Download 25.1-Mathematical

Mathematical Statistics
İST 252
EMRE KAÇMAZ
B4 / 14.00-17.00
Mathematical Statistics
• In probability theory we set up mathematichal models of
processes that are affected by ‘chance’.
• In mathematical statistics or statistics, we check these models
against the observable reality.
• This is called statistical inferance.
Mathematical Statistics
• It is done by sampling, that is , by drowing random samples.
• These are sets of values from a much larger set of values that
could be studied at, called the population.
• An example is 10 diameters of screws drawn from a large lot of
screws.
• Sampling is done in order to see whether a model of the
population is accurate enough for practical purposes. If this is the
case, the model can be used for predictions , decisions, and
actions , for instance, in planning produciton, buying equipment,
investing in a business projects, and so on.
Mathematical Statistics
• Most important methods of statistical inference are estimation of
parameters, determination of confidence intervals, and hypothesis
testing, with application to quality control and acceptance
sampling.
• In the last section we give an introduction to regression and
correlation analysis, which concern experiments involving two
variables.
Random Sampling
• Mathematical statistics consists of methods for designing and
evaluating random experiments to obtain information about
practical problems.
• Such as exploring the relation between iron content and density of
iron ore, the quality raw material or manufactured products, the
efficiency of air conditioning systems, the performance of certain
cars, the effect of advertising, the reaction of consumers to a new
product, etc.
Random Sampling
• Random variables occur more frequently in engineering than one would
think.
• For example, properties of mass-produced articles (screws,
lightbulbs,etc.) always Show random variation, due to small differences
in raw material or manufacturing processes.
• Samples are selected from populations – 20 screws from a lot of 1000,
100, 5000 voters, 8 beavers in a wildlife conservation project – because
inspecting the entire population would be too expensive, timeconsuming, impossible or even senseless
• To obtain meaningful conclusions, samples must be random selections.
Random Sampling
• Each of the 1000 screws must have the same chance of being
sampled at least approximately. Only then will the sample mean
of a sample of size n = 20 be a good
approximation of the population mean μ; and the accuracy of the
approximation will generally improve with increasing n, as we shall
see. Similarly for other parameters (standard deviation, variance,
etc.)
Random Sampling
• Independent sample values will be obtained in experiments with
an infinite sample space S, certainly for the normal distribution.
• This is also true in sampling with replacement.
• It is approximately true in drawing small samples from a large
finite population(for instance, 5 or 10 of 1000 items).
• However, if we sample without replacement from a small
population, the effect of dependence of smaple values may be
considerable.
Random Sampling
• Random numbers help in obtaining samples that are in fact
random selections.
• This is sometimes not easy to accomplish because there are many
subtle factors that can bias sampling (by personal interviews, by
poorly working machines, by the choice of nontypical observation
conditions, etc.).
• Random numbers can be obtained from a random number
generator in Maple, Mathemtica, or other systems.
Example
• To select a sample of size n = 10 from 80 given ball bearings, we number
the bearings from 1 to 80.
• We then let the generator randomly produce 10 of the integers from 1 to
80 and include the bearings with the numbers obtained in our sample,
for example
or whatever.
• Random numbers are also contained in (older) statistical tables.
Example
• Representing and processing data were considered in the first
chapter. İn connection with frequency distributions.
• These are empirical counterparts of probability distributions and
helped motivating axioms and properties in properties in
probability theory.
Example
• The new aspect int his chapter is randomness: the data are
samples selected randomly from a population.
• Accordingly, we can immediately make the connection to first
chapter, using stem-and-leaf plots, box plots, and histograms for
representing samples graphically.
• Also, we now call the mean 𝑥 the sample mean
Example
• We call n the sample size, the variance s², the sample variance
• and its positive square root s the sample standard deviation.
• 𝑥, s², and s are called parameters of a sample; they will be
needed throughout this chapter.