2. GOALS
1. Methods of Estimation
2. Difference between Point and Interval Estimation
3. Defining Level of Confidence
4. Constructing Confidence Intervals
5. Interpretation of these confidence intervals
6. Determine the sample size for attribute and variable
sampling.
7. Explanations using examples /case study
3. What are Estimators?
In statistics, an estimator is a function of the data or
sample that is used to infer the value of an
unknown parameter in population in a statistical
model.
Thus the estimator, the quantity of interest
(the estimand or parameter) and its result (the
estimate) are different from each other.
4. Qualities of Estimators…Statisticians have already
determined the “best” way to estimate a population
parameter.
Qualities desirable in estimators include unbiasedness,
consistency, and relative efficiency:
• 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 the
difference between the estimator and the parameter
grows smaller as the 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.
5. Estimation…
There are two types of inference: estimation and
hypothesis testing; estimation is introduced first.
The objective of estimation is to determine the
approximate value of a population parameter on the
basis of a sample statistic.
E.g., the sample mean ( ) is employed to estimate
the population mean ( ).
6. Point & Interval
Estimation…
For example, suppose we want to estimate the mean summer income of a class of
business students. For n=25 students.
It is calculated and average is found to be 400 $/week.
point estimate interval estimate
An alternative statement is:
The mean income is between 380 and 420 $/week.
10.6
7. Estimation…
The objective of estimation is to determine the
approximate value of a population parameter on the
basis of a sample statistic.
There are two types of estimators:
Point Estimator
Interval Estimator
8. Point Estimator…
A point estimator draws inferences about a population
by estimating the value of an unknown parameter using
a single value or point.
We saw earlier that point probabilities in continuous
distributions were virtually zero. Likewise, we’d expect
that the point estimator gets closer to the parameter
value with an increased sample size, but point estimators
don’t reflect the effects of larger sample sizes. Hence we
will employ the interval estimator to estimate
population parameters…
9. Interval Estimator…
An interval estimator draws inferences about a
population by estimating the value of an unknown
parameter using an interval.
That is we say (with some ___% certainty) that the
population parameter of interest is between some lower
and upper bounds.
10. 5. Statistical Inference: Estimation
Goal: How can we use sample data to estimate values of
population parameters?
Point estimate: A single statistic value that is the “best
guess” for the parameter value
Interval estimate: An interval of numbers around the
point estimate, that has a fixed “confidence level” of
containing the parameter value. Called a confidence
interval.
(Based on sampling distribution of the point estimate)
11. Point Estimators – Most common to use
sample values
Sample mean estimates population mean m
ˆ iy
y
n
m
• Sample std. dev. estimates population std. dev. s
2
( )
ˆ
1
iy y
s
n
s
• Sample proportion p estimates population
proportion p^.
12. Methods of finding
estimators…
Point estimators
Method of moment
Maximum likelihood
estimators
Bayes estimators
The EM algorithm
Interval estimators
Inverting a test statistic
Pivotal quantities
Pivoting the CDF
Bayesian intervals
13. Confidence Interval
Confidence Interval of a parameter consists of an interval of
numbers along with a probability that the interval contains the
unknown parameter
A confidence interval gives a range estimate of values:
Takes into consideration variation in sample statistics from sample to
sample
Based on all the observations from one sample
Gives information about closeness to unknown population parameters
Stated in terms of level of confidence
Example: 95% confidence, 99% confidence
Can never be 100% confident
A confidence interval estimate of 100% would be so wide as to
be meaningless for practical decision making
14. Confidence Interval
The larger the CI , the more confident we can be that the
given interval contains the unknown parameter.
Ideally, we prefer a short interval with high degree of
confidence.
For Example: We will prefer (95,100) with 95% confidence
than (0,100) with 100% confidence.
The value corresponding to a significance level that
determines those test statistics that lead to rejection of null
hypothesis and those that lead to a decision not to reject
null hypothesis is referred to as Critical Value.
15. The general formula for all
confidence intervals is:
Point Estimate ± (Critical Value) (Standard Error)
Sample Mean
or
Sample Proportion
The “z” or “t”
Critical Value
σ / √ n or s / √ n
Confidence Interval Estimates
16. The Level of Significance (α)
Because we only select one sample, and μ or π
are unknown, we never really know whether the
confidence interval includes the true population
mean or proportion, or not.
The level of significance, or “α” risk is the chance
we take that the true population parameter is not
contained in the confidence interval.
Therefore, a 95% confidence interval would have
an “α” of 5%
17. 95% Confidence Interval
“ α “ is the proportion in the tails of the sampling distribution
that is outside the established confidence interval.
