Contents:

• Significance Tests
• The Null and the Alternative Hypothesis

Significance Tests:

Other sections of TimeWeb have outlined how data can be used to make statements about whole populations, based on studying only a small sample drawn from the general population. Because we usually only have full knowledge of the sample we're working with, and not the population from which it is drawn, any inferences we can make about values within the whole population have to be made with some accompanying reservations.

In statistical work these reservations are measured and expressed in terms of probabilities. So, on the basis of some sample evidence, we might say something like: 'The sample mean would appear to be representative of the population mean, but there's a x% chance that it is not.'

Verbal statements such as the one above are equivalent to what statisticians call a hypothesis test. What happens with testing hypotheses is that you start by saying that the population mean, for example, is equal to some constant - let's call the constant k. This is what is known as the null hypothesis.

Then on the basis of some sample evidence, some simple statistics are calculated to test this hypothesis against what is called the alternative hypothesis - let's say that in this case this is that the population mean is not equal to k.

On the basis of this test and the probability of the calculated test values actually happening, either the null hypothesis is accepted or it is rejected. Whatever the result, an inference has been made, so there is a chance that it is wrong. This is quantified in probability terms. So in the verbal example above, this is expressed as x%. The value of x in this kind of statement is important and will be examined in detail later.

The correct way of writing a hypothesis test is shown below:

H₀: population statistic = k

H₁: population statistic k

Simple inferential statistics is about choosing between H₀ and H₁, bearing in mind that this choice always involves some risk of being wrong.

OK, so two options exist for dealing with hypotheses. Let's now try to apply this to a practical scenario; one that you might find in the business world.

Suppose for example, a manufacturer makes a certain claim about a product 'Guaranteed for a minimum of two years!' or 'It does exactly what it says on the packet!' How do can you test these claims from a small sample, and how certain can you really be of your results? When can you justifiably dismiss these manufacturers' claims and when should you accept them?

In another case, suppose you want to compare two populations to find out whether there is a real difference between them. Instances that might be included here may be the impact of different medical treatments for a particular disease, the difference between men and women's performance in specific workplaces and the effectiveness of different training methods for developing skills in sport. To deal with such questions, you will need to take a sample of each population and then compare the results.

From your analysis of each sample and the conclusions you reach about the populations the samples come from, you should be able to say whether the difference between the populations is real, within certain confidence levels.

The Null and the Alternative Hypothesis

You can see that an important part of statistical analysis is testing the claims made about a population or about two seemingly different populations. There are two types of claims that can be made:

1. Group A claims: 'You will see when using our product that': 'it will last as long as we claim'; or that 'it will cost you what we claim to run'; or 'it will rid you of your problem, as we claimed it would'.
2. Group B claims: In terms of some measurable aspect, one part of the population is significantly different from another part. So, you can imagine claims like: 'men handle workplace stress better than do women' or 'Hedeze works better than Paingone' or 'weight-training is better for sprinters than endurance work'.

It is often the case that the logic of a claim works negatively. So that the assertion made is that 'there is no difference between one part of the population and another in terms of this aspect.'

The method of testing such claims begins by collecting the relevant samples. For instance, in testing a claim of the sort made in 1 above, we take a random sample of the product (more than 30), and in case of a claim of the sort mentioned in 2 above we take a random sample of the two populations being compared.

The next stage involves forming two hypotheses, as follows:

1. The Null Hypothesis is, in the case of Group A claims, the assertion that there is no difference between the sample we have collected and the general population established by the claim. For Group B claims, the Null Hypothesis is that there is no difference between the two populations in the study. So, for example, the claim here could be that there is no difference between men and womens' ability to handle workplace stress; or, there is no difference between endurance and weights work in terms of sprinters' performance.
2. The Alternative Hypothesis for Group A claims, is that the sample you have belongs to a population different from that described in the original claim. So the claim is suspect. For Group B claims, the alternative hypothesis is that the samples infer that the two populations studied are, indeed, different.

If you find in the first type of claim, that the sample comes from the population described in the claim, or, in the second type of claim, that the samples you have probably come from the same population, then the Null Hypothesis is upheld. This means that there is no difference; if your samples do not satisfy the requirement that they probably come from the same population, then the Null Hypothesis is rejected, and we may accept the Alternative Hypothesis.

There is also an illustration of significance tests that it may be worth looking at to ensure you understand them.

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