TOPIC 8:
SAMPLING DISTRIBUTIONS
•
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How sample proportions vary around the population
proportion
How sample means vary around the population mean
Populations and samples
• We are often interested in population parameters.
• Since complete populations are difficult (or impossible) to collect data
on, we use sample statistics as point estimates for the unknown
population parameters of interest.
• Sample statistics vary from sample to sample.
• Quantifying how sample statistics vary provides a way to estimate the
margin of error associated with our points estimate.
• For this topic, let’s try to understand how point estimates vary from
sample to sample.
Suppose we randomly sample 1,000 adults from each state in the US.
Would you expect the sample means of their heights to be the same,
somewhat different, or very different?
Populations and samples
• We are often interested in population parameters.
• Since complete populations are difficult (or impossible) to collect data
on, we use sample statistics as point estimates for the unknown
population parameters of interest.
• Sample statistics vary from sample to sample.
• Quantifying how sample statistics vary provides a way to estimate the
margin of error associated with our points estimate.
• For this topic, let’s try to understand how point estimates vary from
sample to sample.
Suppose we randomly sample 1,000 adults from each state in the US.
Would you expect the sample means of their heights to be the same,
somewhat different, or very different?
Not the same, but only somewhat different.
How sample proportions vary around
the population proportion
Example: do you smoke?
30,000 college students were asked if they smoke.
Their answers are summarized below.
25.87% of the college
students reported that
they smoke.
Example: do you smoke?
30,000 college students were asked if they smoke.
Their answers are summarized below.
25.87% of the college
students reported that
they smoke.
Suppose this is our
population, and we
want to estimate this
percentage by choosing
a sample.
A random sample of 1000 students
Our statistic: 27.60%
(proportion of 1000
randomly selected
students that smoke)
Parameter: 25.87%
(proportion of all 30,000
students that smoke).
A random sample of 1000 students
Our statistic: 27.60%
(proportion of 1000
randomly selected
students that smoke)
Parameter: 25.87%
(proportion of all 30,000
students that smoke).
Our statistic is quite
close. Were we lucky, or
is there a reason?
Sampling Distribution
We repeatedly take a random sample of 1000 students and
compute the statistic (proportion that smoke). Below is a histogram
displaying the statistics computed for many samples.
Sampling Distribution
We repeatedly take a random sample of 1000 students and
compute the statistic (proportion that smoke). Below is a histogram
displaying the statistics computed for many samples.
Mean = .2583
Standard Deviation = 0.0140
Sampling Distribution
We repeatedly take a random sample of 1000 students and
compute the statistic (proportion that smoke). Below is a histogram
displaying the statistics computed for many samples.
Mean = .2583
Standard Deviation = 0.0140
By the empirical rule,
95% of the sample
proportions are between
0.2303 and 0.2863
The central limit theorem (proportions)
The distribution of the sample proportion is well approximated
by the normal model:
Where p is the population proportion and n in the sample
size. SE is the standard error, which is the standard
deviation of the sampling distribution.
•It wasn’t coincidence that our sample proportion was “close” to the
population proportion.
•It wasn’t coincidence that the sampling distribution was centered at the
population proportion and approximately normal.
•Note: if the sample size increases, then the standard error decreases.
Central limit theorem (CLT): conditions for proportions
Certain conditions must be met for the CLT to apply:
1. We have a random sample from the population
2. The sample is large enough so that we see at least 5
observations of both possible outcomes
How sample means vary around the
population mean
Example: what do MLB players make?
The salaries of 16,383 Major League Baseball players are displayed in
the histogram below.
Mean salary for these
players is $1,265,466
Suppose this is our
population, and we
want to estimate this
mean by choosing a
small sample.
A random sample of 1000 players
The salaries of 16,383 Major League Baseball players are displayed in
the histogram below.
Our statistic: $1,261,780.70
(mean salary of randomly
selected 1,000 players)
Parameter: $1,265,466
(mean salary for all 16,383
players).
Our statistic is quite close.
Again, were we lucky or is
there a reason?
Sampling Distribution
We repeatedly take a random sample of 1000 players and compute
the statistic (mean salary). Below is a histogram displaying the
statistics computed for many samples.
Sampling Distribution
We repeatedly take a random sample of 1000 players and compute
the statistic (mean salary). Below is a histogram displaying the
statistics computed for many samples.
Mean = $1,266,084.20
Standard deviation = $66,220.87
By the empirical rule, 95% of the
sample means are between
$1,133,642.46 and $1,398,525.94
The central limit theorem (means)
The distribution of the sample mean is well approximated by
the normal model:
Where μ is the population mean and n in the sample size.
SE is the standard error, which is the standard deviation of
the sampling distribution.
•It wasn’t coincidence that our sample mean was “close” to the population
mean.
•It wasn’t coincidence that the sampling distribution was centered at the
population mean and approximately normal.
•Note: if the sample size increases, then the standard error decreases.
Central limit theorem (CLT): conditions for means
Certain conditions must be met for the CLT to apply:
1. Independence: sampled observations must be independent. To help
ensure this, we often follow the rules
• random sampling/assignment is used, and
• n < 10% of the population
2.
Sample size/skew: Either the population distribution is approximately
normal, or if it is skewed, the sample size is large.
• The more skewed the population distribution, the larger sample
size we need for CLT to apply.
• For moderately skewed distributions n > 30 is a widely used rule
of thumb.