If α = .05, then
each tail has
.025 area
a = .025a = .025
.9750.0250
+ 1.96 z- 1.96 z
Z .06
- 1.9 .0250
Z .06
+ 1.9 .9750
The critical values of “z” that
define the “α” areas are
-1.96 and + 1.96
Point Estimate
The Level of Significance (α)
18. Level of Confidence
The Level of Confidence in a confidence interval is
a probability that represents the percentage of
intervals that will contain if a large number of
repeated samples are obtained. The level of
confidence is denoted
For example, a 95% level of confidence
would mean that if 100 confidence intervals were
constructed, each based on a different sample from
the same population, we would expect 95 of the
intervals to contain the population mean.
19. Constructing Confidence Interval
The construction of a confidence interval for the
population mean depends upon three factors
1. The point estimate of the population
2. The level of confidence
3. The standard deviation of the sample mean
20. 1. Select our desired level of confidence
Let’s suppose we want to construct an interval
using the 95% confidence level
2. Calculate α and α/2
(1-α)*100% = 95% α = 0.05, α/2 = 0.025
3. Look up the corresponding z-score
α/2 = 0.025 a z-score of 1.96
Constructing Confidence Interval
21. 4. Multiply the z-score by the standard error to
find the margin of error
5. Find the interval by adding and subtracting this
product from the mean
).,.( 2/2/ errorstdZxerrorstdZx
nn
Z
ss
96.12/
)96.1,96.1(
n
x
n
x
ss
Constructing Confidence Interval
23. Common Confidence Levels
and α values
Here is a table of commonly used confidence
levels, α and α/2 values, and corresponding z-
scores which we are using in our examples:
• (1 - α)*100% α α/2 Zα/2
• 90% 0.1 0.05 1.645
• 95% 0.05 0.025 1.96
• 99% 0.01 0.005 2.58
24. Ex : Suppose we conduct a poll to try and get a sense of the
outcome of an upcoming election with two candidates.
We poll 1000 people, and 550 of them respond that they
will vote for candidate A
How confident can we be that a given person will cast
their vote for candidate A?
1. Select our desired levels of confidence
We’re going to use the 90%, 95%, and 99% levels
Constructing Confidence Interval
Example
25. 2. Calculate α and α/2
Our values are 0.1, 0.05, and 0.01 respectively
Our /2 values are 0.05, 0.025, and 0.005
3. Look up the corresponding z-scores
Our Z/2 values are 1.645, 1.96, and 2.58
4. Multiply the z-score by the standard error to find
the margin of error
First we need to calculate the standard error
Constructing Confidence Interval
Example
26. 5. Find the interval by adding and subtracting this
product from the mean
In this case, we are working with a distribution we have
not previously discussed, a normal binomial distribution
(i.e. a vote can choose Candidate A or B, a binomial
function)
We have a probability estimator from our sample, where
the probability of an individual in our sample voting for
candidate A was found to be 550/1000 or 0.55
We can use this information in a formula to estimate the
standard error for such a distribution:
Constructing Confidence Interval
Example
27. 5. Multiply the z-score by the standard error cont.
• For a normal binominal distribution, the standard
error can be estimated using:
sX =
s
n =
(p)(1-p)
n
=
(0.55)(0.45)
1000
= 0.0157
• We can now multiply this value by the z-scores to
calculate the margins of error for each conf. level
Constructing Confidence Interval
Example
28. 5. Multiply the z-score by the standard error cont.
• We calculate the margin of error and add and subtract that
value from the mean (0.55 in this case) to find the bounds
of our confidence intervals at each level of confidence:
Margin Bounds
CI Z/2 of error Lower Upper
90% 1.645 0.026 0.524 0.576
95% 1.96 0.031 0.519 0.581
99% 2.58 0.041 0.509 0.591
Constructing Confidence Interval
Example
29. Some Myths
What if we can make a“guess”about proportion value?
How can we predict whether it will rain tomorrow?
Be Careful! The following statement is NOT true:
“The probability that µ lies between 143.22 and 162.78 is .95.”
Once you have inserted your sample results into the confidence
interval formula, the word PROBABILITY can no longer be
used to describe the resulting confidence interval.
30. Some comments about CI’S
Effects of n, confidence coefficient true for CIs for other
parameters also
If we repeatedly took random samples of some fixed size n
and each time calculated a 95% CI, in the long run about
95% of the CI’s would contain the population proportion .
The probability that the CI does not contain is called the
error probability, and is denoted by α.
