9. Introduction to Estimation. Confidence
Interval for a Population Mean
Two major problems
Statistical inference is the process by which we acquire information about
populations from samples. There are two procedures for making
inferences:
Estimation
Hypotheses testing
Concepts of estimation
The objective of estimation is to determine the value of a population
parameter on the basis of a sample statistic. There are two types of
estimators
Point Estimator
Interval estimator
Point estimator
A point estimator draws inference about a population by estimating the
value of unknown parameter using a single value or a point.
An unbiased estimator of a population parameter is an estimator whose
expected value is equal to that parameter.
An unbiased estimator is said to be consistent if difference between
estimator and the parameter grows smaller as sample size grows larger.
If there are two unbiased estimators of a parameter, the one whose variance
is smaller is said to be relatively efficient.
Point estimator: three main examples
The sample mean
is the most commonly used unbiased and
consistent estimator of population mean . If data come from a normal
population than it is also most efficient.
The sample variance
is the most commonly used unbiased and
consistent estimator of population variance
.
The sample proportion
is the most commonly used unbiased and
consistent estimator of population proportion .
Confidence interval estimator
A confidence interval (CI) estimator draws inferences about population by
estimating the value of an unknown parameter using an interval.
It is an interval calculated from the observations (which differs in principle
from sample to sample) that frequently includes the parameter of interest
if the experiment is repeated. How frequently the observed interval
contains the parameter is determined by the confidence level.
CI for population mean when sample size is
large ()
Consider a sample
from a population with mean and variance
. When is large, by the Central Limit Theorem, the sample mean
is
approximately normally distributed with mean and variance
.
Using this fact one can show that for any
Since by the Law of Large Numbers
we also get that
CI for population mean when sample size is
large ()
Thus the % confidence interval for population mean when
is the following interval:
The confidence interval is often represented in this form:
The part of the formula,
, is called the margin of error.
Note also that the point estimator of ,
, is the middle point of the CI.
The probability is called the confidence level.
Interpreting the confidence interval
Before the data are collected it is OK to say that the probability of covering
population mean by the % CI is equal to .
However, once the sample mean and the margin error are computed (based
on a specific sample) we cannot talk about probability any longer
because population mean is a number. That is why we use the word
“confidence”.
We usually say “with confidence level %, population mean
lies in the confidence interval”.
-CI for : example
Example 1. The caffeine content (in milligrams) of a random sample of 50
cups of black coffee dispensed by a new machine is measured. The
mean and standard deviation are 100 milligrams and 7.1 milligrams,
respectively. Construct a 98% CI for the true mean caffeine content per
cup dispensed by the machine.
Solution. The sample size is ; the confidence level is 98%,
therefore,
Thus, the 98% CI is given by
So, we can claim that with confidence level 98%, the mean caffeine content
per cup lies in the confidence interval .
Sample size determination
We can control the width of the CI by changing the sample size: the larger
the sample size, the smaller the margin of error. However, the large
sample size will cost more. So sometimes before the data are collected
we can try to estimate what sample size can provide us with targeted
confidence level and margin of error .
This leads us to the following equation:
Solving this equation with respect to we get this estimate for a required
sample size:
Sample size determination
Since and are given, the only issue with the formula is that we also
need to know sample variance
, because we only plan to draw a
sample.
There are two ways how we can address the problem:
We can use the value given by the range approximation:
Or we can use so-called historical value of based on past experience.
For instance, we can draw a small trial sample and use its sample
variance.
Sample size determination: example
Example 2. A research project for an insurance company wishes to
investigate the mean value of the personal property held by urban
apartment renters. A previous study suggested that the sample standard
deviation should be roughly $10000. A 95% confidence interval with
width of $1000 (a plus or minus of $500) is desired. How large a sample
must be taken to obtain such a confidence interval?
Solution. Since , and we have info on the standard
deviation, the required sample size is equal to
CI for population mean when sample size is
small ()
If the sample size is small (less than 30) the confidence interval is given by
where
is the critical value of -distribution with degrees
of freedom.
A random sample is assumed to be taken from a normal population.
-CI for : example
Example 3. A furniture mover calculates the actual weight of shipment as a
proportion of estimated weight for a sample of 25 recent jobs. The
sample mean is 1.13 and the sample standard deviation is .16. Calculate
a 95% CI for the population mean. Assume that data are taken from a
normal population.
Solution. The sample size is , the confidence level is 95%,
therefore,
Thus, the 95% CI is given by
CI for : exercises
Exercise 1. Sixty pieces of a plastic are randomly selected, and the breaking
strength of each piece is recorded in pounds per square inch. Suppose
that:
and pounds per square inch. Find a 99%
confidence interval for the mean breaking strength .
CI for : exercises
Exercise 2. An electrical company tested a new type of oil to be used in its
transformers. Twenty-five readings of dielectric strength were obtained.
Dielectric strength is the potential (in kilovolts per centimeter of
thickness) necessary to cause a disruptive discharge of electricity
through an insulator. The results of the test gave:
kV, kV.
Find a 95% confidence interval for the mean dielectric strength of the oil.
Assume that data are taken from a normal population.
CI for : exercises
Exercise 3. A production manager noticed that the mean time to complete a
job was 160 minutes. The manager made some changes in the
production process in an attempt to reduce the mean time to finish the
job. A stem-and-leaf plot of a sample of 11 times is as follows:
13 | 9
14 | 25
15 | 01356
16 | 24
17 | 0
Note: 14|5 = 145 minutes
The sample mean and standard deviation are 153.36 and 9.47, respectively.
Construct a 95% confidence interval for the mean time.
CI for : exercises
Exercise 4. How many households in a large town should be randomly
sampled to estimate the mean number of dollars spent per household
(per week) on food supplies to within $10 with 90% confidence? Assume
a standard deviation of $50.
