The Normal Model
Ch. 6
“All models are wrong –
but some are useful.”
-- George Box
Approximating with the Normal model
Nearly Normal Condition
• The shape of the data’s distribution is roughly
unimodal and symmetric.
• Check this by making a histogram (or Normal
probability plot)
• YOU MUST CHECK THE NEARLY NORMAL
CONDITION!!!
Statistics vs. Parameters
Statistics
• Summaries of actual data
Parameters
• Values that specify the
model
• 𝜇 -- mean
•
-- standard deviation
• N(𝜇, )—Normal model
with mean 𝜇 and standard
deviation
Rescale the Data
• Standardize the data so that the:
– average is 0
– standard deviation is 1
– Convert the data to z-scores
• Each value is converted to a Standard Score (zscore)
Standard Score (z-score)
𝑦−𝜇
𝑧=
𝜎
The Normal Model
The Normal Model
3
-2
-1
+
1
+
2
+
3
The 68-95-99.7 Rule
a.k.a. The Empirical Rule
68%
95%
99.7%
3
-2
-1
+
1
+
2
+
3
Three Rules of Working with the
Normal model
1. Make a picture
2. Make a picture
3. Make a picture
The Setup
The distribution of fuel efficiency of a particular
vehicle is roughly unimodal and symmetric with
mean 24 mpg and standard deviation 6 mpg.
1.
2.
3.
4.
5.
6.
7.
8.
Check the Nearly Normal Condition
Sketch the Normal model
What percent of all cars get less than 15 mpg?
Describe the fuel efficiency of the worst 20% of
all cars.
What percent of all cars get between 20 mpg
and 30 mpg?
What percent of cars get more than 40 mpg?
What gas mileage represents the third quartile?
Describe the gas mileage of the most efficient
5% of all cars.