STA309 Exam II
Probability and Sampling
•
Random phenomenon:
if individual
outcomes are uncertain
but there is nonetheless a
regular distribution of outcomes in a large number of repetitions
•
Parameters and Statistics:
o
Parameter:
describes the population; usually unknown. written as Greek letters (like µ
and
σ
).
o
Statistic:
describes a sample; computed from the sample data; written w/ letters (like
J and
s
).
J is a random variable
(numerical outcome of a random phenomenon).
J is rarely exactly right.
use statistics to est. parameters. Use J, to est. pop, µ
.
•
Sampling Variability:
value of a statistic (like J) will be different for different samples.
•
Randomness:
individual result can’t be predicted (short run); pattern/regular distribution emerges after repetitions (long run).
o
** To reduce bias, use random sampling. To reduce variability, use a larger sample.**
o
Probability:
probability of any outcome of a random phenomenon is the propor. of times the outcome occurs in a very long series of repetitions.
o
Independence:
the outcome of one trial does not influence the outcome of any other.
To explore randomness, you need a long sequence of independent trials
•
Law of large numbers:
As the number of observations increases, the mean J of the observed values gets closer to the mean µ.
It is reasonable to use J to estimate µ.
The larger
the sample, J will come to the “truer” mean of the population.
•
Sample means:
o
Suppose J is the mean of an SRS of size
n
from a large population.
J is a random variable & unbiased estimator.
o
The distribution of J is called a
sampling distribution
; sampling distribution has mean µ and sd =
σ
/√n.
o
Averages are less variable than individual observations.
o
The results of large samples are less variable than the results of small samples.
o
If a pop. has the N(µ,
σ
) distribution, then sample mean J of
n
observations has the Normal dist: N(µ,
σ
/√n)
•
** Central Limit Theorem:
For any population with mean µ
and finite standard deviation
σ
, when
n
is large the
sampling distribution of the sample mean, J, is approximately
normal.
o
If the histogram is skewed, you probably did something wrong.
In the central limit theorem, even if the population is not normal, the sample population mean will look
normal if n is very large
Statistical Control
•
Statistical Control:
Process results will vary in spite of efforts to be consistent.
Ex: weight, dimensions of manufactured items.
Processes can be “controlled” to be within
acceptable limits.
A variable is said to be in statistical control (or in control) when it continues to be described by the same distribution when observed over time.
•
Statistical process control
o
Purpose: to monitor a process so that any changes can be detected and quickly corrected.
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 Spring '07
 Gemberling
 Statistics, Normal Distribution, Probability, Standard Deviation, Null hypothesis, Statistical hypothesis testing

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