Transcript Lecture 5 - West Virginia University Department of Statistics

Lecture 7
Dan Piett
STAT 211-019
West Virginia University
Last Week
 Binomial Distributions
 2 Outcomes, n trials, probability of success = p,
 X = Number of Successes
 Poisson Distributions
 Occurrences are measured over some unit of time/space with
mean occurrences lambda
 X = Number of Occurrences
 Finding Probabilities
=
 < and ≤
 > and ≥
Overview
 Normal Distribution
 Empirical Rule
 Normal Probabilities
 Percentiles
Continuous Distributions
 Up until this point we have only talked about discrete
random variables.
 Binomial
 Poisson
 Note that in these distributions, X was a countable number.
 Number of successes, Number of occurrences.
 Now we will be looking at continuous distributions
 Ex: height, weight, marathon running time
Continuous Distributions Cont.
 Continuous Distributions are generally represented by a curve
 Unlike discrete distributions, where the sum of the probabilities
equals 1, in the continuous case, the area under the curve is 1.
 One additional important difference is that in continuous
distributions the P(X=x)=0
 Reason for this has to do with the calculus behind continuous
functions.
 Because of this ≥ is the same as >
 Also, ≤ is the same as <
 Therefore, we will only be interested in > or < probabilities.
Normal Distribution
 Unlike the Binomial and Poisson distributions that were
defined by a set of rigid requirements, the only condition for
a normal distribution is that the variable is continuous.
 And that the variable follows normal distribution.
 MANY variables follow normal distribution.
 The normal distribution is one of the most important
distribution in statistics.
 Normal Distribution is defined the mean and standard
deviation
 X~N(mu, sigma)
 If we are given the variance, we will need to take the square
root to get the standard deviation
Normal Distribution Con’t.
 Properties:
 Mound shaped: bell shaped
 Symmetric about µ, population mean
 Continuous
 Total area beneath Normal curve is 1
 Infinite number of Normal distributions, each with its own mu
and sigma
Example: Weight of dogs
 Suppose X, the weight of a full-grown dog is normally
distributed with a mean of 44 lbs and a standard deviation of
8 pounds X~N(44, 8)
20
28
36
44
52
60
68
The Empirical Rule
 The empirical rule states the following:
 Approx. 68% of the data falls within 1 stdv of the mean
 Approx. 95% of the data falls within 2 stdv of the mean
 Approx. 99.7% of the data falls within 3 stdv of the mean
Using the Empirical Rule
 Back to the dog weight example, X~N(44,8)
What percent of dogs weigh between 28 and 60 pounds?
1.

95% by the empirical rule
What percent of dogs weigh more than 60 pounds?
2.


2.5% by the empirical rule
Why is this?
Finding Normal Probabilities
 Like Binomial and Poisson distributions, the cumulative
probabilities for the Normal Distribution can be found using
tables.
 BUT, rather than making tables for different values of mu and
sigma, there is only 1 table.
 N(0,1)
 We will need to convert the normal distribution of our
problem to this normal distribution using the formula:
Examples of Finding Z
 For X~N(44,8)
 Find Z for X =
 52
1
 28
 -2
 68
3
 What do we notice?
 Z measures how many standard deviations we are away from
the mean
Finding Exact Probabilities
 Good news!
 For any X, the P(X=x)=0
 We assume it is impossible to get any 1 particular value
Finding Less Than Probabilities
 To find less than probabilities. We first convert to our z




score then look up the Z value on the normal table.
Remember, since we are using a continuous distribution, < is
the same as <=
For X~N(30, 4), Find
P(X<29)
P(X<40)
P(X≤40)
Greater Than Probabilities
 Similar to less than probabilities, first find the z-score, then




use the table. Just like Binomial and Poisson we will use 1 –
the value in the table.
For X~N(100, 10), Find
P(X>95)
P(X>100)
P(X≥100)
In-Between Probabilities
 To find in-between probabilities, you must first find the z-
score for both points, call them a and b, and then the
probability is just the P(X<b) – P(X<a)
 For X~N(18,2), Find
 P(14<X<22)
 Compare this to the Empirical Rule
Percentiles – Working Backward
 Suppose that we want to find what X value corresponds to a
percentile of the Normal Distribution
 Example: What is the 90th percentile cutoff for SAT Scores?
 How to do this
 Step 1: Find the z value in the z table that matches closest to
.9000.
 Step 2: Put this z in the z-score formula
 Step 3: Solve for x
Example
 Let X be a student’s SAT Math Score with a mean of 500 and
a standard deviation of 100.
 X~N(500,100)
 Find the following percentiles:
 90th
 75th
 50th
 Note that these questions could be asked such that:
 P(X<C)=.9000. Find C