*Double click on grey box*
* Variable name = “heads”*
* Click on change settings TYPE*
* Change decimal places from 2 to 0*
* Click continue*
* OK*

Next establish your column length

Decide on** HOW MANY **simulated results you require( 30? 50 ? 100?
500? ). For 30 results use the cursor keys to go down to the 30th cell
and type in a number (any number). If your column length is to be 50- do
this to cell number 50 and so on. (This is not your group size (n) but
the number of times the experiment is to be repeated.)

The instructions here will produce simulated results for B(10, 0.5). The number appearing in each cell represents the number of heads appearing if 10 coins are tossed.

* Click on Transform then Compute*
* In box with Target Variable - type your
column name (e.g. HEADS).*
* In Function Box (Right hand side) Scroll
down to RV,BINOM [n,p]*
* Highlight and click on arrow to get
it into top box (Numeric Expression)*
* Type in group size n (e.g. 10) in place
of first question mark*
* Type in p (e.g. 0.5) in place of second
question mark.*
* Click on OK*
* Change existing variable - OK*

The number in each cell represent for each group (size n =10), the number
of successes (or heads) obtained.

Look at a new bar graph for larger numbers of trials.

What do you notice?

What are the mean and standard deviation this time?

Is the mean closer to “np” than it was before?

Is the standard deviation closer to “npq”?

Investigate what happens if you increase the number of trials again
(to say 500 ).

2.) Compare p = 0.1 with p = 0.9

3.) What happens if you keep p the same but change n?

NOTE that the bar graphs produced by SPSS here will not take account
of values of x with zero frequencies.

*Click on Transform*

Use the Statistics menu to find the mean and standard deviation.

Use your calculator to square the value of the standard deviation you obtained.

Why would you expect this answer to be similar in value to the mean ?

Try increasing your column length.

Does increasing the number of samples (i.e. the column length) give
values of the mean and variance which are closer in value ?

Why might this happen ?

**Just how close are the distributions ?**

Generate 100 simulated results from a binomial distribution..

n = 10 p = ½

Draw a bar graph of your results.

Now generate 100 simulated results from the poisson distribution with mean = 5.

Draw a bar graph of these results. How similar are your two graphs ?

Now Compare

B(10, 0.1) with P (1)

B(100, 0.1) with P (10)

B(100, 0.01) with p (1)

Which pair of graphs were closest ?

*Click on Transform*

*Click on graphs*

Here are the times taken by 48 students to complete a pencil and paper maze. The normal distribution does not look like a good model for this data. But it can be quite difficult to make any decision on the basis of this kind of graph, particularly for small samples.

The normal PP plot produced by SPSS is similar to (but not quite the same as) the plots drawn on normal probability paper.

A normal PP plot plots the cumulative proportions of your results of your results on the horizontal axis. On the vertical axis it plots the cumulative proportions which would be obtained from a normal distribution with the same mean and standard deviation.

Cumulative proportions are found by

If the normal distribution is a suitable model for your data, the points
plotted should be close to a straight line. You expect a closer fit for
large samples than for small samples.

( A significance test can be obtained by clicking on Statistics - Summarise
- Explore.

Click on grey statistics
box and ask for Normality plots with tests

However, the tests used are NOT ones covered on the A level statistics
syllabus.)

EXAMPLE

The pulse rates for
a group of students were taken (results in beats per minute). Is the normal
distribution a suitable model for this variable ?

Pulse Rates

54 54
56 57 62
63 64 64
66

68 69
69 71 73
76

* *
*Click on graphs*

Next we show a normal Distribution PP Plot for the times taken by 48 students to complete a paper and pencil maze (used in previous section) . Note that the points are not close to a straight line.

The null hypothesis is that the coin is fair and that p = ½ . SPSS will calculate the probability of 12 heads and 8 tails for B(20, ½ ).

*Click on Statistics*

This shows that the probabilityof obtaining twelve heads or more (or twelve tails or more) from a fair coin is 0.5034. Therefore we must retain the null hypothesis and conclude that the coin does not appear to be biased. Only if the “Exact Binomial 2 - tailed P” is less than 0.05, can we reject the null hypothesis and conclude that the coin is biased.