UNDERSTANDING CONFIDENCE
INTERVAL ESTIMATES FOR THE SAMPLE
MEAN
Lesson Objectives
At the end of this lesson, you are expected to:
• define confidence level;
• define confidence interval;
• apply the normal curve concepts in computing
the interval estimate; and
• compute confidence interval estimates.
Pre-Assessment
Pre-Assessment
Lesson Introduction
Suppose we want to know the “true” average weight of
all the students in the population where the students in
this class belong. We can increase the precision of our
guess by getting as many random samples as we can
from the population where the students purportedly
come from.
Form five groups and name each Group A, Group B,
Group C, Group D, and Group E. Assume that these
groups are random samples.
Lesson Introduction
Group Tasks
• Using a weighing scale, find the weight of each group
member carefully.
• Compute the mean weight and the standard
deviation of each group.
• Compute the mean of the group means.
• How would you describe your group based on the
result of the computation?
• What is your estimate of the mean of the population
where your group seems to belong?
• Reflect on your estimation. Are you confident about
it? To what extent are you confident? Express your
confidence as a percentage.
Discussion Points
• An interval estimate, called a confidence interval,
is a range of values that is used to estimate a
parameter. This estimate may or may not contain
the true parameter value.
• The confidence level of an interval estimate of a
parameter is the probability that the interval
estimate contains the parameter. It describes
what percentage of intervals from many different
samples contain the unknown population
parameter
Discussion Points
Formula for Confidence Interval
  
E  z /2 

 n
However, when σ is not known (as is often the case), the
sample standard deviation s is used to approximate σ. So, the
formula for E is modified.
 s 
E  z /2 

 n
Discussion Points
• In computing a confidence interval for a
population mean by using raw data, round
off to one more decimal place than the
number of decimal places in the original
data.
• In computing a confidence interval for a
population mean by using a sample mean
and a standard deviation, round off to the
same number of decimal places as given
for the mean.
Discussion Points
Example 1
Find the estimate of the population mean μ
using the 95% confidence level.
Solution
Point Estimate
Solution
95% Confidence Interval
Example 2
A researcher wants to estimate the number of hours that 5year old children spend watching television. A sample of 50
five-year old children was observed to have a mean viewing
time of 3 hours. The population is normally distributed with a
population standard deviation α = 0.5 hours, find:
•
•
the best point estimate of the population mean
the 95% confidence interval of the population mean
Solution
Point Estimate
Solution
95% Confidence Interval
Exercises
1.
2.
3.
4.
What measure of central value best estimates the population mean?
Why is the interval estimate a preferred value for the population
parameter?
Given the information: the sampled population is normally distributed,
X– = 36.5, σ = 3, and n = 20.
a. What is the 95% confidence interval estimate for μ?
b. Are the assumptions satisfied? Explain why.
A sample of 60 Grade 9 students’ ages was obtained to estimate the
mean age of all Grade 9 students. X– = 15.3 years and the population
variance is 16.
• What is the point estimate for μ?
• Find the 95% confidence interval for μ.
• Find the 99% confidence interval for μ.
• d. What conclusions can you make based on each estimate?
Summary
• An interval estimate, called a confidence interval, is a
range of values that is used to estimate a parameter.
This estimate may or may not contain the true
parameter value.
• The confidence level of an interval estimate of a
parameter is the probability that the interval
estimate contains the parameter. It describes what
percentage of intervals from many different samples
contain the unknown population parameter