I’ve spent much of the last week helping students with “Hypothesis Testing” as they prepare for their A level exams in the next few months.

Fed up of wading through connived examples, and upon stumbling across perhaps the best headline I’ve read in sometime (“Man United regress to the mean after Solskjaer bounce“) I thought I’d use a bit of A level maths to see if the Solskjaer bounce was real or just another Norse myth.

(I’m about to walk through how this may be presented as an A level question, followed by my worked solution, so my less mathematically minded readers may want to skip the next few lines.)

Since taking over as manager of Manchester United in December 2018, Ole Gunnar Solskjaer (OGS) has transformed the club, returning it to winning ways.

In the season to date, in all competitions, Man Utd have won 24 out of 43 games.

Since OGS was appointed manager, Man Utd have won 15 of their 21 games.

Does the data support the theory that OGS has transformed the club at the 5% significance level?

Let p be the probability that Man Utd win a match. They have won 24 out of 43, so the probability of winning is 24/43 = 0.558

H_{o}:p=0.558

H_{1}:p>0.558

The null hypothesis is that the probability is 0.558, the alternative hypothesis is that the probability of a win (under OGS) is greater

Let X be the number of wins in a sample of 21 games (the games that OGS was manager)

If H_{o} is correct then X~B(21,0.558)

We’re modelling the data as a Binomial distribution as there are two outcomes: win, don’t win.

Test statistic: X = 15 (the number of games OGS won)

P(X>=15) = 1 – P(X<=14)

= 1 – 0.8905 = 0.1095

We use our calculator in Binomial CD mode to find the cumulative probability of up to, and including 14, then take that away from 1 to get the probability that Man Utd would win 15 or more of their games under the null hypothesis (i.e OGS has made no difference), which works out to be 0.1095 or 10.95%)

0.1095 is not less than 0.05

Sometimes easier to think in percentages, even if we give answers as decimals. 10.95% is not less than 5% (our significance level)

So there is insufficient evidence to reject H_{o}, our null hypothesis

**(Non-mathematicians, start reading from here)**

At the 5% significance level, there is no evidence to support the theory that OGS has transformed the club.

Or, in other words, the Solskjaer bounce is probably just a myth.

We have shown that even if there had been no change of manager there was a 10.95% chance Man Utd would have won 15 out of the next 21 games.

In fact, even if we widened the significance level to 10%, the data still wouldn’t have supported a Solskjaer bounce. Winning 15 out of the 21 games since taking charge whilst unlikely was not so unlikely that it couldn’t have been due to chance rather than the genius that is OGS.

My theory: if we’d looked for the bounce a little earlier, we may have found evidence for it – perhaps the bounce peaked at around 15 games and Man Utd are, indeed, now regressing to the mean.

As ever, statistics raises as many questions as they answer, but it is good to be able to apply some A level mathematics to answer a “real” question.