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 Four Summary and Key Takeaways
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From the course by University of London
Probability and Statistics: To p or not to p?
ratings
From the lesson
On Your Marks, Get Set, Infer!
Meet the Instructors
Dr James AbdeyAssistant Professorial Lecturer, LSE
University of London
So now, we've concluded week four.
Let's just consider some of the key takeaways from the last few sessions.
So we've started to delve into the world of
statistical inference whereby there is some population of interest,
and we'd like to know something about it.
However, due to the likely size of the population,
it won't be feasible to consider every member of that population.
Hence, we will take a sample and calculate various statistics of interest from
that sample and use these values to infer
the corresponding characteristics in that wider population.
So how do we take a sample from a population?
Well, our key goal is to try and obtain a representative sample.
As we've seen from historic examples,
easier said than done.
But we do have a variety of sampling techniques available
to us with their inevitable merits and limitations.
So at the headline level,
we might wish to distinguish between non-random sampling,
which are non-probability based sampling techniques,
and their random or probabilistic counterparts.
So they will have various pros and cons in terms of time,
money and effort to conduct these different types of
sampling and also the level to which we can apply statistical inference procedures.
So we gave a few examples of things like the convenience, judgment,
quota and snowball sampling on the non-random front and simple random sampling,
systematic, stratified and cluster sampling on the random front.
We then moved into some, perhaps,
slightly more abstract territory whereby
we introduced the concept of a sampling distribution.
So to appreciate that as we obtain different samples,
we'll get different members of the population within those samples.
And hence, any statistics we calculate such as the sample mean x-bar,
which has been our main focus as far as this course is concerned, but of course,
we could relate this to things like
sample variances or sample standard deviations as well.
Inevitably, the values of
these descriptive statistics will vary from one sample to another.
So a sampling distribution is simply the probability distribution of some statistic,
recognizing the variation which typically occurs within these statistics.
So of interest to us, as we said,
was the primary parameter of interest,
the mean of a population, i.e.,
that main measure of central tendency,
that measure of location.
I'm going to use that simple descriptive statistic of x-bar,
the sample mean as our estimator of mu.
So we looked at a very simple example of deriving a sampling distribution from scratch,
and then we considered the more generic case whereby there was
an assumption of a normally distributed population,
and we considered that theoretical sampling distribution of x-bar there.
So having derived to sampling distribution,
I invited you to consider any observed value of x-bar to
be viewed as a random drawing from said sampling distribution.
We know that on average,
the expectation of x-bar is equal mu,
which simply equates to in the long run, i.e.,
on average point estimate as given by
the sample mean is correct in estimating the population mean but only on average.
There's inevitable uncertainty in any point
estimates we derive due to the potential that,
by chance, our random sample does happen to
deviate a bit from the characteristics of the population.
So we rounded off week four with
a brief discussion of how we could convert a point estimate,
the sample mean x-bar into an interval estimate or a confidence interval.
We briefly examined the margin of error associated with
a confidence interval for a mean and we considered
the three parameters which affected the width of the interval,
some of which are under our control, others not, namely,
the level of confidence and hence,
the Z value in the formula seen that is within our control.
And other things equal, we prefer to be more confident than less,
the sample size and also within our control.
And other things equal, the larger the sample size,
the more precise any estimate is likely to be and hence,
the narrower, the less wide the confidence interval would be.
And thirdly, the level of variation which exists in the population or when
unknown proxied by the variation within our sample that is not within our control,
but other things equal, the more heterogeneous,
the more, the greater the variation which exists in the population.
Then of course, that leads to more uncertainty in any estimation,
and hence, the wider the confidence intervals tend to be.
So armed now with these concepts of sampling distributions and confidence intervals,
we now look ahead to week five where we consider
another major branch of statistical inference, namely hypothesis testing.
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