Download Estimation

Topics: Inferential Statistics
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Inference
Terminology
Central Limit Theorem
Estimation
– Point Estimation
– Confidence Intervals
• Hypothesis Testing
Inferential Statistics
• Research is about trying to make valid
inferences
• Inferential statistics: the part of statistics
that allows researchers to generalize their
findings beyond data collected.
• Statistical inference: a procedure for
making inferences or generalizations about
a larger population from a sample of that
population
How Statistical Inference Works
Basic Terminology
• Population: any collection of entities that
have at least one characteristic in common
• Parameter: the numbers that describe
characteristics of scores in the population
(mean, variance, s.d., etc.)
Basic Terminology (cont’d)
• Sample: a part of the population
• Statistic: the numbers that describe
characteristics of scores in the sample
(mean, variance, s.d., correlation
coefficient, reliability coefficient, etc.)
Basic Statistical Symbols
Basic Terminology (con’t)
• Estimate: a number computed by using the
data collected from a sample
• Estimator: formula used to compute an
estimate
The Process of Estimation
Types of Samples
• Probability
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Simple Random Samples
Simple Stratified Samples
Systematic Samples
Cluster Samples
• Non Probability
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Purposive Samples
Convenience Samples
Quota Samples
Snowball Samples
Limits on Inferences and Warnings
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Response Rates
Source of data
Sample size and sample quality
“Random”
Estimation
• Point Estimation
• Interval estimation
– Sampling Error
– Sampling Distribution
– Confidence Intervals
Interval Estimation
• Interval Estimation: an inferential
statistical procedure used to estimate
population parameters from sample data
through the building of confidence intervals
• Confidence Intervals: a range of values
computed from sample data that has a
known probability of capturing some
population parameter of interest
Sampling Error
• Samples rarely mirror exactly the
population
• The sample statistics will almost always
contain sampling error
• The magnitude of the difference of the
sampling statistic from the population
parameter
Sampling Distribution
• Sampling Distribution: a theoretical distribution
that shows the frequency of occurrence of values
of some statistic computed for all possible samples
of size N drawn from some population.
• Sampling Distribution of the Mean: A
theoretical distribution of the frequency of
occurrence of values of the mean computed for all
possible samples of size N from a population
Sampling Distribution of Mean
Sampling Distribution of Means and
Standard Error of the Means
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Population mean
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Central Limit Theorem
• The sampling distribution of means, for samples
of 30 or more:
– Is normally distributed (regardless of the shape of the
population from which the samples were drawn)
– Has a mean equal to the population mean, “mu”
regardless of the shape population or of the size of the
sample
– Has a standard deviation--the standard error of the
mean--equal to the population standard deviation
divided by the square root of the sample size
Sampling Distribution of 1000 Sample
Means
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Confidence Intervals
• A defined interval of values that includes the
statistic of interest, by adding and subtracting a
specific amount from the computed statistic
• A CI is the probability that the interval computed
from the sample data includes the population
parameter of interest
Factors Affecting Confidence Intervals
Various Levels of Confidence
• When population standard deviation is
known use Z table values:
– For 95%CI: mean +/- 1.96 s.e. of mean
– For 99% CI: mean +/- 2.58 s.e. of mean
• When population standard deviation is not
known use “Critical Value of t” table
– For 95%CI: mean +/- 2.04 s.e. of mean
– For 99% CI: mean +/- 2.75 s.e. of mean
95%Confidence Interval
95 times out of 100 the interval constructed
around the sample mean will capture
the population mean. 5 times out of 100 the
interval will not capture the population mean
95%
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99%Confidence Interval
99 times out of 100 the interval constructed
around the sample mean will capture
the population mean. 1 time out of 100 the
interval will not capture the population mean
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Effects of Sample Size
Process for Constructing Confidence
Intervals
• Compute the sample statistic (e.g. a mean)
• Compute the standard error of the mean
• Make a decision about level of confidence that is
desired (usually 95% or 99%)
• Find tabled value for 95% or 99% confidence
interval
• Multiply standard error of the mean by the tabled
value
• Form interval by adding and subtracting calculated
value to and from the mean