Success in business involves taking decisions about the future but how do you know that the decision you took has made any difference?
You must be able to detect changes that are taking place if you are ever going to react sensibly. Now part of the problem of detecting change is that you can not afford the time or the money to keep carrying out a complete census of your target market. You are going to have to take samples and you know from the work of Prof. Gauss that every sample you take will be different.
You could end up basing a costly decision on the variation of one sample from another. After all the means of samples are normally distributed so we know that any results we get will be spread out. Under no circumstances do we expect two measures taken at different times from the same population to give exactly the same answer but by how much is it reasonable for them to be different?
Fortunately we can make use of Gauss's work here because his normal distribution lets us answer that last question. His Central Limit Theorem said that if we keep taking samples from the same population than 95 times out of a hundred the mean value of the samples will be all be within two standards errors of the mean. We can work out the standard error by adjusting the standard deviation of the sample for the sample size. So this gives us a way of checking if anything is changing. Because it implies that if our samples have come from a different mean they will be further then 2 standard errors from the mean.
All that work we did learning how to work out 'z' values is now about to become really useful. Lets try it with some of our real data from out own survey.
I'm going to launch a new unisex eye-shadow range especially aimed at people with blue eyes.
My question is are males or females more likely to have blue eyes?
To try and find out I counted the females in the sample and there were 87 of which 32 had blue eyes.
I did the same to the males and I counted 76 males and of those 32 also had blue eyes.
I turned these results into percentages and had 37% of females had blue eyes and 43% of males had blue eyes.
Does this mean that males are more likely than females to have blue eyes? Just looking at the raw figures this might be true. But you lot are not the entire world, you are only a sample of it. Perhaps I got this result just because of the sample I happened to choose. If I'd chosen a different sample I might have got a different result. I really could not we being a little more certain. Perhaps Prof. Gauss and his central limit theorem can help. It can if we think about it first.
One of the most important limitations of statistics is that is totally unable to prove anything at all. It can only fail to prove something. If we want to know if something might be true we have to try and prove the opposite. If we fail then we can say that the opposite is likely. We still don't know if it is true but we can be a lot more comfortable that it might be. We call this way of doing things 'setting up a null hypothesis'. An hypothesis is an idea that we want to test. If we set up two conflicting ones then one will be rejected. When this happens we know that the other one must be reasonable but we cannot say we have proved it. We can only say we failed to disprove it.
Lets apply this idea to the question about male and female blue eyes.
First I'll set up my null hypothesis which is that there is no difference in the proportion of males and females with blue eyes.
Now if that is true than the difference between the proportion of males with blue eyes P1 and the proportion of Females with blue eyes is really zero. Statisticians call the null hypothesis H0
So H0 is P1-P2 =0 this is our Null Hypothesis.
Our Alternative Hypothesis is that there is a difference between the two proportions. Statisticians call the alternative hypothesis H1
So H1 is P1-P2<>0 this is our Alternative Hypothesis
The 'z' value is worked as just as we did it in the tutorial last week.
z=(P1-P2)/Sqrt[(p(1-p)/sample size]
The p in the Standard Error calculation is the total proportion of Blue eyes in the whole sample because we are assuming that there is no difference between males and females.
Now we had a total of 163 valid questionnaires with 64 students with blue eyes. As a percentage this is 39%.
The standard error then is the square root of 39*61/163 which is the square root of 14.6 which works out as 3.82
We can now work out the 'z' value as (37-43)/3.82 that works out as -1.57. Now we know from the work of the good Dr Gauss that 95% of all samples are between 'z' values of +2 and -2. This means that we have failed to disprove the null hypothesis and we can find no difference at the 95% confidence level that there is any variation in the proportion of blue eyes between males and females.
The difference we saw in the first results are well within the range of variation we can expect from that size of sample.
I can go ahead with my Unisex eye shadow for blue eyed people in confidence that I have as many potential female customers as I have male ones.
This is a very useful technique because I can not just use it to see if there are differences between groups but I can also test the same group at different times and see if there are any changes taking place. This trick is very useful for both market research and for quality control.
