Constructing and Interpreting
Confidence Intervals
Confidence Intervals
In this power point, you will learn:
Why confidence intervals are important in evaluation research
How to interpret a confidence interval
How to construct a confidence interval yourself
Confidence Intervals
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Purpose
• The purpose of a confidence interval is to estimate a
population mean or proportion based on a sample mean or proportion.
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Relationship with Margin of Error
• When estimating the population mean or proportion, we do not try to estimate the exact value. We construct an
interval within which we are 95% sure the population
means or proportion lies. The width of the interval on each side of the mean or proportion is referred to as the margin of error.
• Example…….If you estimate that the percent of farmers who will increase crop yield using the new fertilizer is 75%
and the margin of error is 5%, more technically you are stating that you are 95% sure the population percent is somewhere between 70% and 80%.
• Example using a Mean…..If you estimate the mean
number of bushels of corn to be 400 for farmers who used the new fertilizer and our margin of error is 10, then you are stating you are 95% sure the mean is somewhere between 390 to 410.
Constructing a Confidence Interval
• Constructing a confidence interval by hand
• Before you can construct a confidence interval by hand, you must have data from a random sample. You need:
• The sample size
• The proportion or mean you want to estimate
• The standard deviation of the mean you want to estimate
• A critical value associated with the probability
• For a proportion this is a Z value
• The value associated with a 95% confidence interval is 1.96
• Other values can be found in a Z table
• For a mean this is a T value
• The value associated with a 95% confidence interval is 1.98
• Other values can be found in a T table
Constructing a Confidence Interval
• It is relatively easy to construct a confidence interval.
The following slides tell you how to:
Construct a confidence interval using a proportion or percent
Statistical formula (most precise)
Formula used by Opinion-Polls (most commonly used)
Simplified formula used by Opinion-Polls (easiest and most conservative)
Construct a confidence interval using a mean
EXAMPLE of Confidence Interval for a Proportion
50% of our sample of 36 individuals indicated they would vote for Obama in this election.
Research Question: What percent of population will vote for Obama?
Relevant Information
p = .50 z = 1.96 for 95% confidence interval n = 36
Simple Instructions
1.96 times the square root of .50 (1 - .50) divided by 36 (number of class members who voted)
Take this number and add it to .50 and then subtract it from .50 (proportion who voted for Obama in sample)
Or Step by Step
Step 1 1-.50 = .50
Step 2 .50 times .50 = .25
Step 3 .25 divided by 36 (n) = 6.944 Step 4 Square root of 6.944 = .08333 Step 5 1.96 times .08333 = .16333 Step 6 .50 plus .16333 = .66333 Step 7 .50 minus .16333 = .33667
Step 8 Confidence interval is .33667 to .66333
34 to 66% of the population will vote for Obama. NOTE,
our confidence interval is so wide primarily because our sample is so
small.
We are 95% sure that the mean score is between 73.1 and 78.5.
Example of Confidence Interval for a Mean
Research Question
What is the mean GPA of participants after participating in our tutoring program?
Relevant Information Sample GPA = 3.224 Sample Size = 40
T Value = 1.98 (value for a 95% confidence interval.)
Standard Deviation of sample = .1864
Step by Step Instructions
Step 1 1.98 times .1864=.3691 Step 2 Square root of 40 = 6.3246 Step 3 .3691/6.3246=.0584
Step 4 3.224 plus .0584=3.2824 Step 5 3.224 minus .0584=3.1656
Conclusion: We are 95% sure that the mean GPA
for our entire group is between 3.1656-3.2824
Width of Confidence Intervals
Factors that influence the width of the interval:
Standard Error
Standard error estimates the dispersion of the data in the population. The more dispersed your data the wider is your confidence interval.*
Sample Size
The smaller the sample size that you used to construct a confidence interval then the wider the confidence interval. In layman’s terms, you can estimate a population mean or
proportion more accurately if you have a larger sample size.
Probability
The greater the probability, the wider will be your confidence interval.** Thus a 99% confidence interval is wider than a 95%
confidence interval. In layman’s terms, the more certain you want to be that the proportion or mean actually does lie within the interval the wider the interval must be.
Proportion as a Special Case
When estimating a proportion then the closer to the middle (.5) the proportion is the wider will be your confidence interval.
*The margin of error is directly related to the width of the confidence interval. The wider the interval, the greater the margin of error.
**Rule of thumb is to use a 95% confidence interval, but you can construct an interval with any degree of certainty.
Contact Information
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Dr. Carol Albrecht
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USU Extension
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Assessment Specialist
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[email protected]
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