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