Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Lab2: Probability Sampling
and Discrete Distributions
Michael Akritas
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Simulating the Binomial Random Variable
Sampling From a PMF
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To obtain a sample of size 100 from the sample space
population 1-5 with respective probabilities 0.1, 0.2, 0.4, 0.2
and 0.1, do:
1. Enter the numbers 1-5 in C1. Enter the corresponding
probabilities in C2.
2. Calc>Random Data>Discrete>Enter ”100” into
Generate; enter ”C3” into Store in column(s); enter ”C1”
into Values in; enter ”C2” in Probabilities in>OK
To view the empirical probabilities (or sample proportions)
use: Stat>Basic Statistics>Display Descriptive
Statistics>Enter C3 in Variables and also in By variables
(optional), Click on Statistics and select ”N total” ”Percent”,
”Cumulative percent”. OK, OK
Copy and paste the output on a word document.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Simulating the Binomial Random Variable
Bar Graph and Dot Histogram of a PMF
Columns C1 and C2 of the previous slide contain the pmf
x
1
2
3
4
5
.
p(x)
0.1
0.2
0.4
0.2
0.1
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To do a bar graph of it use: Graph> Bar Chart > Select ”A
function of a variable”, OK> Enter ”C2” in Graph variables,
enter ”C1” in Categorical variable, OK
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Copy the bar graph by right-clicking on its margin and paste
on a word document.
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An alternative visual graph of a pmf is the dot histogram:
Graph>Scatterplot> select ”simple”> OK> Enter ”C2” (the
probabilities column) for Y and ”C1” (the sample space
column) for X>OK
Copy the dot graph and paste it on a word document.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Simulating the Binomial Random Variable
Bar Graph of the Empirical Probabilities
We will use the raw data given in column C3 to do a bar graph of
the empirical probabilities (or sample proportions):
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Graph> Bar Chart > Select ”Counts of unique values”, OK>
Enter ”C3” in Categorical variables, click on Chart options
and select Show Y as Percent, OK, OK
Copy and paste the graph in the word document
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An alternative way to do the bar graph of the empirical
probabilities is to do a Histogram:
Graph> Histogram>Choose ”Simple”, OK>Enter ”C3” in
Graph variables, click on Scale and on Y-Scale Type and
select Percent>OK, OK
Copy and paste the graph in the word document.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Simulating the Binomial Random Variable
Probability Sampling from the Binomial PMF
If each of 50 people toss a coin 10 times, and each records the
number of heads, how variable do you expect the 50 numbers to
be? We can answer this question by simulation.
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Generate 50 observations from the Bin(n=10,p=0.5)
distribution:
Calc>Random Data>Binomial> Enter ”50” into Generate;
enter ”C1” into Store in column(s); enter ”10” into Number
of trials; enter ”0.5” into Probability success> OK
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View the sample of 50 observations numerically and
graphically as described in the context of probability sampling.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
Simulating the Binomial Random Variable
Comparison of Empirical and True Probabilities
Since empirical probabilities imitate the true ones, the answer to
the previous question (how variable do we expect the 50 numbers
to be) can be answered by a bar graph of the Bin (10, 0.5) pmf.
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Obtain the pmf of X ∼ Bin (10, 0.5). Enter the numbers 0-10
in rows 1-11 of column C2 and then:
Calc>Probability Distribution>Binomial> click on
”Probability”, enter ”10” in Number of trials, enter ”0.5” in
Probability of success, enter ”C2” in Input column, enter
”C3” in Optional storage> OK.
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Do a probability dot histogram as described in the context of
probability sampling.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions
Probability Sampling
Binomial Approximation to Hypergeometric Probabilities
1. Use commands as before to generate the pmf of
Binomial(10,0.3), in c2, c4.
2. Use commands similar to the above to generate the pmf of
Hypergeometric(n=10, N=100, M=30), in c2, c5.
3. Use commands similar to the above to generate the pmf of
Hypergeometric(n=10, N=1,000, M=300), in c2, c6.
4. Use commands similar to the above to generate the pmf of
Hypergeometric(n=10, N=10,000, M=3,000), in c2, c7.
Copy and paste in a word document the above pmf’s. Comment
on the rule n ≤ 0.05 × N for satisfactory approximation, as well as
the quality of the approximation as N increases. Comment on
whether the Bin(10, 0.3) pmf provides a good approximation to the
hypergeometric(n=10, N=10,000, M=4,000) pmf.
Michael Akritas
Lab2: Probability Sampling and Discrete Distributions