S1600 #8
The Normal Percentile
February 4, 2016
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Outline
1
The Normal Percentile
Percentile in N(0,1)
General Normal Percentile
2
Calculating symmetric tail areas
Calculating symmetric tail areas
3
Empirical Rule
Using Empirical Rule
(WMU)
S1600 #8
S1600, Lecture 8
2 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Revisiting TV+More Example
Question 2
Assumption: demand “follows” normal (bell-shaped) curve with mean
36 and SD 8 TV sets per month. Given the cost of running out of stock
and the storage cost of keeping too many TVs, the manager decides to
order enough TV sets to satisfy customer demands 90% of the time.
How many TV sets should they order?
(WMU)
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S1600, Lecture 8
3 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile
The (100p)th percentile for N(0, 1) can be found by
1
look up entries in the table and find the probability nearest p
2
and trace its z value
3
(optional) if two probabilities are found in (1), average their
respective z values
(WMU)
S1600 #8
S1600, Lecture 8
4 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile
The (100p)th percentile for N(0, 1) can be found by
1
look up entries in the table and find the probability nearest p
2
and trace its z value
3
(optional) if two probabilities are found in (1), average their
respective z values
(WMU)
S1600 #8
S1600, Lecture 8
4 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile
The (100p)th percentile for N(0, 1) can be found by
1
look up entries in the table and find the probability nearest p
2
and trace its z value
3
(optional) if two probabilities are found in (1), average their
respective z values
(WMU)
S1600 #8
S1600, Lecture 8
4 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile
The (100p)th percentile for N(0, 1) can be found by
1
look up entries in the table and find the probability nearest p
2
and trace its z value
3
(optional) if two probabilities are found in (1), average their
respective z values
(WMU)
S1600 #8
S1600, Lecture 8
4 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile, Examples
1
2
90th percentile, p = 0.9, the closest probability is 0.8997 when
z = 1.28, so 90th percentile is 1.28
95th percentile, p = 0.95, the two (equally) nearest probabilities
are 0.9495 when z = 1.64 and 0.9505 when z = 1.65, so 95th
percentile is 1.64+1.65
= 1.645
2
(WMU)
S1600 #8
S1600, Lecture 8
5 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile, Examples
1
2
90th percentile, p = 0.9, the closest probability is 0.8997 when
z = 1.28, so 90th percentile is 1.28
95th percentile, p = 0.95, the two (equally) nearest probabilities
are 0.9495 when z = 1.64 and 0.9505 when z = 1.65, so 95th
percentile is 1.64+1.65
= 1.645
2
(WMU)
S1600 #8
S1600, Lecture 8
5 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Standard Normal Percentile, Examples
1
2
90th percentile, p = 0.9, the closest probability is 0.8997 when
z = 1.28, so 90th percentile is 1.28
95th percentile, p = 0.95, the two (equally) nearest probabilities
are 0.9495 when z = 1.64 and 0.9505 when z = 1.65, so 95th
= 1.645
percentile is 1.64+1.65
2
(WMU)
S1600 #8
S1600, Lecture 8
5 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
General Normal Percentile
percentile for N(mean, SD) = mean + SD × percentile for N(0, 1)
(WMU)
S1600 #8
S1600, Lecture 8
6 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Answer to Question 2 in TV+More Example
In order to satisfy customer demands 90% of the time, the store
manager should order x(= 90th percentile) TV sets. So,
x = |{z}
36 + |{z}
8 ×1.28 = 46.24.
mean
SD
That is, at least 46 TV sets should be ordered.
(WMU)
S1600 #8
S1600, Lecture 8
7 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
iClicker Question 8.1
Suppose that cashiers have average weekly earnings of $380 with an
SD of $40. Assume that weekly earnings is approximately normally
distributed. A cashier who earns $380 a week, falls on what
percentile?
A. 6.68th percentile
B. 50th percentile
C. 93.32th percentile
D. 75th percentile
E. 77.34th percentile
(WMU)
S1600 #8
S1600, Lecture 8
8 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Outline
1
The Normal Percentile
Percentile in N(0,1)
General Normal Percentile
2
Calculating symmetric tail areas
Calculating symmetric tail areas
3
Empirical Rule
Using Empirical Rule
(WMU)
S1600 #8
S1600, Lecture 8
9 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Calculating Combined Area of Two Symmetric Tails
a normal variable falls within 1 SD of center 68% of the time,
hence outside of 1 SD only 32% of the time.
