Point and Interval Estimators

1
Point and Interval Estimators
2
Introduction

• Statistical inference is the process by which we
  acquire information about populations from
  samples.
• There are two procedures for making inferences
• Estimation.
• Hypotheses testing.

3
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

4
Point Estimator

• A point estimator draws inference about a
  population by estimating the value of an unknown
  parameter using a single value or a point.
• For example
• The sample mean is used to estimate the
  population mean. We refer to the sample mean as
  the estimator of the population mean. Once the
  sample mean has been computed, its value is
  called the estimate.

5
Point Estimator

• A point estimator draws inference about a
  population by estimating the value of an unknown
  parameter using a single value or a point.

Parameter
Population distribution
?
Sample distribution
Point estimator
6
Point Estimator

• We can compute the value of the estimator and
  consider that value as the estimate of the
  parameter. This is called a point estimator.
• When drawing inferences about a population, we
  expect a large sample to produce more accurate
  results because it contains more information than
  a smaller sample.
• This principle is illustrated by the Central
  Limit Theorem which says that more information
  reduces variation.
• Point estimates do not have the capability to
  reflect the effects of larger sample sizes

7
Interval Estimator

• Second way Interval estimator
• Interval estimator draws inferences about a
  population by estimating the value of an unknown
  parameter using an interval
• Interval estimators are affected by sample size.
• For example
• We want to know the average salary of a Lamar
  engineering graduate after their first year. We
  selected 25 people at random. The mean annual
  income is 50,000. This is a point estimate. Using
  an interval estimate we say that the mean annual
  income is between 45,000 and 55,000.

8
Interval Estimator

• An interval estimator draws inferences about a
  population by estimating the value of an unknown
  parameter using an interval.
• The interval estimator is affected by the sample
  size.

Interval estimator
9
Estimators desirable characteristics

• Selecting the right sample statistic to estimate
  a parameter value depends on the characteristics
  of the statistics.
• Estimators desirable characteristics
• Unbiasedness An unbiased estimator is one whose
  expected value is equal to the parameter it
  estimates.
• Consistency An unbiased estimator is said to be
  consistent if the difference between the
  estimator and the parameter grows smaller as the
  sample size increases.
• Relative efficiency For two unbiased estimators,
  the one with a smaller variance is said to be
  relatively efficient.

10
The Estimator Must Be Unbiased

• An unbiased estimator of a population parameter
  is an estimator whose expected value to that
  parameter.
• For example x-bar is an unbiased estimator of ?,
  because E(x-bar) ?
• This means that, if you were to take an infinite
  number of samples, calculate the value of the
  estimator in each sample, than average these
  values, the average value would equal the
  parameter.

11
The Estimator Must Be Consistent.

• Knowing that an estimator is unbiased only
  assures us that its expected value equals the
  parameter it does not tell us how close the
  estimator is to the parameter.
• We want the estimator to be as close to its
  parameter as possible.
• As the sample size grows larger, the sample
  statistic should come closer to the population
  parameter. This is called consistency.
• An unbiased estimator is consistent if the
  difference between the estimator and the
  parameter grows smaller as the sample size grows
  larger.

12
Estimating the Population Mean when the
Population Standard Deviation is Known

• How is an interval estimator produced from a
  sampling distribution?
• To estimate ?, a sample of size n is drawn from
  the population, and its mean is calculated.
• Under certain conditions, is normally
  distributed (or approximately normally
  distributed.), thus

13
Estimating the Population Mean when the
Population Standard Deviation is Known

• We know that

• This leads to the relationship

1 - a of all the values of obtained in
repeated sampling from this distribution,
construct an interval that includes (covers)
the expected value of the population.
14
Estimating the Population Mean when the
Population Standard Deviation is Known
1 - a
Upper confidence limit
Lower confidence limit
See simulation results demonstrating this point
Interval Estimator
15
Estimating the Population Mean when the
Population Standard Deviation is Known

• The confidence interval are correct most, but
  not all of the time.

