Hypothesis Testing

The reason we mess with statistics is to help make decisions.  Experience teaches us that many aspects of day-to-day life are subject to random variations.  For example, on most days the bus to work arrives within a few minutes of the advertised time (buses can be early as well as late, however, most people only notice when its late).  There will be days when the bus appears to be very late.  Most people will form an opinion if the late arrival of the bus is typical, in which case they will do something else, e.g. catch an earlier bus, walk to work, become a monk etc..  However, if the late arrival is judged to be exceptional, they will keep catching the the bus each day.  Hypothesis testing is simply the formalisation of this process.

In practice, hypothesis tests are reduced to testing to see if an observed value is either less than or greater than a critical value.  Students are expected to appreciate the origins of this process which is based on two mutually exclusive hypotheses, the Null Hypothesis and the Alternative Hypothesis.

The Null Hypothesis (H₀)

The Null Hypothesis states that any difference between the observed value and the predicted value is due to chance alone.  Using the bus as an example, variations in its arrival time are simply due to the vagaries of city traffic and nothing to do with UFO's landing in the bus lane to empty the chemical toilet.

The Alternative Hypothesis (H₁)

It follows that if the difference between the expected and observed is not due to chance, it is the result of some process.  The identity of the process may not be known.

The key skill is designing a test which allows the acceptance or rejection of the Null hypothesis.  Each case will have its own test, what works for a week's experience of hanging round bus shelters may not be suitable determining if a pharmaceutical product has any therapeutic value.

Level of Significance

The outcome of a significance test will be influenced by the threshold value used to test against.

Example 1 - Population Mean and standard deviation known, Is a sample mean equal to the population mean.

The average weight of a apples from an orchard is 200gm with a standard deviation of 24gm..  The average weight of nine apples from a shady part of the orchard is 175gm, we  want to know if the average weight of the sample apples is less than that for the whole orchard.  The Hypothesises are:

The Null hypothesis is that the Sample Mean = Population Mean, thus any difference is due to chance.

The Alternative hypothesis is that the Sample mean is less than the population mean.

The central limit theorem tells us that the sample mean will be normally distributed with a standard deviation of:

StDev of Sample = StDev of Population / square root (sample size)

= 24/ square root (9)

= 8

Thus the 5% confidence limits for the sample are:

175 ± 1.96 * 8

175 ± 15.7

159.3 - 190.7

Thus as the population mean is greater than the upper bound of the confidence interval, we reject the null hypothesis.  Thus the average weight of the sample is less than that for the whole orchard.

Example 2 - Is a Single value typical of a Population

The example is based on winter rainfall in southern England.  This was reportedly low for the winter of 05/06.  The histogram shows data for the period 1959 to 2005.

The dataset is a left skewed distribution which suggests a modal value of 325mm.  The 90% confidence interval is 250mm to 550mm.  The value for the winter of 05/06 is approximately 260mm.

The null hypothesis is that the experience of 05/06 due to chance (i.e. rainfall is subject to random variation) and the alternative hypothesis is that 05/06 was an exceptional winter.

The Null hypothesis is accepted because the 05/06 value is within the 90% confidence interval.  This means that whilst the winter of 05/06 is drier than common experience (i.e. its less than the modal value), it is not exceptional in the context of a 90% confidence interval.

Page updated: 13-May-2006

For more information: info@brighton-webs.co.uk