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Estimation & Hypotheses Testing (Chapter 2/EQT271)
ESTIMATION
1. Statistical inference is a process of drawing an inference about the data statistically. It
concerned in making conclusion about the characteristics of a population based on
information contained in a sample. Since populations are characterized by numerical
descriptive measures called parameters, therefore, statistical inference is concerned in
making inferences about population parameters.
2. There are two types of inference: estimation and hypothesis testing
3. The objective of estimation is to determine the approximate value of a population
parameter on the basis of a sample statistic.
4. The objective of estimation is to determine the approximate value of a population
parameter on the basis of a sample statistic.
There are two types of estimators:
a. Point Estimator
b. Interval Estimator
5. A point estimator draws inferences about a population by estimating the value of an
unknown parameter using a single value or point.
We saw earlier that point probabilities in continuous distributions were virtually zero.
Likewise, we’d expect that the point estimator gets closer to the parameter value with an
increased sample size, but point estimators don’t reflect the effects of larger sample sizes.
Hence we will employ the interval estimator to estimate population parameters
Estimation & Hypotheses Testing (Chapter 2/EQT271)
6. For example, the sample mean X is a point estimate of the population mean  . Similarly,
the sample proportion P̂ is a point estimate of the population proportion P .
7. Choosing the right point estimators to estimate a parameter depends on the properties of
the estimators it selves. There are four properties of the estimators that need to be satisfied
in which it is considered as best linear unbiased estimators. The properties are:
i.
Unbiased
ii.
Consistent
iii. Efficient
iv. Sufficient
UNBIASEDNESS: An estimator is said to be unbiased if in the long run it takes on
the value of the population parameter. That is, if you were to draw a sample,
compute the statistic, repeat this many, many times, then the average over the
entire sample statistics would equal the population parameter.
CONSISTENCY: A sequence of estimators is said to be consistent if it converges in
probability to the true value of the parameter
EFFICIENCY: An estimator is said to be efficient if in the class of unbiased
estimators it has minimum variance
SUFFICIENCY: We say that an estimator is sufficient if it uses all the sample
information. The median, because it considers only rank, is not sufficient. The
sample mean considers each member of the sample as well as its size, so is a
sufficient statistic. Or, given the sample mean, the distribution of no other
statistic can contribute more information about the population mean. We use the
factorization theorem to prove sufficiency. If the likelihood function of a random
variable can be factored into a part which has as its arguments only the statistic
and the population parameter and a part which involves only the sample data, the
statistic is sufficient.
8. An interval estimator draws inferences about a population by estimating the value of an
unknown parameter using an interval.
That is we say (with some ___% certainty) that the population parameter of interest is
between some lower and upper bounds.
Estimation & Hypotheses Testing (Chapter 2/EQT271)
For example, a < X < b is an interval estimate of the population mean  . It indicates that
the population mean is greater than a but less than b.
The population parameter is likely to occur within that range at a specified probability.
Specified probability is called the level of confidence.
Ex: States how much confidence we have that this interval contains the true population
parameter. The confidence level is denoted by (1   )100%
9. The width of the confidence interval estimate is a function of the confidence level, the
population standard deviation, and the sample size.
10. A larger confidence level produces a wider confidence interval:
11. Larger values of standard deviation produce a wider confidence interval
12. Increasing the sample size decreases the width of the confidence interval while the
confidence level can remain unchanged.
Note: this also increases the cost of obtaining additional data
Estimation & Hypotheses Testing (Chapter 2/EQT271)
HYPOTHESIS TESTING
1. Hypothesis Testing: A hypothesis is a claim (assumption) about one or more population
parameters.
a. Average price of a six-pack in the U.S. is μ = $4.90
b. The population mean monthly cell phone bill of this city is: μ = $42
c. The average number of TV sets in U.S. Homes is equal to three; μ = 3
For more, please refer to Notes Hypothesis Testing (Allan Bluman)-download at portal
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