Statistical Analysis for Education and Psychology Researchers

Discrete Probability Distribution

Example 4.2: The Binomial Probability Distribution

If we choose a simple random sample of 10 schools from the population of all secondary

schools in a Local Education Authority and for each school we assigned it into a ‘better’

or ‘worse’ category, depending on the answer to the following question, ‘Is the

percentage of 15-year-old pupils achieving 5 or more GCSEs at grades A to C ‘better’ or

‘worse’ than the median national (population) pass rate of 39.9 per cent?’ We may find

that our sample of 10 schools has the following per cent pass rates (+ or − indicates

whether it is better or worse than the national average):

19(−); 37(−); 52(+); 11(−); 13(−); 31(−); 100(+); 25(−); 41(+); 18(−)^

When a discrete random variable is a count, X, of the ‘successes’ in n independent

trials or observations which each has the same probability of success, it is said to be a

binomial random variable. If we consider a sampled school that is above the median to

be a success then an appropriate statistical model to describe the probability

distribution of the random variable (school=success/fail) is the binomial probability
distribution, sometimes called the binomial model. This probability distribution enables
determination of the probability for any number of successes r, (r=0, 1, 2...10), in n=10
schools provided they were selected at random.
We can use the binomial sampling distribution to answer questions like, ‘Could it
happen by chance that we obtain 3 schools in our sample that have above the national
median pass rate?’

Pascal’s Triangle
Pascal’s triangle, named after the Mathematician Blaise Pascal who discovered it when
he was only 16, is used to illustrate how probabilities can be calculated for a binomial
variable. Let us consider selecting one school at random. This is equivalent to one flip of
a coin. What is the probability that the selected school lies below the population median?
Think about the probability of obtaining a ‘head’ with one flip of a coin. In both cases the
answer is one-half.
Intuitively, it is easy to answer the question about flipping the coin. Assuming the coin
is fair there is an even chance that heads will turn up. The reason why the probability that
the selected school lies below the median is one-half is because by definition half the
population lies below the median. Think about it this way, there are only two possible
outcomes, the selected school is either below or above the population median (we are
assuming for now that none of the schools sampled has a pass rate that is equal to the
population median).
If we select 10 schools at random this is equivalent to flipping 10 coins. We can work
out, for each school selected, the probability of falling above the national median pass
rate in the following way. If one school is selected there is an even chance that it will be
above, using (A) for above, or below, using (B) for below, the population median. If a
second school is selected there are four possible outcomes: both schools are above the

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