UCL
LCL
Not all the confidence intervals cover the real
expected value of 100.
100
0
The selected confidence level is 90, and 10 out
of 100 intervals do not cover the real m.
16

• Four commonly used confidence levels

The mean values obtained in repeated draws of
samples of size 100 result in interval
estimators of the form sample mean - .28,
Sample mean .28 90 of which cover the real
mean of the distribution.
za/2
17

• Recalculate the confidence interval for 95
  confidence level.
• Solution

• The width of the 90 confidence interval
  2(.28) .56
• The width of the 95 confidence interval
  2(.34) .68
• Because the 95 confidence interval is wider,
  it is more likely to include the value of m.

.95
.90
18
Television Example

• The number and the types of television programs
  and commercials targeted at children is affected
  by the amount of time children watch TV.
• A survey was conducted among 100 North American
  children, in which they were asked to record the
  number of hours they watched TV per week.
• The population standard deviation of TV watch was
  known to be s 8.0
• Estimate the watch time with 95 confidence
  level.

19
Television Example

• What do we have?
• ? 8.0 hours
• What do we need?
• ? , n, x-bar, z
• Interval equation

20
Television Example

• What is our first step?
• Determine x-bar

• What is our confidence level? 95
• Thus ? .05 and ?/2 .025
• Determine Z
• z?/2 z.025 1.96
• Determine confidence interval

21
Television Example Solution

• The parameter to be estimated is ?, the mean time
  of TV watch per week per child (of all American
  Children).
• We need to compute the interval estimator for ?.
• From the data provided in file XM09-01, the
  sample mean is

Since 1 - a .95, a .05. Thus a/2 .025.
Z.025 1.96
22
Television Example

• Now use Minitab
• Type or import the data into one column (XM09-01)
• Click Stat, Basic Statistics, and 1-Sample z
• Type variable name (Time or C1)
• Use the cursor to select Confidence interval
• Hit tab and type the confidence level (95)
• Hit tab and type the value of the population
  standard deviation (sigma 8.0)
• Click O.K.
• As a result of working the problem by hand and
  with Minitab, we have the following result
• LCL 25.622 and UCL 28.760
• x-bar 27.191
• Our result gives us the 95 confidence interval
  estimate of the mean number of hours that
  children watch television per week to be LCL
  25.622 and UCL 28.760.

23

• Using Excel (Data analysis plus)

Observe the histogram of the raw data. It seems
that the amount of TV watch time is normally
distributed. Using normal distribution for
calculation of the confidence interval looks
appropriate. (Recall the central limit theorem)
24

• Interpreting the interval estimate
• It is wrong to state that the interval
  estimator is an interval for which there is 1 - a
  chance that the population mean lies between the
  LCL and the UCL.
• This is so because the ? is a parameter, not a
  random variable.

25
m

• Note that LCL and UCL are random variables.
• Thus, it is correct to state that there is 1 - a
  chance that LCL will be less than m and UCL will
  be greater than m.

26
m
m
27
Discussion of Example

• This interval does not mean that there is a 95
  probability that the population mean lies between
  25.622 and 28.760.
• This interpretation is wrong because it implies
  that the population mean is a variable about
  which we can make probability statements.
• In fact, the population mean is a fixed but
  unknown quantity. We cannot interpret the
  confidence interval estimate of ? as a
  probability statement about ?.
• We can say that 95 of the sample means will
  produce a LCL that is less than 27.191 and a UCL
  that is greater then 27.191. We note that this
  statistical procedure is correct 95 of the time
  in this case.
• The higher the level of confidence, 1 - ?, the
  wider the interval will be. As 1 - ? increases,
  ? decreases and z ?/2 becomes larger.
• As n increases, the confidence width decreases.
• The smaller the population variance, ?2, the
  smaller the confidence interval will be.

28
Doll Computer Example

• To lower inventory costs, the Doll Computer
  company wants to employ an inventory model.
• Lead time demand is normally distributed with
  standard deviation of 50 computers.
• It is required to know the mean in order to
  calculate optimum inventory levels.
• Estimate the mean demand during lead time with
  95 confidence.

