Download (that is, calculate the mean and variance of a binomial

Lesson 8 - 1
Discrete Distribution
Binomial
Taken from http://www.pendragoncove.info/statistics/statistics.htm
And modified slightly
Knowledge Objectives
• Describe the conditions that need to be present to
have a binomial setting.
• Define a binomial distribution.
• Explain when it might be all right to assume a
binomial setting even though the independence
condition is not satisfied.
• Explain what is meant by the sampling distribution
of a count.
• State the mathematical expression that gives the
value of a binomial coefficient. Explain how to find
the value of that expression.
• State the mathematical expression used to calculate
the value of binomial probability.
Construction Objectives
• Evaluate a binomial probability by using the
mathematical formula for P(X = k).
• Explain the difference between binompdf(n, p, X) and
binomcdf(n, p, X).
• Use your calculator to help evaluate a binomial
probability.
• If X is B(n, p), find µx and x (that is, calculate the
mean and variance of a binomial distribution).
• Use a Normal approximation for a binomial
distribution to solve questions involving binomial
probability
Vocabulary
• Binomial Setting – random variable meets binomial conditions
• Trial – each repetition of an experiment
• Success – one assigned result of a binomial experiment
• Failure – the other result of a binomial experiment
• PDF – probability distribution function; assigns a probability to
each value of X
• CDF – cumulative (probability) distribution function; assigns
the sum of probabilities less than or equal to X
• Binomial Coefficient – combination of k success in n trials
• Factorial – n! is n  (n-1)  (n-2)  …  2  1
Binomial Mean and Std Dev
A binomial experiment with n independent trials and
probability of success p has
Mean μx = np
Standard Deviation σx = √np(1-p)
Example 1
Find the mean and standard deviation of a
binomial distribution with n = 10 and p = 0.1
Mean:
μx = np = 10(0.1) = 1
Standard Deviation:
σx = √np(1-p)
= 10(0.1)(0.9)
= 0.9
= 0.9487
Using Normal Apx to Binomials
As binominal’s number of trials increases the
formula for a binomial becomes unworkable (a
situation alleviated with statistical software).
So statisticians developed a procedure to use a
continuous distribution, the normal, to estimate
a discrete distribution.
This procedure is used later with proportions.
Example 2
Sample surveys show that fewer people enjoy shopping than in the
past. A survey asked a nationwide random sample of 2500 adults if
shopping was often frustrating and time-consuming. Assume that
60% of all US adults would agree if asked the same question, what is
the probability that 1520 or more of the sample would agree?
P(X ≥ 1520) = 1 – P(X ≤ 1519)
= 1 – binomcdf(2500, 0.6, 1519) = 0.2131
𝜇x = np = 2500 (0.6) = 1500
σx = √np(1-p) = √ 2500(0.6)(0.4) = 600 = 24.49
P(X ≥ 1520) = using normCDF(1520, E99, 1500, 24.49) = 0.2071
a difference of 0.0061 or less than 0.6%
Example 2 cont
Histogram showing normal apx to binomial
Simulating Binomial Events
• To simulate a binomial event, we must know:
– how random variable X and “success” is defined
– probability of success
– number of trials
• Calculator has a randbin(1,p,n) function that
will generate results with “1”s as successes
• We can store the results in lists and do
statistics on the lists as usual
Example 3
Each entry in a table of random digits like Table B in our
book has a probability of 0.1 of being a zero.
a) Find probability of finding exactly 4 zeros in a line 40
digits long.
P(X = 4) = binompdf(40, 0.1, 4) = 0.206
b) What is the probability that a group of five digits from
the table will contain at least 1 zero?
P(X ≥ 1) = 1 – P(X=0)
= 1 ‒ binompdf (5, 0.1, 0) = .4095
= 1 – P(no zeros)5 = 0.4095 (before binomials)
= 1 – (0.9)5 = 0.4095
Example 4
A university claims that 80% of its basketball players
get their degree. An investigation examines the fates
of a random sample of 24 players who entered the
program over a period of several years. Of these
players, 12 graduated and 12 are no longer in school. If
the university's claim is true,
a) What is the probability that at most 12 out of 24
players graduate?
P (x < 12) = binomcdf (24, .8, 12) = .000978
b) What is the probability that all 24 graduate?
binompdf (24, .8, 24) = .0047
Summary and Homework
• Summary
– Binomial experiments have 4 specific criteria that
must be met
– Means and Variance for a Binomial
• E(X) = np and V(X) = np(1-p)
– Normal distribution (continuous) can approximate
a Binomial (discrete)
– Calculator has a random binomial generator