Title: Conclusion: Probability
Course: UGQuants (SSPS) > Section: Probability

Week 7 Conclusion

There are three key ideas you learnt about in this module.

1 | Understanding the nature and properties of a normal distribution, and how these can be thought of as representing the shape of chance or purely random variation.
2 | Understanding that estimates of a population parameter obtained from repeated random samples will take a normal distribution.
3 | Understanding that, because of this, we can calculate the probability that an estimate for a population parameter found in a random sample will lie within a number of standard deviations from the (unknown) value of the population parameter itself. We can therefore calculate the confidence interval that expresses the probability that the population parameter will lie within a range of values.

It is important that you understand the logic behind the calculations we have just gone through, and the best way to become familiar with this is to actually do them, using different examples. However, you can usually just leave them to the computer. Statistical packages will happily calculate confidence intervals for you (in SPSS, for example, using the explore function) by assuming that the cases in your dataset form a random sample drawn from a population.

It is important that you understand what a confidence interval is and how to interpret it. In particular, it is important to grasp what it means when we say that we have a certain probability of the true population parameter lying within a range of results: e.g. a 95% confidence that a mean lies between two values. As we have seen, we establish the size of this interval using the known properties of probability. We know that if we were able to draw random samples again and again and plot the result for a population parameter in those samples we would get a normal curve with the value of the population parameter (e.g. the mean age of the population) lying at the peak of our curve. In any one random sample it is the most likely individual result.

But it is also highly unlikely that if we were to draw only one random sample, we would get exactly this result. Since this is the situation researchers normally find ourselves in, we can then calculate a confidence interval for our result, since we know that 95% of the results we get were we to sample repeatedly would lie within two standard errors. We know that the real and unknown population parameter value has a 95% chance of lying within our confidence interval. When we thus give our estimate of the population parameter's value we have a 5% chance of being wrong - that is of having been unlucky enough to have drawn one of those samples in the tails of the normal distribution of all samples in the area beyond that 95% of samples within two standard errors. And of course, nothing in our sample can tell us where in the distribution it might lie.