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Questions about the Assignment
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Population parameter versus sample statistic
Uncertainty in estimates
Sampling distribution
Confidence interval
Understanding Inference:
Confidence Intervals I
Population
Sample
Sampling
Statistical
Inference
The Big Picture
A sample statistic is a number computed from sample data.
(e.g., sample mean: mean income of the people in the sample)
A population parameter is a number that describes some
aspect of a population.
(e.g., population mean: mean income of the entire population)
We usually have a sample statistic and want to make
inferences about the population parameter.
Statistic vs. Parameter
Population
Sample
Sampling
Statistical
Inference
The Big Picture
Statistic
Parameter
Statistic vs. Parameter
Sample
Statistics
Population
Parameters
Mean ݔഥ^ μ^ (mu)
Proportion ̂^ p
Std. Deviation s^ ߪ^ (sigma)
Correlation r^ ρ^ (rho)
Slope b^ β^ (beta)
Gallup surveyed 1,500 Americans between June 9th-11th^ 2012 and 49% of these people approved of the job Barack Obama is doing as president.
What is the population?
What is the sample size?
Is this categorical or quantitative variable?
For categorical variables, what sample statistic are we
interested in?
Sample statistic: (sample proportion)
Based on this sample statistic, what do you think is the true proportion of Americans who approve of the job Barack Obama is doing as president?
Population parameter: (population proportion)
Obama’s Approval Rating
http://www.gallup.com/poll/113980/Gallup-Daily-Obama-Job-Approval.aspx
~330million (All Americans) 1, Categorical
Sample proportion
p =?
The sample statistic gives a point estimate (a single number)
for the population parameter.
Usually, it is more useful to provide an interval estimate
which gives a range of plausible values for the population
parameter:
interval estimate = point estimate േ margin of error
How do we determine the margin of error???
Point and Interval Estimates
Point Estimate: ො =.
Interval Estimate: 0.49 േ 0.
Between 46% and 52% of Americans currently approve of
the job Obama is doing as president.
Obama’s Approval Rating
point estimate
margin of error
The population parameter is a fixed value.
Sample statistics vary from sample to sample. They will not
match the population parameter exactly.
For a given sample statistic, what are plausible values for the
population parameter? How much uncertainty surrounds the
sample statistic?
It depends on how much the sample statistic varies from
sample to sample!
Important Points
What proportion of Reese’s pieces are orange?
Reese’s Pieces
The sample size does not affect the shape of the sampling distribution.
The sample size does not affect the center of the sampling distribution.
The sample size does affect the spread of the sampling distribution.
As the sample size increases, the spread decreases.
The Importance of Sample Size
n = 1,
n = 200
n = 50
Each dot represents a sample statistic. The number of samples taken to generate these sampling distributions is the same. What varies for each sampling distribution is the size of the sample taken to calculate the sample statistic.
3 Sampling
Distributions The sample size influences the spread of the sampling distribution(i.e., the variation in sample statistics), which influences the margin
of error for our estimate of the population parameter.
If we increased the sample size to 100, the standard deviation of the sampling distribution will... A. increase B. decrease C. remain the same
and the margin of error for our point estimate will… A. increase B. decrease C. remains the same
Sample Size
For each sample, the sample statistic (i.e., the proportion of orange pieces) would be closer to the proportion of the population and thus closer to each other.
Increasing the sample size will cause the standard deviation of the
sampling distribution to decrease.
Let’s Test Our Hypothesis
http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html
Hypothesis
If you take random samples, the sampling distribution will be
centered around the true population parameter.
If sampling bias exists (if you do not take random samples),
the sampling distribution may provide inaccurate information
about the true population parameter.
Random Samples
A confidence interval for a population parameter estimate is an interval computed from sample data that will contain the true population parameter for a specified proportion of all samples.
The confidence level is the proportion of samples whose intervals contain the true population parameter.
The confidence level indicates how confident we are that our interval contains the population parameter.
A 95% confidence interval will contain the true population parameter for 95% of all samples. We are 95% confident that the true population parameters falls within this range.
Confidence Intervals
The population parameter () is fixed. It is typically not known. The sample statistic ( x (^) i ) is random. It depends on the sample. The confidence interval ( x (^) i േ2SD)* is random. It depends on the sample statistic. The sampling distribution is comprised of the sample statistics and is centered on the population parameter. 95% of the sample statistics will fall within 2 standard deviations of the population parameter. 95% of the sample intervals will contain the population parameter.
Confidence Intervals
Sample Statistic
2 SDs
http://bcs.whfreeman.com/ips4e/cat_010/applets/confidenceinterval.html
Sampling Distribution Population Parameter
Confidence Interval p
----------95%--------
*The standard deviation used to calculate the confidence interval is the standard deviation of the sampling distribution (not the sample distribution).
Population Proportion
A 95% confidence interval can be created by:
sample statistic േ 2 ൈ standard deviations
point estimate േ margin of error
The point estimate is calculated from our sample.
The margin of error is calculated from the sampling distribution.
Confidence Intervals
The standard deviation of the sampling distribution (i.e., the
distribution of sample statistics) is called the standard error (SE).
This is done to clearly distinguish it from the standard deviation
of the sample distribution.
Standard Error:
The Standard Deviation of the Sampling Distribution
To create a plausible range of values for a parameter:
- Take many random samples from the population, and compute the sample statistic for each sample.
- Compute the standard error as the standard deviation of all these statistics.
- Use: sample statistic ± 2 × standard error
One small problem…
Often we only have one sample!
How can we calculate the variation in sample statistics,
if we only have one sample?
Summary
Part I: Graded Problems
3.12, 3.16, 3.24, and 3.
Part II: (Type up this assignment in a Word document)
Goto http://sda.berkeley.edu/cgi-bin/hsda?harcsda+gss Find 3 quantitative variables and for each variable find another quantitative variable that you think is associated with it. Conduct a correlation test to see how correlated they are. For each pair of variables provide the following information: Variable names Question related to the variable Explain in your own words what this variable is measuring The unit used to measure the variable (e.g., years, dollars, inches, etc.) Min, Max, Mean, Median, Standard Deviation (Std Dev) The correlation score An interpretation of the correlation score
Assignment
Calculating Correlations from the GSS
Under the “Analysis” tab, click on the “Correlation matrix” tab.
Enter the names of two quantitative variables here.
Click on this button and the correlation statistics will open up in a new window.
Calculating Correlations from the GSS
This is what will pop up in the new window. This is the correlation ( r ) score for the two variables
