We live in an uncertain and complex world, yet we continually have to make decisions in the present with uncertain future outcomes. Indeed, we should be on the look-out for "black swans" - low-probability high-impact events. To study, or not to study? To invest, or not to invest? To marry, or not to marry? While uncertainty makes decision-making difficult, it does at least make life exciting! If the entire future was known in advance, there would never be an element of surprise. Whether a good future or a bad future, it would be a known future. In this course we consider many useful tools to deal with uncertainty and help us to make informed (and hence better) decisions - essential skills for a lifetime of good decision-making. Key topics include quantifying uncertainty with probability, descriptive statistics, point and interval estimation of means and proportions, the basics of hypothesis testing, and a selection of multivariate applications of key terms and concepts seen throughout the course.
Week Five Summary and Key Takeaways
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來自 University of London 的課程
Probability and Statistics: To p or not to p?
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從本節課中
To p Or Not To p?
與講師見面
Dr James AbdeyAssistant Professorial Lecturer, LSE
University of London
[MUSIC]
We've now reached the end of week 5 of this introductory MOOC looking at
probability and statistics.
So what are the key takeaways from week five?
So here we've moved into the realm of a decision theory,
specifically hypothesis testing.
So we began with a simple legal analogy, which hopefully everyone can
easily relate to, whereby a jury has to decide on a defendants guilt or
innocence based on the evidence i.e., the data provided to them in the courtroom.
So of course the jury was making a binary decision, finding the defendant guilty or
not guilty, just as in statistical hypothesis testing
we are deciding whether or not to reject some null hypothesis.
But we also noted that juries don't always get it right, and
sometimes they make mistakes, not ideal, but it's not an ideal world, and
we introduce the concepts of Type I errors and Type II errors.
Effectively, we could think of these as false positives and
false negatives, respectively.
Now which of these is worse, which is more problematic,
we are now going into subjective terrain, however, on balance,
we tend to consider Type I errors to be more problematic than Type II errors.
So when we design a hypothesis test we seek to control four
of the probability of committing a Type I error, and we saw that we achieved this
using our significance level denoted generically by alpha.
Now what is this appropriate significance level to choose?
Well remember, we were looking for
the statistical equivalent of being beyond a reasonable doubt.
Now again, very much a subjective decision here, but nonetheless convention
tends to suggest a 5% significance level is often quite appropriate.
So having considered those main mechanics of hypothesis testing, and
the possible errors which could occur, we then went into our to p, or
not to p discussion, i.e., the introduction of the term of a p-value.
So this is an instrument which allow us to very easily make this a binary
decision of whether or not to reject some null hypothesis,
whereby a p-value is simply a probability.
So a value between zero and one, and
we compare this to our chosen significance level,
which is also a probability i.e., the probability of committing a Type I error,
and we introduce that very simple decision rule, that should the p-value be below
the significance level we would deem this to reflect statistically significant
evidence and hence we would be justified in rejecting our null hypothesis.
But be warned, of course, rejection of the null hypothesis
either leads to a correct decision, but of course, it might be a Type I error.
Unfortunately, we won't know which of those two events has occurred.
Similarly, if our p-value exceeded, so it was above the significance level,
we would not reject the null hypothesis.
Of course, be warned, yet again, we hope we are right and
have reached the correct decision, but of course an error may have occurred,
and it might have been a Type II error.
Again, we will not know which of those two eventualities has taken place.
Of course we will proceed to, assuming we've reached the correct decision,
but please keep in mind the risk that a Type I or
Type II error could have been committed.
Whenever the jury returns a verdict of guilty,
they will hope they got it right, but there'll be those niggling doubts
in some of those juror's minds about what if that person truly was innocent.
Just as when they return the verdict of not guilty, they hope they're right, but
potentially they may just have let an offender back out onto the streets.
So having introduced the p-values and those two key influences,
which affect the magnitude of a p-value, namely the effect size influence and
the sample size influence, we've then moved onto a simple,
numerical illustration of conducting a hypothesis test about a population mean.
So for that we gave the sort of ever day example of the volume of water within
a mineral water bottle, and considered how we may test the claim made
by a manufacturer, based on a random sample of those particular products.
Then the wrap up week five looked at introducing the central limit theorem,
looking at a particular case of the sample mean when viewed as a sample proportion.
Very useful when applied to Bernoulli sampling.
A natural case and point
is when we consider political preferences in the run up to some election.
So, having viewed the sample proportion as a special case of the sample mean,
through the central limit theorem, we noted the sum of distribution of P,
upon which we subsequently presented confidence interval firmly,
and the hypothesis testing procedure related to statistical
inference about a single population proportion.
So this wraps up week five.
We have one week remaining, or week six, where we're going to see an eclectic
mix of possible applications of statistical methods to sort of incentivize
you to prolong your studies of probability and statistics beyond this MOOC.
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