8.3 - Estimating a Population Mean, mu
Constructing Confidence Intervals for mu
Assumptions: In order to construct a confidence interval the following conditions must be
satisfied
 The sample is a simple random sample
 Either or both of the following are satisfied:
i) The population is normally distributed, or
ii) n ≥ 30 (The sample has 30 or more values)
Procedure for Constructing a Confidence Interval for μ (with Known σ)
1) Verify that the required assumptions are satisfied.
2) Decide if you will use a z- or a t-score (z-table or t-table)
i.
Use z when given the standard deviation of the population (σ)
ii.
Use t when given the standard deviation of a sample OR if the problem contains
data, use 1-Var-Stats to find x-bar and s. (ignore the sigma you see in the
calculator)
3) Find the margin of error E
4) Construct the interval:
(E = z c

n
) or (E = t c
s
)
n
x  E    x  E or ( x  E , x  E )
Properties of the t-distribution:
1. The t-distribution is different for different sample sizes
2. The mean is t = 0 and the standard deviation is greater than one
3. The t-distribution is bell shaped like the SND but extends further out because it has a
larger variability
4. The area under the curve is 1
5. As the sample size n gets larger, the t-distribution gets closer to the standard normal
distribution
Using the TI-83 to Construct Confidence Intervals for μ:
STAT>>TESTS select 7:ZInterval or 8:TInterval
If you are given DATA, use the Data option, otherwise use the Stats option
8.4 - SAMPLE SIZE FORMULAS for estimating population means

n
 2 z2
,
E2
where E is the margin of error (Note: our book use the symbol m for the margin of
error)
If sigma is not available, use a sample standard deviation from a similar study or use
another educated guess. Read example 10, page 383 where sigma is estimated using a
reasonable value for the range since we can consider that Range ~ 6 
1
1) The health of the bear population in Yellowstone National Park is monitored by periodic
measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of
182.9 lb. Assuming that sigma is known to be 121.8 lbs., find a 95% confidence interval
estimate of the mean of the population of all such bear weights.
a) What is the point estimate for mu?
b) Verify that the requirements for constructing a confidence interval about x-bar are
satisfied.
c) Construct a 95% confidence interval estimate for the mean weight of all such beaars.
(Are you using z or t? Why?)
d) Now check with the calculator feature
e) The statement “95% confident” means that, if 100 samples of size ______ were taken,
about _____ intervals will contain the parameter μ and about ____ will not.
f) Complete the following: We are _____% confident that the mean weight of all such
bears is between _____________ and _____________
g) With ______% confidence we can say that the mean weights of the bears is ______
with a margin of error of _________________
h) For ______% of intervals constructed with this method, the sample mean would not
differ from the actual population mean by more than _______________
i) How can you produce a more precise confidence interval?
j) Section 8.4 - If we want an estimate which is within 25 lbs. of the actual population
mean mu, what should be the sample size selected?
2
2) In order to correctly diagnose the disorder of hydrocephalus, a pediatrician investigates
head circumferences of two month old babies. 100 two-month old babies are selected at
random and the sample mean observed is 40.573 cm with a sample standard deviation of
1.649.
a) What is the point estimate?
b) Verify that the requirements for constructing a confidence interval about x-bar are
satisfied.
c) Construct a 99% confidence interval estimate for the head circumference of all two
months old babies. (Are you using z or t? Why?)
d) The statement “99% confident” means that, if 100 samples of size _____ were taken,
about _____ intervals will contain the parameter μ and about ____ will not.
e) Complete the following: We are _____% confident that the mean head circumference
of all two months old babies is between __________ and __________
f) With ______% confidence we can say that the mean head circumference of all two
months old babies is ______________ with a margin of error of ____________
g) For ______% of intervals constructed with this method, the sample mean would not
differ from the actual population mean by more than __________
h) How can you produce a more precise confidence interval?
i) Section 8.4 - How large of a sample should be selected in order to be 99% confident
that the point estimate x-bar will be within 0.2 cm of the true population mean? (We
will use the standard deviation of this study, s = __________ as an estimate for sigma)
3