Math 1107
Introduction to Statistics
Lecture 11
The Normal Distribution
Math 1107 – The Normal Distribution
Analyzing Specific Data Drawing Conclusions
Looking for Outliers from Representative Data
Looking for Relationships Making Decisions
Looking for Relationships
Descriptive Statistics Inferential Statistics
Visualization, Summarization, Sampling & Central Limit Theorem
Outliers
Confidence Intervals, Hypothesis
Categorical Data Analysis Testing, Regression, ANOVA, etc.
Math 1107 – The Normal Distribution
There are many types of distributions:
• Binomial – 2 outcomes (success or failure…H or T);
• Poisson – Infinite possibilities, with discrete
occurrences;
• Normal – Bell Shaped continuous distribution
Math 1107 – The Normal Distribution
A family of continuous random variables whose outcomes
range from minus infinity to plus infinity.
Bell shaped and symmetric about the mean μ.
Mean = μ, Median = μ, Mode = μ.
The standard deviation is σ .
The area under the normal curve below μ is .5.
• The area above μ is also .5.
Probability that a Normal Random Variable Outcome:
• Lies within +/- 1 std dev of the mean is .6826
• Lies within +/- 2 std dev of the mean is .9544
• Lies within +/- 3 std dev of the mean is .9974
Math 1107 – The Normal Distribution
Height for 1107
8
7
Frequency
6
5
4
3
2
1
0
58
60
62
64
66
68
70
72
74
e
or
M
Height in Inches
Math 1107 – The Normal Distribution
68%
95%
99%
-3 -2 -1 0 1 2 3
Math 1107 – The Normal Distribution
The Standard Normal Distribution looks like a
Normal Distribution, but has important statistical
properties:
• mean = 0
• std dev = 1
Remember from earlier in the semester that:
• The Std Normal Distribution enables the calculation of Z-
scores
• Z-Scores can be compared against ANY populations using any
scale
Math 1107 – The Normal Distribution
Remember from earlier in the semester that:
• The Std Normal Distribution enables the calculation
of Z-scores;
• Z-Scores can be compared against ANY populations
using any scale;
•Z-scores are stated in units of standard deviations;
• So, typical Z-scores will range from 0 (the mean) to
3 and can be negative or positive.
And…most importantly…we can use Z-scores to
determine the associated probability of an outcome.
Math 1107 – The Normal Distribution
How do we use a z-score to find a probability?
Z=(x-mu)/std dev
Where,
X is a value of interest from the distribution;
Mu = the average of the distribution;
Std dev = the std dev of the distribution.
Math 1107 – The Normal Distribution
Prior to solving any Normal Distribution problem using
Z-scores, ALWAYS draw a sketch of what you are
doing. This will provide you with a guide for what is a
“reasonable” answer.
Math 1107 – The Normal Distribution
Example:
Watts Corporation makes lightbulbs with an average life of
1000 hours and a std dev of 200 hours. Assuming the life of
the bulbs is normally distributed, what is the probability of
buying a bulb at random that lasts for up to 1400 hours?
X=1400
Mu = 1000
Std dev = 200
So, Z=(1400-1000)/200 = 2.
A z-score of 2 equals .4772. We add .5 to this and get a
probability of .9772.
Math 1107 – The Normal Distribution
Example:
Unlucky Larry bought a Watts Corporation bulb and it only
lasted 800 hours. What is the probability that a bulb selected
at random would last between 800 and 1000 hours?
X=800
Mu = 1000
Std dev = 200
So, Z=(800-1000)/200 = -1.
A z-score of -1 equals .3413. So, there is a 34.13% chance of
selecting a bulb at random that generates between 800 and
1000 hours of light.
Math 1107 – The Normal Distribution
Example:
What is the probability of selecting a bulb at random that
generates less than 800 hours?
The total area under the curve less than the average is .50 or
50%. So, if we know the area between 800 and 1000 is
.3413, then the area less than 800 is .5-.3413 or .1587.
What is the probability of selecting a bulb at random that
generates more than 800 hours?
The total area under the curve more than the average is .50
or 50%. So, if we know the area between 800 and 1000 is
.3413, then the area less than 800 is .5+.3413 or .8413.
Math 1107 – The Normal Distribution
Example:
Coca Cola Bottlers produce millions of cans of coke a year.
The average can holds 12 ounces with a std dev of .2
ounces. What is the probability of getting a coke with
between 11.8 and 12 ounces?
X=11.8 ounces
Mu = 12
Std dev = .2
So, Z=(11.8-12)/.2 = -1.
A z-score of -1 equals .3413.
Math 1107 – The Normal Distribution
Example:
Coca Cola Bottlers produce millions of cans of coke a year.
The average can holds 12 ounces with a std dev of .8
ounces. What is the probability of getting a coke with
between 11.8 and 12 ounces?
X=11.8 ounces
Mu = 12
Std dev = .8
So, Z=(11.8-12)/.8 = -.25.
A z-score of -.25 equals .0987, or 9.87%
Math 1107 – The Normal Distribution
Example from Page 243:
Airlines have designed their seats to accommodate the hip width
of 98% of all males. Men have hip widths that are normally
distributed with a mean of 14.4 inches and a standard deviation
of 1.0. What is the minimum hip width that airlines cannot
accommodate? This is the 98th percentile.
Math 1107 – The Normal Distribution
In this example, we are working “backward”. We know the
Probability (98%) and we want to know the value that generates
this probability. Given the Z formula, we now solve for x.
Z=(x-mu)/std dev
2.05=(x-14.4)/1
2.05 = x-14.4
2.05+14.4 = x-14.4+14.4
16.45 = x