Estimating Population Parameters using Confidence Intervals (Part 1) &
Evaluating Population Parameters using Tests of Significance (Part 2)
Population parameters are unknown fixed values about populations, such as m and p. To make an
inference about a population that cannot be surveyed entirely, sample statistics can be taken from an
SRS of the population and then used to estimate population parameters.
 The population mean, , is estimated using (x-bar), the sample mean.
 The population proportion, p, is estimated using (p-hat), the sample proportion.
These point estimates are the best estimates of the population parameters, as they are unbiased
estimators (because the mean of the sampling distribution is equal to the parameter).
Confidence Intervals show a range of where some population parameter lies based on confidence
levels.
Tests of Significance (aka Hypothesis Testing) show the strength of evidence of some claim about a
parameter.
Part 1—Confidence Interval & Confidence Level
Because the point estimate (the mean of the sample or the proportion of the sample ) will vary
with each new sample and will not necessarily be the exact value of the population parameter, a
Confidence Interval (CI) must be determined.
This confidence interval, based on a confidence level, is a range of values constructed around the point
estimate, which is at the center of the interval, with a margin of error, above and below the point
estimate.
A confidence interval has the general form:
point estimate
margin of error
(standardized score) · (standard error)
z or t score
standard deviation of sample
The z score (or t score) is a standardized score that is also referred to as a critical value when working
with confidence intervals (the “boundaries” of the interval). The level of confidence determines its
specific value. The Confidence Level (CL) is a predetermined percentage that represents how
confident we are that the confidence interval constructed will actually capture the true population
parameter.
If asked to construct a confidence interval based on a 95% confidence level (for mean of a normally
distributed population when standard deviation is known), we would sketch a normal curve and label
the critical values by finding the z score that corresponds to the area to the left of the lower bound and
to the right of the upper bound using a z score sheet or a calculator:

Using the z-score sheet, look in the body of the table for area .0250
obtain a -1.960 z score… critical values
point estimate).

to
(as standard deviations below and above the
Using the calculator, press “2nd,” “Distribution,” “invNorm(.025),” “Enter” to obtain a 1.960 z score… critical values
(as standard deviations below and above the point
estimate).
Interpret the interval you have constructed in context of the problem. Be sure to correctly interpret the
meaning of the confidence interval:

A 95% confidence level is interpreted as, “We are using a method that will provide correct
results in 95% of all confidence intervals constructed using randomly obtained data,” or
“We are 95% confident that the (true parameter) lies between (lower limit) and (upper
limit).”

The remaining 5%, split equally between the highest 2.5% and the lowest 2.5%, is where
the true parameter could be when our interval does not include the actual parameter.
PRACTICE: Sketch and label a normal curve with the z scores for the given confidence intervals.
a. 85%
b. 90%
Margin of Error & Required Sample Size
Sample size depends on the desired confidence level and margin of error. Use the formulas for margin
of error, and round up to the next larger integer.
To estimate p
m Z
pˆ (1  pˆ )
n
To estimate  with
m  Z

To estimate  without

n
m  t

s
n
PRACTICE: Find the margin of error that corresponds to the given statistics and confidence level.
1.
x = 70, n = 100, 95% confidence, s =12
2. sample size 500, 20% successes, 99% confidence
PRACTICE: How many doctors must be randomly selected for IQ tests if we want to estimate the
mean IQ score with 95% confidence that the sample mean is within two IQ points of the population
mean?
PRACTICE: What is the minimum sample size needed in order to be 99% confident that the margin of
error is at most 6% when the value is estimated to be .76?
Estimating Population Parameters using Confidence Intervals
A four-step system is used to organize the inference process when estimating means and proportions:
I.
Identify the population and the parameter of interest.
II.
Verify the conditions/assumptions (a-d), and identify the procedure (e).
a. Condition: Data must be obtained from an SRS. (Otherwise, this condition may
be assumed, which may limit the ability to generalize the results to the
population.)
b. Condition: Population is at least 10 times n. (written in context)
(With groups: OR random allocation)
(With proportion, add Condition: n p̂ 10, n(1- p̂ ) 10)
c. Assumption: Population is normally distributed and/or n > 30.
d. Assumption: Individuals/groups/data are independent.
e. “We have verified the conditions
for a (type

 of interval) for (parameter).”
III.
With conditions met, carry out the selected procedure.
Find confidence interval: estimate ± margin of error
(show all work)
IV.
Interpret results in context of the problem.
“We are (confidence level)% confident that the true (parameter) of (the specific
population) lies between (the interval bounds).
Confidence Interval for Population Mean m given Standard deviation s
EXAMPLE 1: A sample of 54 bears from Yellowstone National Park has a mean weight of 182.9 lb.
Assuming that s is known to be 121.8 lb., find a 99% CI estimate of the mean of the population of all
such bear weights.
The Confidence Interval, CI, is found using:
x  Z

n
m, margin of error for mean given 
x , the sample mean, the best point estimate of the

population mean

(Continue with EXAMPLE 1 on your paper…)
Confidence Interval for Population Mean m without Standard Deviation s
EXAMPLE 2: A sample of 54 bears from Yellowstone National Park has a mean weight of 182.9 lb.
Assuming that s is not known, but the sample standard deviation is 111.2 lb., construct a 99% CI
estimate of the mean of the population of all such bear weights.
The Confidence Interval, CI, is found using: x  t s
n
s, standard deviation of sample
m, margin of error for mean without s
x , the sample mean, the best point estimate of the
population mean

Use t-distribution to obtain larger critical
values when the standard deviation is not known. When
determining “degrees of freedom,” choose one fewer than sample size n on t score sheet (or in
calculator).
Work for EXAMPLE 2:
PRACTICE: Find critical t value for the following:
1. n = 15; SRS; normally distributed population; 95% confidence level; s = 2.1
2. Construct a CI (without 4-step process) for estimating mean:
n = 20 (from a normally distributed population)
x = 4.4 in
s = 4.2 in
99% confidence level