Let's try a test like this with your heights.
You remember that the mean height of the group was 172 cms and the standards deviation of the group was 9.82 cms.
Now I've selected 30 students at random and asked them their height. The mean height is 176.2 cms and the standard deviation is 9.5.
Is the mean height of this group different from the mean height of your group. In other words is this sample of 30 students taken from a taller population?
Again I'll set up a null and an alternative hypothesis.
The null hypothesis is that there is no difference between the two groups. The alternative is that there is a difference.
We need to work out the standard error of the mean. This again is related to the standard deviation of the population and size of the sample.
The standard error is the std dev of the main population multiplied by the square root of (1/n1)+(1/n2) where n1 is the number of the main population and n2 is the number in the sample.
The standard error is 9.82* SQRT [(1/163)+(1/30)] which works out as 1.95
Now I can work out the 'z' value which is (176.2-172)/1.95 which is 2.15
This is outside the plus or minus 2 std errors so I can say the at the 95% confidence level I reject the null hypothesis that the two means are really the same. This means it is very likely that my sample is on average taller than the first year business studies students. Just out of interest I took my sample from the thirty tallest males I could find so it is not really surprising that they are taller than your population that has many less tall (i.e. shorter) females in it.
I've looked at ways of using hypothesis testing to investigate how things might be changing but there are sometimes unusual circumstances where you may want to try and understand the motives of somebody who is no longer available to questioned.
Let me give you an example.
In one of my other incarnations I am interested in the origins of Freemasonry. The official line of the Craft (as Freemason's refer to their order) is that it started in London in 1717. For many years rumours circulated among the wacky fringe of romantic writers that the Craft may really be a remnant of a medieval order of chivalry called the Knights Templar. This order was arrested throughout France and charged with heresy on 13 Oct. 1307. The Templar fleet escaped and was never found. Legend says that a large group of Templars fled to Scotland where they helped turn the tide against the English at the Battle of Bannockburn.
What has this gibberish got to do with statistics you ask? Well, let me explain. At the time of the destruction of the Templars the Grand Prior of Scotland was Sir William St. Clair of Roslin. He was eventually killed taking the heart of the dead King Robert of Scotland on a last crusade to Jerusalem. A hundred years later the direct descendant of Sir William, also called Sir William (The St. Clair family were very conservative in their choice of Christian names) built a very strange building near the family's castle at Roslin. Known as Rosslyn Chapel it is an ornately carved stone building that is a perfect copy of the ground plan of the now destroyed Jewish Temple of Jerusalem.
Sir William adopted the first recorded instance of quality control because he made his masons produce all the stone work wood before cutting it in stone. Only when he had inspected and passed the work suitable was it cut in to stone. From this practise we know that nothing carved into the building is accidental. The building is full of Templar, Celtic and Masonic symbols.
On the south wall of this building Sir William had carved a little figure which seems to show a link between Templarism and Freemasonry. The figure shows a man kneeling between two pillars. He is blindfolded and has a running noose about his neck. His feet are in a strange and unnatural posture and in his left hand he holds a bible. The end of the rope about his neck is held by another man who is wearing the mantle of a Knight Templar.
Now for those of you are not familiar with the strange ways of the craft, when somebody is admitted they are called the Candidate. Men are admitted to men's lodges and women to separate women's lodges. The candidate is dressed in a very odd manner but will only be admitted when properly dressed for the ceremony. The way of dressing is to wear a rough white clothing folded back to reveal particular parts of the body. The candidate is blindfolded and has a running noose about his neck. Here is a picture of a Masonic candidate who is correctly dressed for the ceremony standing before the two pillars that appear in every Masonic Lodge.
The statue at Rosslyn shows a number of features which are now considered to be Masonic. It is in a Templar building built by a direct descendant of the last Grand Prior of the Templars in Scotland and it shows a Templar carrying out what is now the first ceremony of Freemasonry. Here is a photograph of the statue
and a drawing of its main features.