What about 2.25 SD’s?
In general
Hence, the combined area of two tails beyond ±2.25 SD’s for a
normal variable is
2 × (1 − 0.9878) = 0.0244.
(WMU)
S1600 #8
S1600, Lecture 8
10 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Calculating Combined Area of Two Symmetric Tails
a normal variable falls within 1 SD of center 68% of the time,
hence outside of 1 SD only 32% of the time.
What about 2.25 SD’s?
In general
Hence, the combined area of two tails beyond ±2.25 SD’s for a
normal variable is
2 × (1 − 0.9878) = 0.0244.
(WMU)
S1600 #8
S1600, Lecture 8
10 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Calculating Combined Area of Two Symmetric Tails
a normal variable falls within 1 SD of center 68% of the time,
hence outside of 1 SD only 32% of the time.
What about 2.25 SD’s?
In general
Hence, the combined area of two tails beyond ±2.25 SD’s for a
normal variable is
2 × (1 − 0.9878) = 0.0244.
(WMU)
S1600 #8
S1600, Lecture 8
10 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Calculating Combined Area of Two Symmetric Tails
a normal variable falls within 1 SD of center 68% of the time,
hence outside of 1 SD only 32% of the time.
What about 2.25 SD’s?
In general
Hence, the combined area of two tails beyond ±2.25 SD’s for a
normal variable is
2 × (1 − 0.9878) = 0.0244.
(WMU)
S1600 #8
S1600, Lecture 8
10 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Calculating Combined Area of Two Symmetric Tails
a normal variable falls within 1 SD of center 68% of the time,
hence outside of 1 SD only 32% of the time.
What about 2.25 SD’s?
In general
Hence, the combined area of two tails beyond ±2.25 SD’s for a
normal variable is
2 × (1 − 0.9878) = 0.0244.
(WMU)
S1600 #8
S1600, Lecture 8
10 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Outline
1
The Normal Percentile
Percentile in N(0,1)
General Normal Percentile
2
Calculating symmetric tail areas
Calculating symmetric tail areas
3
Empirical Rule
Using Empirical Rule
(WMU)
S1600 #8
S1600, Lecture 8
11 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Empirical Rule
If data histogram ≈ bell-shaped, you should expect the following:
68% of the observations will fall within 1 SD of the mean
95% of the observations will fall within 2 SD of the mean
99.7% of the observations will fall within 3 SD of the mean
(WMU)
S1600 #8
S1600, Lecture 8
12 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Empirical Rule
If data histogram ≈ bell-shaped, you should expect the following:
68% of the observations will fall within 1 SD of the mean
95% of the observations will fall within 2 SD of the mean
99.7% of the observations will fall within 3 SD of the mean
(WMU)
S1600 #8
S1600, Lecture 8
12 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Empirical Rule
If data histogram ≈ bell-shaped, you should expect the following:
68% of the observations will fall within 1 SD of the mean
95% of the observations will fall within 2 SD of the mean
99.7% of the observations will fall within 3 SD of the mean
(WMU)
S1600 #8
S1600, Lecture 8
12 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
Empirical Rule
0.05
TV+More Example revisited
0.03
0.02
µ ± 3σ
0.02
0.03
µ ± 2σ
0.00
0.01
density
0.04
0.050.00
0.01
density
0.04
µ±σ
20
30
40
50
60
20
# TVs sold
(WMU)
30
40
50
60
# TVs sold
S1600 #8
S1600, Lecture 8
13 / 14
The Normal Percentile
Calculating symmetric tail areas
Empirical Rule
iClicker Question 8.2
Suppose that cashiers have average weekly earnings of $380 with an
SD of $40. Assume that weekly earnings is approximately normally
distributed. Using only the empirical rule, what percent of cashiers
earn between $340 and $420 a week?
A. 68%
B. 95%
C. 99.7%
D. 18.94%
E. 50%
(WMU)
S1600 #8
S1600, Lecture 8
14 / 14