29
Doll Computer Example

• The parameter to be estimated is ?.The interval
  estimator is
• Demand during 60 lead times is recorded514, 525,
  ., 476.
• The sample mean is calculated
• The 95 confidence interval is

Now try it in Minitab
30
Information and the Width of the Interval

• Wide interval estimator provides little
  information.
• For example, estimating the mean income of Lamar
  University students during the Summer break falls
  between 300 - 10,000 provides very little
  information.

Where is ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Ahaaa!
Here is a much smaller interval. If the
confidence level remains unchanged, the smaller
interval provides more meaningful information.
31

• The width of the interval estimate is a function
  of
• the population standard deviation
• the confidence level
• the sample size.

32
Suppose the standard deviation has increased by
50.
90 Confidence level
To maintain a certain level of confidence, changin
g to a larger standard deviation requires a
longer confidence interval.
33
Let us increase the confidence level from 90
to 95.
Increasing the confidence level produces a wider
interval
90 Confidence level
95
There is an inverse relationship between the
width of the interval and the sample size
Increasing the sample size decreases the width
of the interval estimate while the confidence
level can remain unchanged.
34
Selecting the Sample size

• Consider the Doll Computer example. The interval
  estimate was 487.1 to 512.4. The manager needs
  more precision. He states that he needs to
  estimate the mean demand to within 5 units of its
  true value.

• The formula for interval estimate of ?

• Therefore

35
Selecting the Sample size

• After some algebra

• The level of confidence is 95, thus z ?/2 1.96
  and ? 50

• If the sample mean is 510 the interval estimate
  becomes 510 /- 5.

36
Selecting the Sample size

• We can control the width of the interval estimate
  by changing the sample size.
• Thus, we determine the interval width first, and
  derive the required sample size.
• The phrase estimate the mean to within W units,
  translates to an interval estimate of the form

37

• The required sample size to estimate the mean is
• Example
• To estimate the amount of lumber that can be
  harvested in a tract of land, the mean diameter
  of trees in the tract must be estimated to within
  one inch with 99 confidence.
• What sample size should be taken? (assume
  diameters are normally distributed with s 6
  inches.

38
Solution

• The estimate accuracy is /-1 inch. That is w
  1.
• The confidence level 99 leads to a .01, thus
  za/2 z.005 2.575.
• We compute

39
Electronic Component Example

• The operations manager of a large production
  plant would like to estimate the average amount
  of time a worker takes to assemble a new
  electronic component. After observing a number of
  workers assembling similar devices, she noted
  that the shortest time taken was 10 minutes,
  while the longest time taken was 22 minutes. How
  large a sample of workers should she take if she
  wishes to estimate the mean assembly time to
  within 20 seconds? Assume that the confidence
  level is to be 99.
• What do we know?
• The confidence level is 1 - ? .99
• Z ?/2 z .005 2.575

40
Electronic Component Example

• The error bound is B 20 seconds
• The range 22 10 12 minutes 720 seconds
• We approximate ? as ? range/4 720/4 180
  seconds
• Now we can solve for n

Which rounds up to 538.
41
What about a confidence interval for the
variance?

• An equal tail 1 - ? confidence interval on ?2

• We note that this confidence interval is neither
  symmetric nor the shortest possible interval for
  this level of confidence.
• Because the Chi-square distribution is skewed to
  the right, the lower confidence limit is
  necessarily closer to the sample estimator than
  the upper confidence limit.
• This equation is used because it is easy to
  compute.

42
What if we did not know ? and used s?

• We still assume that we are sampling from a
  normal population.
• Look at Figure 8-1 in Barnes.
• For unknown s and small n, we can use

Which leads to
43
What if we did not know ? and used s?

• For unknown s and large n, we can use

• The t distribution is more variable than the
  standardized normal distribution.
• Therefore, if "s" is approximately the same
  magnitude as ?, we expect that the confidence
  interval using the t distribution will be wider
  than the confidence interval using the standard
  normal distribution.