REVIEWING NORMAL
DISTRIBUTIONS AND
Z-SCORES
You will need:
• White board & marker
• Calculator
• Z-score Table
• FFM Sheet (eventually)
from Math III
REMINDER:
 The Standard Normal Distribution is a normal distribution with
a mean of 0 and a standard deviation of 1 . To more easily
compute the probability of a particular observation given a
normally distributed variable, we can transform any normal
distribution to this standard normal distribution using the
following formula:
𝑋−𝜇
𝑧=
𝜎
 𝑋 = 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 ;
𝜇 = 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 ; =
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
 When you find this value for a given value, it is referred to as
the z-score.
 The z-score is a standard score for a data value that indicates
the number of standard deviations that the data value is away
from its respective mean.
1.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 20 𝑎𝑛𝑑 𝜎 = 3,
a.) What data value corresponds to a
z-score of 1.5?
1.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 20 𝑎𝑛𝑑 𝜎 = 3,
b.) What data value corresponds to a
z-score of −2.5?
2.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Calculate the z-score for:
a.) X = 65
2.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Calculate the z-score for:
b.) X = 90
2.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Calculate the z-score for:
c.) X = 82.5
3.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Determine the X value which corresponds to the
following z-scores:
a.) Z = −𝟐. 𝟓
HINT: You will need to work backwards to find X…plug values
into z-score formula and solve for x.
3.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Determine the X value which corresponds to the
following z-scores:
b.) Z = 0.75
HINT: You will need to work backwards to find X…plug values
into z-score formula and solve for x.
3.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Determine the X value which corresponds to the
following z-scores:
c.) Z = 1.9
HINT: You will need to work backwards to find X…plug values
into z-score formula and solve for x.
4.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Using the empirical rule,
a.) what percent of the data falls between 𝒛 =
− 𝟏 and 𝒛 = 𝟏 ?
b.) what x-values would correspond to (a) to
contain that same percentage of data?
4.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Using the empirical rule,
c.) what percent of the data falls between 𝒛 =
− 𝟐 and 𝒛 = 𝟐 ?
d.) what x-values would correspond to (c) to
contain that same percentage of data?
4.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Using the empirical rule,
e.) What percent of the data falls to
the LEFT of 𝒛 = 𝟎?
4.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Using the empirical rule,
f.) What percent of the data falls to
the LEFT of 𝒛 = 𝟑?
4.) GIVEN A NORMAL DISTRIBUTION WITH
A 𝜇 = 75 𝑎𝑛𝑑 𝜎 = 5,
Using the empirical rule,
g.) What percent of the data falls to
the RIGHT of 𝒛 = 𝟐?
FFM 8/16/13:
In a normal distribution with
𝝁 = 𝟒𝟎 and 𝝈 = 𝟓,
1.) find X if 𝒛 = −𝟎. 𝟒
2.) find P(𝒛 < −𝟎. 𝟒) = _____
3.) P (𝒙 > 48.2)
A.M. III
HOMEWORK
Section 9.2, pg. 588-590
# 9.20
#9.22
#9.24
#9.28