CriticalCriticalProbabilityProbability
When analyzing data, a scientist draws conclusions by observing that some results from the data are not explainable by chance alone. There are criteria for what results would be unlikely enough to have occurred purely by chance. These criteria are widely, but not universally agreed upon. In our course, we will use the conventions used in the biological sciences.
In summary, to draw conclusions and do hypothesis testing we need to agree on what is
The probability of winning the Lotto 6/49 jackpot on one ticket is$$P=\frac{1}{13\,983\,816}=0.0000000715$$This event is considered to be extremely unlikely.
Probability of getting killed by lightning in a year in Canada is (85% of the fatalities are men)$$P=\frac{1}{4\,000\,000}=0.00000025$$This event is also considered to be extremely unlikely.
Probability that one out of $24$ people will be lefthanded is$$P(X=1)={}_{24}C_1\;(0.1)^1\, (0.9)^{23}=0.2127 $$This event is not considered to be unlikely.
Statisticians conventionally adopt three critical probability values:
An event which is predicted to occur with probability $ P \leq 0.05 $ is considered to beunlikelyorstatistically significant.
An event which is predicted to occur with probability $ P \leq 0.01 $ is considered to bevery unlikely orstatistically highly significant.
An event which is predicted to occur with probability$ P \leq 0.001 $ is considered to beextremely unlikelyorstatistically very highly significant.
On average, there are $2.7 $ sizeable potholes per km on the roads of Mirabel in mid-March. Within $ 1\, \mathrm{km} $ of Yvan's house there are$\, 6\, $sizeable potholes. How likely is this to happen at random? Is the result statistically significant?
This is a Poisson distribution with $ \mu =2.7 $. The probability of having exacly 6 potholes in $ 1\, \mathrm{km}
$ is$$P(X= 6)=\frac{e^{-2.7}\cdot 2.7^6}{6!}=0.036$$Since $0.05>0.036>0.01$ this result is statistically significant but not highly significant.
The historical probability for a significant (daily) snowfall in Montreal in March is $0.074$. What is the probability that there will be a significant snowfall in March in Montreal on 5 days?
This is a binomial distribution with $n =31$ and $p =0.074$. The probability of having 5 days of snowfall in March is $$P(X=5)= {}_{31}C_5\; (0.074)^5\, (1-0.074)^{26}=0.051 $$Sive $0.051>0.05$ this result is not unlikely (not statistically significant).