Lecture - 9 EstimationRM (ECON 1005 2011-2012)
Lecture - 9 EstimationRM (ECON 1005 2011-2012)
Lecture - 9 EstimationRM (ECON 1005 2011-2012)
INTRODUCTION TO STATISTICS
ESTIMATION
Introduction
• In the last two lectures, we discussed the characteristics and
properties of the probability distributions of random variables
First approach -
• Perform a complete enumeration of the population
(also known as a census) and calculate the mean and
variance from the dataset so derived. Unfortunately:
– It can be expensive
– It can be time consuming
– It consumes large quantities of resources
– It can be destructive to the elements of the population
– It may yield a level of accuracy that is not cost effective
when compared with the results of an appropriately sized
sample.
Second Approach -
Λ
Proportion p Sample Proportion p
Example
The mean and standard deviation of the teaching experience of
faculty members in a department at a University are unknown. A
random sample of 5 faculty members were selected; their teaching
experience in years were as follows: 7 8 14 7 20
2. We can use the sample standard deviation (s) as the point estimator for the
population standard deviation.
On the basis of the three estimators declared in 1. above, we can compute three
point estimates.
– Sample Mode = 7
– Sample Median = 8
• This total error between the point estimate and the true
value of the population parameter can be the result of both
sampling error and non-sampling error.
Suppose that a random sample of three students was drawn i.e. 70, 80 & 95.
• Use the sample data and the sample mean to estimate the population
mean. ( = 81.67)
• What is the new difference between the population mean and the point
estimate? (1.73)
Example (cont’d)
• It is this difference of 1.73 that we call the total error in
the estimate. It is subdivided into two components:
– The sampling error of 1.07
– The non-sampling error of 0.66
1. Consider all possible samples of three scores from this population; there
are 10 such samples.
76.00 2
76.67 1
79.33 1
81.00 1
81.67 2
84.33 2
85.00 1
∑f = 10
4. The Relative Frequency Distribution of
Sample Means
X Relative Frequency
76.00 0.2
76.67 0.1
79.33 0.1
81.00 0.1
81.67 0.2
84.33 0.2
85.00 0.1
∑Rel. Freq. = 1
5. The Probability Distribution of Sample Means
(or The Sampling Distribution of the Mean)
X Probability
76.00 0.2
76.67 0.1
79.33 0.1
81.00 0.1
81.67 0.2
84.33 0.2
85.00 0.1
∑Probability = 1
Sampling Distributions in this Course
• In general, the probability distribution of a
Sample Statistic is called its sampling distribution.
• We will focus on two sampling distributions:
– Sampling Distribution of the Mean
– Sampling Distribution of the Proportion
• In the Sampling Distribution of the Mean, the
random variable is the sample mean .
• In the Sampling Distribution of the Proportion,
the random variable is the sample proportion pΛ.
The Mean of the Sampling Distribution of the Mean
• Class Activity
Compute the mean of the Sampling Distribution of
the Mean History Score based on the ten random
samples of size 3.
Show that it is indeed equal to the population mean.
The Standard Deviation of the Sampling Distribution
of the Mean
• The Standard Deviation of the Sampling Distribution of
Mean is given by σx where
σx = σ /√n.
• The Student t Distribution has only one parameter i.e. the number of degrees of
freedom abbreviated df
• If n ≥ 30 the CLT allows us to use the Normal Distribution N(μ , s/√n ) as the
Sampling Distribution
44 52 31 48 46 39 47 36 41 57
Exhibit I
91% Confidence Interval for the mean Group B score is given by:
Therefore, if random samples of size 108 are drawn a large number of times
and the sample mean calculated is 51.8, then 91% of the times, the
corresponding population mean would lie between 49.8 and 53.8.
End of Lecture 8
• We have reviewed the Confidence Intervals
that form an integral part of the 5 stages of a
statistical analysis.
• Next we move on to another level of
investigation with respect to sample data.
• This involves Hypothesis testing.