Dan PiettSTAT 211-019
West Virginia University
Lecture 7
Last WeekBinomial Distributions
2 Outcomes, n trials, probability of success = p,
X = Number of SuccessesPoisson Distributions
Occurrences are measured over some unit of time/space with mean occurrences lambda
X = Number of OccurrencesFinding Probabilities
=< and ≤> and ≥
OverviewNormal DistributionEmpirical RuleNormal ProbabilitiesPercentiles
Continuous DistributionsUp until this point we have only talked
about discrete random variables.BinomialPoissonNote 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 curveUnlike 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)=0Reason 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 DistributionUnlike 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 deviationX~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 shapedSymmetric about µ, population meanContinuousTotal area beneath Normal curve is 1Infinite number of Normal distributions, each
with its own mu and sigma
Example: Weight of dogsSuppose 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 RuleThe 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 RuleBack to the dog weight example,
X~N(44,8)
1.What percent of dogs weigh between 28 and 60 pounds? 95% by the empirical rule
2. What percent of dogs weigh more than 60 pounds? 2.5% by the empirical rule Why is this?
Finding Normal ProbabilitiesLike 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 ZFor X~N(44,8)Find Z for X =52
128
-268
3What do we notice?Z measures how many standard deviations
we are away from the mean
Finding Exact ProbabilitiesGood news!For any X, the P(X=x)=0
We assume it is impossible to get any 1 particular value
Finding Less Than ProbabilitiesTo 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), FindP(X<29)P(X<40)P(X≤40)
Greater Than ProbabilitiesSimilar 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), FindP(X>95)P(X>100)P(X≥100)
In-Between ProbabilitiesTo 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), FindP(14<X<22)
Compare this to the Empirical Rule
Percentiles – Working BackwardSuppose that we want to find what X value
corresponds to a percentile of the Normal DistributionExample: What is the 90th percentile cutoff
for SAT Scores?How to do thisStep 1: Find the z value in the z table that
matches closest to .9000.Step 2: Put this z in the z-score formulaStep 3: Solve for x
ExampleLet 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