1. Methodology: Confidence Intervals for Mean and Proportion
Example 1: CI for the Mean (with summary statistics)
Suppose average pizza delivery times are normally distributed with an unknown population mean
and a population standard deviation of six minutes. A random sample of 28 pizza delivery
restaurants is taken and has a sample mean delivery time of 36 minutes.
Find a 90% confidence interval estimate for the population mean delivery time.
Step Discussion Example
Step 1: What is the
parameter?
Identify if you want to
estimate the mean or
proportion of the population
We want to estimate the
mean
Step 2: Are the requirements
met?
For the Mean you need a
random sample with
(a) or
(b) Population normal
with no outliers
The sample size is greater
than 30.
Step 3: What distribution
will you use?
If you know the population
standard deviation the use
the normal distribution,
otherwise use the t-
distribution.
We know the population
standard deviation so we
will use the normal
distribution.
Step 4: Identify the summary
statistics.
Find or calculate the sample
mean, standard deviation,
sample size and confidence
level.
̅
Step 5: Calculate Confidence
Interval
Use the formula or
technology to calculate the
confidence interval
TI-84:
STAT>TESTS>Z-Interval
Choose STATS
Excel:
z=-NORMSINV((1-C)/2)
̅
√
( )
Step 6: Interpret Confidence
Interval
Write a complete sentence
that includes the confidence
level, limits of interval, and
units of random variable.
We are 90% confident that
the mean pizza delivery time
is between 34.14 minutes
and 37.87 minutes.
Example 2: CI for the Mean (with sample data)
You do a study of hypnotherapy to determine how effective it is in increasing the number of hours
of sleep subjects get each night. You measure hours of sleep for 12 subjects with the following
results. Construct a 95% confidence interval for the mean number of hours slept for the population
(assumed normal) from which you took the data.
8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10.5
Step Discussion Example
Step 1: What is the
parameter?
Identify if you want to
estimate the mean or
We want to estimate the
mean
2. proportion of the population
Step 2: Are the requirements
met?
For the Mean you need a
random sample with
(a) or
(b) Population normal
with no outliers
The population is normal.
Step 3: What distribution
will you use?
If you know the population
standard deviation the use
the normal distribution,
otherwise use the t-
distribution.
We do not know the
population standard
deviation so we will use the
t-distribution.
Step 4:Identify the summary
statistics.
Find or calculate the sample
mean, standard deviation,
sample size and confidence
level.
TI-84: STAT>EDIT
Enter Data in L1
̅
Step 5: Calculate Confidence
Interval
Use the formula or
technology to calculate the
confidence interval
TI-84:
STAT>TESTS>T-Interval
Choose DATA
Excel:
t=TINV(1-C)
̅
√
( )
Step 6: Interpret Confidence
Interval
Write a complete sentence
that includes the confidence
level, limits of interval, and
units of random variable.
We are 95% confident that
the mean amount of sleep
for patients undergoing
hypnotherapy is between
8.96 hours and 9.01 hours.
Example 3: CI for the Proportion
Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250
surveyed, 98 reported owning a tablet. Using a 95% confidence level, compute a confidence interval
estimate for the true proportion of people who own tablets.
Step Discussion Example
Step 1: What is the
parameter?
Identify if you want to
estimate the mean or
proportion of the population
We want to estimate the
proportion
Step 2: Are the requirements
met?
For the proportion you need
a random sample with
(a) ( ) and
(b)
( )
Step 3: What distribution
will you use?
If the sample size is
sufficiently large then the
We will use the normal
distribution.
3. normal distribution
approximates the binomial
distribution quite well.
Step 4:Identify the summary
statistics.
Find or calculate the sample
proportion, sample size and
confidence level.
Step 5: Calculate Confidence
Interval
Use the formula or
technology to calculate the
confidence interval
TI-84:
STAT>TESTS>1-
PropZInterval
Excel:
z=-NORMSINV((1-C)/2)
√
( )
( )
Step 6: Interpret Confidence
Interval
Write a complete sentence
that includes the confidence
level, limits of interval, and
units of random variable.
We are 95% confident that
the proportion of people
who own tablets between
33% and 45%
