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Sampling Distributions
Note: Homework 3 due 3/23
Probability distributions
• If we measure a random variable many times,
we can build up a distribution of the values it can
take.
• Imagine an underlying distribution of values
which we would get if it was possible to take
more and more measurements under the same
conditions.
• This gives the probability distribution for the
variable.
Continuous probability distributions
• Because continuous random variables can take
all values in a range, it is not possible to assign
probabilities to individual values.
• Instead we have a continuous curve, called a
probability density function, which allows us to
calculate the probability a value within any
interval.
• This probability is calculated as the area under
the curve between the values of interest. The
total area under the curve must equal 1.
Normal (Gaussian) distributions
• Normal (also known as Gaussian) distributions
are by far the most commonly used family of
continuous distributions.
• They are ‘bell-shaped’ –and are indexed by two
parameters:
– The mean m – the distribution is symmetric about this
value
– The standard deviation s – this determines the
spread of the distribution. Roughly 2/3 of the
distribution lies within 1 standard deviation of the
mean, and 95% within 2 standard deviations.
The probability of continuous
variables
• IQ test
– Mean = 100 and sd = 15
• What is the probability of randomly
selecting an individual with a test score of
130 or greater?
– P(X ≤ 95)?
– P(X ≥ 112)?
– P(X ≤ 95 or X ≥ 112)?
The probability of continuous
variables (cont.)
• What is the probability of randomly
selecting three people with a test score
greater than 112?
– Remember the multiplication rule for
independent events.
Introduction to Statistical
Inference
Chapter 11
Populations vs. Samples
• Population
– The complete set of individuals
• Characteristics are called parameters
• Sample
– A subset of the population
• Characteristics are called statistics.
– In most cases we cannot study all the
members of a population
Inferential Statistics
• Statistical Inference
– A series of procedures in which the data
obtained from samples are used to make
statements about some broader set of
circumstances.
Two different types of procedures
• Estimating population parameters
– Point estimation
• Using a sample statistic to estimate a population parameter
– Interval estimation
• Estimation of the amount of variability in a sample statistic
when many samples are repeatedly taken from a population.
• Hypothesis testing
– The comparison of sample results with a known or
hypothesized population parameter
These procedures share a
fundamental concept
• Sampling distribution
– A theoretical distribution of the possible
values of samples statistics if an infinite
number of same-sized samples were taken
from a population.
Example of the sampling
distribution of a discrete variable
p(x)
Binomial sampling distribution of an
unbiased coin tossed 10 times
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
Number of heads in 10 tosses
9
10
Continuous Distributions
• Interval or ratio level data
– Weight, height, achievement, etc.
• JellyBlubbers!!!
Histogram of the Jellyblubber
population
Repeated sampling of the
Jellyblubber population (n = 3)
Repeated sampling of the
Jellyblubber population (n = 5)
Repeated sampling of the
Jellyblubber population (n = 10)
Repeated sampling of the
Jellyblubber population (n = 40)
For more on this concept
• Visit
– http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
Central Limit Theorem
• Proposition 1:
– The mean of the sampling
distribution will equal the
mean of the population.
mx  m
• Proposition 2:
– The sampling distribution of
means will be approximately
normal regardless of the
shape of the population.
• Proposition 3:
– The standard deviation
(standard error) equals the
standard deviation of the
population divided by the
square root of the sample
size. (see 11.5 in text)
sx 
s
N
Application of the sampling
distribution
• Sampling error
– The difference between the sample mean and the population
mean.
• Assumed to be due to random error.
• From the jellyblubber experience we know that a
sampling distribution of means will be randomly
distributed with
mx  m
sx 
s
N
Standard Error of the Mean and
Confidence Intervals
• We can estimate how
much variability there
is among potential
sample means by
calculating the
standard error of the
mean.
s.e.  s x 
s
N
Confidence Intervals
• With our Jellyblubbers
– One random sample (n = 3)
• Mean = 9
– Therefore;
• 68% CI = 9 + or – 1(3.54)
• 95% CI = 9 + or – 1.96(3.54)
• 99% CI = 9 + or – 2.58(3.54)
6.132
s.e.  s x 
 3.54
3
Confidence Intervals
• With our Jellyblubbers
– One random sample (n = 30)
• Mean = 8.90
– Therefore;
• 68% CI = 8.90 + or – 1(1.11)
• 95% CI = 8.90 + or – 1.96(1.11)
• 99% CI = 8.90 + or – 2.58(1.11)
6.132
s.e.  s x 
 1.11
30
Hypothesis Testing (see handout)
1.
2.
3.
4.
5.
6.
State the research question.
State the statistical hypothesis.
Set decision rule.
Calculate the test statistic.
Decide if result is significant.
Interpret result as it relates to your
research question.