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## Testing Some examples

#### Example: The round die

 In a specialty shop on games and games attributes I bought a nice round die. It is fun but is it fair? We can run a little test to figure that out. To do so I have rolled the die 100 times and have recorded the number of dots that faced up (like the 5 dots in the picture). The results can be found in the file rounddie.sav. Now we all know how an ideal fair die is supposed to roll. There are six possible outcomes that all have the same probability of 1/6 of facing up when the die is rolled. The expected average of the outcomes equals 3.5 We will run two different tests to assess whether or not this round die is fair. First of all we will do a one-sample t-test to find out if the average deviates from 3.5. And then we will run a chi-square goodness-of-fit test to see whether the distrubution of the dots is indeed a discrete uniform one.  #### The one-sample t-test for the population mean

Like with any test we start by specifying the hypotheses.
We write μ = the population average number of dots facing up when rolling this round die.

H0: μ = 3.5    and     HA: μ ≠ 3.5; the test is two-tailed since we have no indication about a direction for possible bias.

Now we run the test using SPSS; choose Analyze > Compare Means > One-Sample T-Test.

Note that we have to specify the Test Value for the mean in the appropriate field. The default setting is μ = 0, but most of the time that is not the correct value for the mean. In our case it is 3.5. By means of the Options button we can set the confidence interval for the mean, which is automatically calculated by SPSS. Here is the output: We see that the sample mean is equal to 3.4. That is not precisely equal to the expected mean of 3.5 specified by the null hypothesis, but the deviation doesn't seem to be large.
"Doesn't seem to be large" is quantified in the two-tailed significance reported by SPSS. It is 0.572, which is a large significance.
Hence there is no evidence against H0. We have no indication that our die is biassed. We keep our faith in the die and in the shop that sold it.

As a useful extra we have the 95% confidence interval that the average number of dots facing up when rolling this die is between 3.05 and 3.75. #### The chi-square goodness-of-fit test

Again we start by specifying the hypotheses.
H0: Each number of dots has a probability of 1/6 to face up when we roll the die   and
HA: Not all dots have the same probability to face up when we roll the die.

Our test can be found through Analyze > Nonparametric tests.
In SPSS 20 there is a new approach to this group of tests. But also the old way of asking for them is still available through the Legacy Dialogs Note that both for the dialog boxes and for the output there are substantial differences between the old and the new way of performing these tests. In the new approach the default setting is that you specify the type of data (One Sample, Independent Samples or Related Samples) and the variables involved. Based on the structure of the data SPSS now choses the most logical test to perform. It is named: "Automatically choose the tests based on the data". You can customize it if you want to. But of course you have to make sure that the measurement level and the role of each variable is set properly! In the old approach you need to know what all the various tests in the menu are about, under which conditions they are valid and what they do.

Here is the output from the new approach: It shows little details. Only the name of the test and the resulting significance are reported.
SPSS ends with a decision based on the default setting of a 0.05 level of significance.
The conclusion is that there is no evidence for any bias of our round die.

Warning: When the test is executed in this way SPSS does not check the chi-square conditions.

When you consult the SPSS help it doesn't mention any conditions for chi-square. However, when you check the same chi-square test through the Legacy Dialogs the SPSS help tells you:

Assumptions: Nonparametric tests do not require assumptions about the shape of the underlying distribution. The data are assumed to be a random sample. The expected frequencies for each category should be at least 1. No more than 20% of the categories should have expected frequencies of less than 5.

Technical note: Consult the statistical literature and you will find that the test statistic Σ(O-E)2/E only asymptotically can be approximated by a chi-square distribution. The scientists differ about when the approximation is good enough. The strict ones demand all expected frequencies to be larger than 5. The more lenient ones use the assumption mentioned in the SPSS help.

 Here is the output from the old approach: There is far more information about the test results in this output. First of all there is information about the conditions for the chi-square test. This is shown in footnote a. In our example there are no expected frequencies below 5, so it is safe to use the asymptotic significance. We see (of cource) the same significance reported as above. We have to draw our own conclusion. Since the significance is 0.416 we stick to the null hypothesis. There is no reason to doubt the fairness of the die. Finally in this output we see the residuals. This gives us extra information about the differences between the sample results and the expected counts for a fair die. In this case the deviations are small. The excess of ones is not worrying. These things happen when we have a small sample like n = 100. Last modified 30-10-2012 © Jos Seegers, 2009; English version by Gé Groenewegen.