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

• Purpose

• The purpose of a confidence interval is to estimate a population mean or proportion based on a sample mean or proportion.

• 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 l.96• Other values can be found in a Z table

• For a proportion this is a T value• The value associated with a 95% confidence

interval is l1.98• 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 do it.

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

Examples of Constructing Confidence Intervals

EXAMPLE

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

Research Question What is the mean GPA of participants after participating in our tutoring program?

Relevant InformationSample GPA = 3.224Sample Size = 40T Value = 1.98 (value for a 95% confidence interval.)Standard Deviation of sample = .1864

Step by Step InstructionsStep 1 1.98 times .1864=.3691Step 2 Square root of 40 = 6.3246Step 3 .3691/6.3246=.0584Step 4 3.224 plus .0584=3.2824Step 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 IntervalsFactors that influence the width of the interval.

Standard ErrorStandard error estimates the dispersion of the data in the population.

The more dispersed your data the wider is your confidence interval*

Sample SizeThe 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.

ProbabilityThe 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 CaseWhen estimating a proportion then the closer to the middle (.5) the

proportion is the wider will be your confidence interval.

*The margin or 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

• Dr. Carol Albrecht• USU Extension• Assessment Specialist• Carol.albrecht@usu.edu• 979-777-2421