The problem for English Freemasonry was that the stature was carved five hundred and fifty years ago and two hundred and seventy years before the claimed founding of the Craft in England. Publishing this information caused a tremendous fuss in English Freemasonry. Articles were written by various Masonic writers saying that these similarities were simply co-incidence. Freemasonry could not possibly have come from such an outlandish place as Scotland!
That counter claim can be tested using the techniques of Hypothesis testing and this is how I did it.
There are seven points of congruence (agreement) between the carving and the modern Masonic ceremony. These are
1. The man is blindfolded. This is unusual in medieval statues and the only other example it the figure of blind justice. There is no other blindfold figure carved in Rosslyn.
2. The man is kneeling. This is fairly common in medieval carvings and there are other kneeling figures in Rosslyn.
3. The man is holding a bible in his left hand. There are a number of other carvings showing figures holding books or scrolls within Rosslyn.
4. The man has a noose about his neck. There are few known figures of the period showing nooses about their necks. The best known is the statue called 'The dying Gaul'. There is one other figure in Rosslyn which has a noose in it and that is the figure of the hanged man which represents the angel Shemhazai whose sins caused God to send the Flood and who was so afraid to face God that he hung himself between heaven and earth with his face away from God. Shemhazai is carved with a noose about his feet but there is no other noose carved in Rosslyn.
5. The man has his feet in the posture that is still used today by Masonic candidates. This is a very unusual position and does not occur in any other carvings in Rosslyn.
6. The ceremony is being carried out between two pillars as it is in a Masonic Lodge. Pillars figure in a lot of the carvings at Rosslyn.
7. The noose is being held by a man clearly dressed as a Templar. There are many Templar symbols and images of Templars carved in Rosslyn.
So what is the chance of all these factors coming together by chance? I set up a null hypothesis that it was pure co-incidence that all these elements linking Templarism and Freemasonry occurred in the same carving and then set out to test the probability of the idea.
1. The probability that the figure is blindfolded by chance is 0.5 as it can only be blindfolded or not blindfolded. This is a worst case probability that gives the null hypothesis the best chance of succeeding as there is no other blindfolded figure in Rosslyn.
2. The probability that the figure is kneeling by chance is 0.5 as it again can only be kneeling or not kneeling.
3. The probability that the figure is holding a bible by chance is 0.5 as there are again only two possibilities. Holding a bible or not holding a bible.
4. The probability that the figure has noose about its neck by chance is 0.5 even though it is the only figure in Rosslyn with a noose about its neck. Again I am giving the Null Hypothesis the best possible chance of succeeding.
5. The probability that the figure has his feet in a Masonic posture (which the ritual says is the only way a Candidate will be admitted to Freemasonry) by chance is 0.5 because he can have them that way or not. No other figure in Rosslyn holds in feet in this strange symbolic way so again the Null Hypothesis is being given the full benefit of any doubt.
6. The probability that the ceremony is taking place between two pillars by chance is 0.5 because the alternative would be not to place the two pillars there.
7. The probability that a Templar is holding the noose by chance is 0.5 and this is generous towards the Null Hypothesis because the rope could be loose or held by somebody who is not a Templar. In the modern Masonic ceremony the rope is held by the senior deacon whilst the candidate takes his oath hold the bible.
I now needed to consider the possibility of all these seven probabilities occurring at the same time. As you know from the exercises in probability to find the composite probability I must multiply the separate probabilities together.
So the highest possible probability of the null hypothesis being true is
(0.5)*(0.5)*(0.5)*(0.5)*(0.5)*(0.5)*(0.5) which works out as 0.0078
So there are only eight chances in a thousand that all these elements linking Freemasonry to Templarism and Sir William St Clair are there by co-incidence. This probability is less than the 95% confidence level one in 20 and less than the 99% confidence level of one in 100. There is only one chance in 128 of the links being co-incidence. On this evidence I reject the null hypothesis, that leaves me with a strong claim that Sir William was linked to Freemasonry in 1440.
So hypothesis testing can be used for all sorts of useful purposes both business and otherwise.
Return to Main Index