What is the first question to ask when deciding between a binomial and a normal model?

Whether the variable counts or measures . A variable that counts is discrete , and counting successes is what a binomial distribution does; a variable that measures is continuous , such as a mass, a time or a length, and that points to a normal distribution. That settles which family you are in, but the specific conditions for each model then have to be checked separately, because being a count does not by itself make something binomial.

You have real data rather than a description. How can you tell whether a normal model is reasonable?

Draw a histogram of the data and look at its shape: if the outline is roughly symmetrical and bell-shaped , a normal model is reasonable. If the variable really is normally distributed, then the more data you collect the smoother that outline should become, settling towards the shape of a normal distribution curve. A histogram that is clearly lopsided, or that has two humps, is evidence against the model, whatever the context suggests.

A LevelMathsOCRMaths AStatisticsFlashcardsStatistical DistributionsChoosing Distributions

Choosing Distributions (OCR A Level Maths A: Statistics): Flashcards

Exam code: H240

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FrontModelling with Distributions
What is the first question to ask when deciding between a binomial and a normal model?
BackModelling with Distributions
Whether the variable counts or measures.
A variable that counts is discrete, and counting successes is what a binomial distribution does; a variable that measures is continuous, such as a mass, a time or a length, and that points to a normal distribution.
That settles which family you are in, but the specific conditions for each model then have to be checked separately, because being a count does not by itself make something binomial.
BackModelling with Distributions
Draw a histogram of the data and look at its shape: if the outline is roughly symmetrical and bell-shaped, a normal model is reasonable.
If the variable really is normally distributed, then the more data you collect the smoother that outline should become, settling towards the shape of a normal distribution curve.
A histogram that is clearly lopsided, or that has two humps, is evidence against the model, whatever the context suggests.
BackModelling with Distributions
With two distributions, one feeding the other, starting with the normal distribution for the probability that a single cow is beefy:
$p = P (M > 700) = 0.030396 \dots$
That probability then becomes the $p$ of a binomial distribution for the sample, because each of the 10 cows either is beefy or is not:
$X ~ B (10, 0.030396 \dots), P (X \leq 1) = 0.965$
Carry plenty of decimal places in $p$ : rounding it to 0.03 before the second stage shifts the final answer.

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What is the first question to ask when deciding between a binomial and a normal model?
Whether the variable counts or measures.
A variable that counts is discrete, and counting successes is what a binomial distribution does; a variable that measures is continuous, such as a mass, a time or a length, and that points to a normal distribution.
That settles which family you are in, but the specific conditions for each model then have to be checked separately, because being a count does not by itself make something binomial.
You have real data rather than a description. How can you tell whether a normal model is reasonable?
Draw a histogram of the data and look at its shape: if the outline is roughly symmetrical and bell-shaped, a normal model is reasonable.
If the variable really is normally distributed, then the more data you collect the smoother that outline should become, settling towards the shape of a normal distribution curve.
A histogram that is clearly lopsided, or that has two humps, is evidence against the model, whatever the context suggests.
Cow masses are modelled by $N (550, 80^{2})$ , and a cow is called beefy if it weighs more than 700 kg. A random sample of 10 cows is taken. How do you find the probability that at most one is beefy?
With two distributions, one feeding the other, starting with the normal distribution for the probability that a single cow is beefy:
$p = P (M > 700) = 0.030396 \dots$
That probability then becomes the $p$ of a binomial distribution for the sample, because each of the 10 cows either is beefy or is not:
$X ~ B (10, 0.030396 \dots), P (X \leq 1) = 0.965$
Carry plenty of decimal places in $p$ : rounding it to 0.03 before the second stage shifts the final answer.
In a question that uses two distributions, what must you write down before calculating anything?
Exactly what each variable and each parameter stands for, in words, and then its distribution.
So, for example:
- let $M$ be the mass of a cow, with $M ~ N (550, 80^{2})$
- let $X$ be the number of beefy cows in the sample, with $X ~ B (10, p)$ , where $p$ is the probability that a cow is beefy
Without that the two variables are easy to confuse, and the 10 in the binomial gets mixed up with quantities from the normal distribution; saying what $p$ means also tells you it must be calculated rather than read off.
When can a binomial distribution be approximated by a normal distribution?
When $n$ is large and $p$ is close to 0.5.
Those two conditions between them make the binomial's vertical line graph roughly symmetrical and closely packed, which is the shape a normal curve has.
A binomial distribution $X ~ B (n, p)$ is to be approximated by a normal distribution. Complete the mean and variance of the approximating distribution:
$μ =______and σ^{2} =______$
The completed parameters are:
$μ = n p and σ^{2} = n p (1 - p)$
They are simply the mean and variance of the binomial distribution itself, so the normal distribution chosen is the one matching it at the centre and in its spread.
What kind of distribution is being replaced by what kind, when a binomial distribution is approximated by a normal one?
A discrete distribution is being replaced by a continuous one.
The binomial gives a probability to each whole number from 0 to $n$ and is drawn as a set of separate vertical lines, while a normal curve spreads probability continuously across a whole range of values.
The approximation works when those vertical lines are numerous enough and symmetrical enough that their tops trace out the shape of a normal curve.
True or False?
A binomial distribution with a very large number of trials can always be approximated by a normal distribution.
False.
A large number of trials is only one of the two conditions: $p$ must also be close to 0.5.
With $p$ very close to 0 or to 1 the binomial stays strongly lopsided however large $n$ becomes, and a symmetrical normal curve cannot follow that shape.
$X ~ B (1250, 0.4)$ is to be approximated by a normal distribution.
What is that normal distribution?
The approximating distribution is $N (500, 300)$ in the usual notation.
The mean is $1250 \times 0.4 = 500$ and the variance is $1250 \times 0.4 \times 0.6 = 300$ so the standard deviation is $\sqrt{300}$ or about 17.3.

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