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The Normal Distribution

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Title: The Normal Distribution


1
Math 1107 Introduction to Statistics
  • Lecture 11
  • The Normal Distribution

2
Math 1107 The Normal Distribution
Drawing Conclusions from Representative
Data Making Decisions Looking for Relationships
Analyzing Specific Data Looking for
Outliers Looking for Relationships
  • Descriptive Statistics
  • Visualization, Summarization, Outliers
  • Categorical Data Analysis
  • Inferential Statistics
  • Sampling Central Limit Theorem
  • Confidence Intervals, Hypothesis Testing,
    Regression, ANOVA, etc.

3
Math 1107 The Normal Distribution
  • There are many types of distributions
  • Binomial 2 outcomes (success or failureH or
    T)
  • Poisson Infinite possibilities, with discrete
    occurrences
  • Normal Bell Shaped continuous distribution

4
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 s .
  • 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

5
Math 1107 The Normal Distribution
6
Math 1107 The Normal Distribution
0
1
2
3
-1
-2
-3
7
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

8
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.
  • Andmost importantlywe can use Z-scores to
    determine the associated probability of an
    outcome.

9
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.
10
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.
11
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?
X1400 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.
12
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?
X800 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.
13
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.
14
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?
X11.8 ounces Mu 12 Std dev .2 So,
Z(11.8-12)/.2 -1. A z-score of -1 equals
.3413.
15
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?
X11.8 ounces Mu 12 Std dev .8 So,
Z(11.8-12)/.8 -.25. A z-score of -.25 equals
.0987, or 9.87
16
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.
17
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 2.0514.4
x-14.414.4 16.45 x
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