2. • Confidence interval: used to describe the amount of uncertainty
associated with a sample estimate of a population parameter
• How to interpret a CI:
You try first: How would you interpret this: “There is a 95% CI that
states that the population mean is less than 150 but greater than 75.”
incorrect interpretation: there is a 95% chance that the population mean falls between 75
and 150
**since the population mean is a population parameter, the population mean is a
constant, not a random variable
Let’s take a look at what the confidence level means first before we talk about the correct
interpretation
Confidence Level: describes the uncertainty associated with a sampling method
What does this mean?
Suppose we use a sampling method to select different samples and compute the
interval estimate for each sample. Then a 95% confidence level says that we should expect
95% of the interval estimates to include a population parameter and the same for an other
confidence level.
3. Confidence Interval Data
Requirements
Need ALL three:
• Confidence Level
• Statistic
• Margin of Error
• With the above information, the range of the CI is defined by:
sample statistic + margin of error
• Uncertainty associated with the CI is specified by the CL
If Margin of Error not given, must calculate:
ME= Critical Value x Standard Deviation of Statistic
OR
ME= Critical Value x Standard Error of Statistic
4. How to Construct a Confidence
Interval
1. Identify a sample statistic
{choose the statistic that you will use to estimate
the population parameter (sample mean, sample
proportion, etc.)}
2. Select a confidence level (usually 90%, 95%, 99%)
3. Find Margin of Error (ME)
4. Specify Confidence Interval:
Sample statistic + ME
5.
6.
7.
8.
9. ** where z* or t* can be found using the tables and represents the standard error
10.
11.
12. How confidence intervals behave
Typically a person performing an observational study chooses the confidence he
desires and the margin of error follows from this choice. We usually want high
confidence and a small margin of error, but we cannot have both. There is usually
a trade-off.
• If we ask for high confidence, we have to allow ourselves a large margin of error.
Example: If I want to predict your average in a course with 99% confidence, I
might say that I am 99% confident that you will get a 75% with a margin of error
of 25%.
That is saying that we are 99% confident that you will get between 50% and 100%. Notice that
this doesn’t say much other than you will probably pass the course.
• If we want a small margin of error, we have to ask for a smaller confidence level.
Example: If I want to predict your average with a margin of error of 2 points, I
might say that I am 50% confident that you will get a 92% with a margin of error
of 2 percentage points.
That is saying that I am 50% confident that you will get between a 90 and 94 in the course.
Again, the small range is impressive but with 50% confidence, I am not very confident at all. It is
a coin flip.
13.
14. Sample Size for Desired Margin of Error
• Sometimes we wish to establish a specified margin of error for a certain confidence level. That fixes
z* and σ certainly cannot change. The only way we can achieve what we want is to change n, the
sample size.
