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Statistics - Maths of Real Life

This pilot collection of resources is designed to introduce key statistical ideas and help students to deepen their understanding.

Binomial Conditions

When is an experiment described by the binomial distribution? Why do we need both the condition about independence and the one about constant probability?

Binomial or Not?

Are these scenarios described by the binomial distribution?


Age 16 to 18 Challenge Level:
Let $S$ be the number of seats in the aircraft;
$T$ be the number of tickets sold;
$p$ be the probability that any given passenger arrives for the flight.

Let $X$ be the number of passengers that arrrive for a given flight.

Then $X$ has the Binomial distribution for $T$ trials with the probability of success $0.95$. You can calculate the mean $\mu$ and variance $\sigma^2$ for this distribution.

In order to avoid lengthy calculations in the discrete case we approximate the Binomial distribution by a Normal distribution with the same mean and variance, i.e. by $$N(\mu,\sigma^2)$$

Thus we now assume that $X$ has distribution $N(\mu,\sigma^2)$. In order to use the standard Normal probability tables $N(0,1)$ we have to put $$Y = {X-\mu\over \sigma}$$ then $Y$ has distribution $N(0,1)$.

So as to allow for the approximation to the discrete data by the continuous Normal distribution, we want to find $\text{Prob}[X\leq 400.5$] and look up the probability for the corresponding value of $Y$ in the Normal table.

If you use a Normal distribution table you need to check to see if it gives the area $\Phi(Y)$ under the Normal curve to the left of $Y$, that is the probability that the variable is less than $Y$, or to the right of $Y$.