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# Secondary Magazine - Issue 127: Sixth Sense

Created on 03 November 2015 by ncetm_administrator
Updated on 18 November 2015 by ncetm_administrator

# Sixth Sense

It’s clear from reading examiners’ reports that, for many AS and A2 students, binomial hypothesis testing is the topic that is least securely grasped compared to most of the other material usually assessed on Statistics 1 papers; conditional probability is probably (no pun intended) in second place. If students are to understand conceptually the logic of a binomial hypothesis test (and without that they will find it hard to answer any procedural question other than the most standard and straightforward), they need to have a fluent grasp of all the conceptual prerequisites. Next month we’ll talk about teaching the test (just it, not “to” it!); but first, let’s get the building blocks in order. I suggest this tree-like diagram (PDF).

By doing lots of similar examples, students should get to the position to generalise by themselves: if $\dpi{80} \fn_jvn \small P$(“success each time”) = $\inline \dpi{80} \fn_jvn \small p$ and I carry out the experiment $\dpi{80} \fn_jvn \small n$ times, and each time I do so I can assume that the outcome is independent of previous outcomes, then we write $\dpi{80} \fn_jvn \small P$(I get $\dpi{80} \fn_jvn \small r$ successes in total out of $\dpi{80} \fn_jvn \small n$) as $\inline \dpi{80} \fn_jvn \small P\left ( X=r \right )=p^{r}\left ( 1-p^{n-r} \right )\times \frac{n!}{r!\left ( n-r \right )!}$”. I’d much rather have a scheme of work that allows me to do all of this preparatory work than walk into an AS lesson and start by writing this last result on the board: the students having a secure understanding of the processes involved is far better than their learning mathematics as a series of rules of “things to do in a given situation”. There should then be some time given to consolidating how to calculate, for example, $\inline \dpi{80} \fn_jvn \small P\left ( X\leq 2 \right )$$\inline \dpi{80} \fn_jvn \small P\left ( X< 4 \right )$$\inline \dpi{80} \fn_jvn \small P\left ( X\geq 13 \right )$$\inline \dpi{80} \fn_jvn \small P\left ( 3< X\leq 6 \right )$,both using the formula and then by using the statistical tables: being familiar and confident with the tables and also the formula is a crucial skill to develop separately from, and prior to, hypothesis testing.

Finally, your students need to identify the structural characteristics of situations that are likely to suit modelling with a binomial distribution:

• an experiment is carried out $\dpi{80} \fn_jvn \small n$ times
• each time, the experiment is either successful or not
• each time, the outcome of the experiment can be assumed to be independent of previous outcomes, and so we can say that there is a fixed $\dpi{80} \fn_jvn \small P$(“success each time”).

It’s well worth looking at situations where one or more of these aren’t likely to hold, to help students understand where they are likely to do so.

If you’ve never written a Scheme of Learning, this is the kind of thinking required, so have some sympathy with your Subject Lead! Why not pick another topic and ask yourself “what needs to be secure before I can teach this successfully?” and see if you can draw a similar picture? Share your examples with us, by email to info@ncetm.org.uk or Twitter @NCETMsecondary.

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