α = 1 – confidence coefficient
(1-a)100% a a/2 za/2
90% .10 .050 1.645
95% .05 .025 1.96
99% .01 .005 2.58
31. Comments about CI for population mean µ
The method is robust to violations of the assumption
of a normal population distribution
(Be careful if sample data distribution is very highly
skewed, or if it contains severe outliers)
Greater confidence requires wider CI
Greater n produces narrower CI
32. To determine sample size:
Know the desired confidence level, which determines the
value of Z (the critical value from the standardized
normal distribution. Determining the confidence level is
subjective.
Know the acceptable sampling error, e. The amount of
error that can be tolerated.
Know the standard deviation, σ. If unknown, estimate
by past data or make an educated guess
estimate σ: [σ = range/4] This estimate is derived from
the empirical rule stating that approximately 95% of the
values in a normal distribution are within +/- 2σ of the
mean, giving a range within which most of the values
are located.
33. Selecting the Sample
Size…
We can control the width of the interval by determining the
sample size necessary to produce narrow intervals.
Suppose we want to estimate the mean demand “to within 5
units”; i.e. we want to the interval estimate to be:
Since:
It follows that
Solve for n to get requisite sample size!
34. Selecting the Sample
Size…
The amount of sampling error you are willing to accept and the
level of confidence desired, determines the size of your
sample.
n = Z2σ2 / e2
e = Z (σ / √ n )
35. Choosing the Sample Size
Ex. How large a sample size do we need to estimate
a population proportion (e.g., “very happy”) to
be within 0.03, with probability 0.95?
i.e., what is n so that margin of error of 95%
confidence interval is 0.03?
Set 0.03 = margin of error and solve for n
ˆ0.03 1.96 1.96 (1 )/ ns
36. Some comments about CIs and sample size
We’ve seen that n depends on confidence level
(higher confidence requires larger n) and the
population variability (more variability requires
larger n)
In practice, determining n not so easy, because (1)
many parameters are to be estimated, (2) resources
may be limited and we may need to compromise,
due to several constraints.
CI’s can be formed for any parameter (i.e., for
median, mode etc.)
37. How large must a sample be for the Central
Limit theorem to apply?
The sample size varies according to the shape of the
population. However, for our use, a sample size of 30 or
larger will suffice.
Q. Must sample sizes be 30 or larger for populations that are
normally distributed?
Ans: No. If the population is normally distributed, the
sample means are normally distributed for sample sizes as
small as n=1.
Q. How large is large?
Q. Why not just always pick a sample size of 30?
Aim: Finding an optimum size for the sample
38. How can I tell the shape of the underlying
population?
CHECK FOR NORMALITY:
Use descriptive statistics. Construct stem-and-leaf plots for small
or moderate-sized data sets and frequency distributions and
histograms for large data sets.
Compute measures of central tendency (mean and median) and
compare with the theoretical and practical properties of the
normal distribution. Compute the interquartile range. Does it
approximate the 1.33 times the standard deviation?
How are the observations in the data set distributed? Do
approximately two thirds of the observations lie between the
mean and plus or minus 1 standard deviation? Do approximately
four-fifths of the observations lie between the mean and plus or
minus 1.28 standard deviations? Do approximately 19 out of
every 20 observations lie between the mean and plus or minus 2
standard deviations?
39. Interpreting a Confidence Interval
For the previous 95% confidence interval, the following conclusions are
valid:
I am 95% confident that the average length of a call for the
population µ, lies between 143.22 and 162.78 minutes.
If I repeatedly obtained samples of size 85, then 95% of the
resulting confidence intervals would contain µ and 5% would
not.
QUESTION: Does this confidence interval [143.22 to 162.78]
contain µ?
ANSWER: I don’t know. All I can say is that this procedure leads
to an interval containing µ 95% of the time.
I am 95% confident that my estimate of µ [namely 153 minutes] is
within 9.78 minutes of the actual value of µ. RECALL: 9.78 is the
margin of error.
40. Interpretations (contd.)
Therefore, 7.93% of the time, a random sample of 40
customers from this population will yield a mean
expenditure of 8700 or more.
OR
From any random sample of 40 customers, 7.93% of them will
spend on average 8700 or more.
41. Interpretations (contd.)
The point estimate for this problem is 13.56 hours, with an
error of +/- 3.20 hours.
I am 90% confident that the average amount of time
accumulated by a manager per week in this industry is
between 10.36 and 16.76 hours.
We are 95% confident that the population proportion of
telemarketing firms that use their operation to assist
order processing is somewhere between .29 and .49.
There is a point estimate of .39 with a margin of error of +/-
.10.
42. Assumptions necessary to use t-
distribution
Assumes random variable x is normally
distributed
However, if sample size is large enough ( > 30),
t-distribution can be used when σ is unknown.
But if sample size is small, evaluate the shape of
the sample data using a histogram or stem-and-
leaf.
As the sample size increases, the t-distribution
approaches the Z distribution.
