Download Understanding the Binomial Distribution as a Sampling Distribution and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity!
Sampling Distributions
You have seen probability distributions of various types. The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as weights, heights, etc. The binomial distribution is an example of a discrete distribution because the possible outcomes are only a “discrete” set of values, 0, 1, … , n. The value of the binomial random variable is the number of “successes” out of a random sample of n trials, in which the probability of success on a particular trail is π.
The Binomial Distribution
as a Sampling Distributions
The nature of the binomial distribution makes it a sampling distribution. In other words, the value of the binomial random variable is derived from a sample of size n from a population. You can think of the population as being a set containing 0’s and 1’s, with a 1 representing a “success” and 0 representing a “failure.” (The term “success” does not necessarily imply something good, nor does “failure” imply something bad.) The sample is obtained as a sequence of n trials, with each trial being a draw from the population of 0’s and 1’s. (Such trials are called “Bernoulli” trials.)
On the first trial, you draw a value from {0, 1}, with P(1)=π. Then you draw again in the same way, and do this repeatedly a total of n times. So your sample will be a set such as {1, 1, 0, 1, 0, 1} in the case for n=6. This set has 4 1’s and 2 0’s, so the value of the binomial random variable is y =4. The value of the binomial distribution is the sum of the outcomes from the trials; that is, the number of 1’s.
The Binomial distribution
as a Sampling Distribution
Consider a population that consists 60% of 1’s and 40% of 0’s. If you draw a value at random, you get a 1 with probability .6 and a 0 with probability .4. Such a draw would constitute a Bernoulli trial with P(1)=.6.
Suppose you draw a sample of size n and add up the value you obtain. This would give you a binomial random variable. Now suppose you do this again and again for a very large number of samples. The conceptual results make up a conceptual population. Here are the histograms that correspond to the population for values of n equal to 1, 2, 3, 6, 10, and 20.
Notice how the shapes of the histograms change as n increases. The distributions become more “mound-shaped” and symmetric.
Histotram of B(1, .6) Distribution
0
1
2
3
4
5
6
7
0 1 Successes
Relative Frequency
Histogram of B(2, .6) Distribution
0
0 1 2 Successes
Relative Frequency
Histogram of B(3, .6) Distribution
0
0 1 2 3 Successes
Relative Frequency
The Binomial Sampling Distribution
and Statistical Inference
Here are the values of P( y=k ) and P( y<=k ) for n=20 and k=6- (P( y=k )<.005 for n<6 or n>17):
P( y=k ) .005 .015 .035 .071 .117 .160 .180 .165 .124 .075 .035.
P( y ≤ k ) .006 .021 .057 .128 .245 .404 .584 .750 .874 .949 .984.
Remember that these are the probabilities for π=.6.
Suppose you are sampling from a distribution with unknown π.
If you drew sample with y =12, would you have reason to doubt that π=.6? (Note: y /n=.6)
If you drew sample with y =10, would you have reason to doubt that π=.6? (Note: y /n=.5)
If you drew sample with y =6, would you have reason to doubt that π=.6? (Note: y /n=.3)
The Binomial Sampling Distribution
and Statistical Inference
Here are the numeric values of P( y=k ) for n=10, 3 decimal places:
P( y= k ) .000 .002 .011 .042 .111 .201 .251 .215 .121 .040.
P( y ≤ k ) .000 .002 .012 .055 .166 .367 .618 .833 .954 .994 1.
Remember that these are the probabilities for π=.6.
Suppose you are sampling from a distribution with unknown π.
If you drew sample with y =6, would you have reason to doubt that π=.6? (Note: y /n=.6)
If you drew sample with y =5, would you have reason to doubt that π=.6? (Note: y /n=.5)
If you drew sample with y =3, would you have reason to doubt that π=.6? (Note: y /n=.3)
Normal Distributions and
Sampling Distributions
If the distribution from which the samples were obtained is
not normal, then the sampling distribution of y^ is only
approximately normal, but the distribution becomes more nearly normal as n increases. This is the Central limit Theorem.
Central Limit Theorem : If y is a random variable with mean μ
and variance σ^2 , then the sampling distribution of y^ is
approximately normal with mean μ and variance σ^2 /n.
