) to denote the mean,
σ ( sigma) to denote the standard deviation of a normal distribution.
 A small σ leads to a tall, narrow curve, because most of the data is close to
the mean.
 A large σ means the data is very spread out, producing a flat, wide curve.
 The area of the shaded region under the normal curve from a to b is
the probability that an observed data value will be between a and b.
 This area have already been calculate for the normal curve with
mean 0 and standard deviation of 1 in the table 2 at the back of the
textbook.(call standard normal curve) Also the equation of the
standard normal curve is
Ex1. Find the given areas under the standard normal curve. (Use table 2)
a) The area between Z=0 and Z=1, the shaded region.
b) The area between Z= -2.43 and Z=0
Ex2. Find the percent of the total area for the given areas under the standard
normal curve. (Use table 2)
a) The area between 0.88 standard deviations below the mean and 2.35
standard deviations above the mean ( that is, between Z=-0.88 and
Z=2.35)
b) The area between 0.58 standard deviations above the mean and 1.94
standard deviation abouve the mean
c) The area to the right of 2.09 standard deviations above the mean
The Z-Score for x is the number of standard deviations that x lies
from the mean. (Positive if x is above the mean, negative if x is below the
mean)
Ex3. If a normal distribution has mean 60 and standard deviation5, find
the given z-scores.
a) The z-score for x=65
b) The z-score for x=52.5
Ex4. Dixie Office Supplies finds that its sales force drives anaverage of
1200miles per month per person, with a standard deviation of 150 miles.
Assume that the number of miles driven by a salesperson is closely
approximated by a normal distribution.
a) Find the probability that a salesperson drives between 1200 and
1600 miles per month.
b) Find the probability that a salesperson drives between 1000 and
1500 miles per month.
Ex5. With data from the 2005-2006 National Health and Nutritional
Examination Survey, we can use 196 (mg/dL) as an estimate of the mean
total cholesterol level for all Americans and 44 (mg/dL) as an estimate of
the standard deviation. Assuming total cholesterol levels to be normally
distributed, what is the probability that an American chosen at random
has a cholesterol level higher than 250?
If 200 Americans are chosen at random, how many can we expect to have
total cholesterol higher than 250?
Boxplots
Where Q1 is called the first quartile, the medianQ2 is called the second
quartile, and Q3 is called the third quartile. The minimum, Q1, Q2, Q3,
and the maximum are often called the five number summary, and they
are used to construct a boxplot.
Ex6. The following table gives the revenues (in billions of U.S. dollars) for
Apple and Microsoft Corporations for the given years
a) Construct a boxplot for the Apple revenue data.
b) Construct a boxplot for the Microsoft revenue data.