Unformatted text preview: Lessons in Business Statistics
Prepared By
P.K. Viswanathan Chapter 7: Estimation Introduction
Marketing Manager in an organization
needs to estimate the likely market share
his company can achieve in the market
place. Quality Assurance Manager may be
interested in estimating the proportion
defective of the finished product before
shipment to the customer. Manager of the
credit department needs to estimate the
average collection period for collecting
dues from the customers. How confident
are they in their estimates? This chapter
provides some insights into point
estimation and interval estimation that are
essential in business planning. Please
remember that the three componentspoint
estimation, interval estimation, and
hypothesis testing together constitute the
allimportant inferential statistics. 1) Point Estimation
Point Estimation deals with the task of
selecting a specific sample value as an
estimate for a population parameter. 1) Point EstimationContinues
A point estimate is a
specific value of a
sample statistic that is
used to estimate a
population parameter. 1)Point Estimation
Population Mean The sample mean X is an unbiased estimator of
the population mean μ . An unbiased estimator
is one whose expected value is equal to the
population parameter. That is, E( X ) is equal to μ . Of course the
samples drawn must be independent random
samples from the population of interest. 1)Point Estimation
Population Proportion Sample proportion is an unbiased
estimator of the population proportion.
That is p is an unbiased estimator of P.
So E(p) = P. A particular value of p based on a
sample survey becomes a point
estimate. 2) Interval Estimation A point estimate cannot be expected to coincide exactly
with the population parameter. Suppose in a survey you
find that the average income of a household is Rs. 300000
per year. Is it that the income of every household is
Rs.300000 per year? Certainly not. Some households may
have more than Rs. 300000 and some may have less than
this amount. In other words point estimate will not
coincide with the population parameter. How to cope with
this problem? Interval Estimation comes to our help in this
regard. Interval Estimation establishes an interval
consisting of a lower limit and an upper limit in which the
true value of the population parameter is expected to fall.
This interval is called “ Confidence Interval ” in the
parlance of inferential statistics. 2) Interval Estimation
Picture 3) Confidence Interval for Population
MeanLarge Sample
(1 Confidence Interval for the
)
Population Mean is given by The 3) Confidence Interval for Population
MeanLarge SampleExample
A machine produces components, which have a
standard deviation of 1.6 cm in length. A
random sample of 64 parts is selected from the
output and this sample has a mean length of 90
cm. The customer will reject the part if it is
either less than 88cm or more than 92cm. Does
the 95% confidence interval for the true mean
length of all the components produced ensure
acceptance by the customer? 3) Confidence Interval for Population
MeanSolution to the Example
To answer the question of acceptance by the
customer, you should first work out the 95%
confidence interval for the population mean μ
(Here μ is the mean length of the components in
the population). When you want 95% confidence
level, the Z value is 1.96 from the standard normal
distribution. See diagram in the next slide. 3) Confidence Interval for Population MeanLarge Sample Solution to the Example
Picture Showing Z value
for 95% confidence level 3) Confidence Interval for Population MeanLarge Sample Solution to the Example
Picture Showing for 95% confidence level for μby using the formula
X
1.96 σ
n μ X 1.96 σ
n 3) Confidence Interval for Population
ProportionLarge Sample
The (1 Confidence Interval for the Population
)
Proportion P is given by p(1 p)
p(1  p)
p
Z PpZ
n
n
Where P is the population proportion,
p is the sample proportion
In particular, 95% confidence interval for the population
proportion is given by
p(1  p)
p(1  p)
p
1.96
P p 1.96
n
n 3) Confidence Interval for Population
ProportionLarge SampleExample
In a health survey involving a random sample of
75 patients who developed a particular illness,
70% of them are cured of this illness by a new
drug. Establish the 95% confidence interval for
the population proportion of all the patients who
will be cured by the new drug. This would help
assess the market potential for this new drug by a
pharmaceutical company. 3) Confidence Interval for Population
ProportionLarge SampleSolution to
the Example
Applying the formula for
p
1.96
0.70 1.96 p(1  p)
p(1  p) P p 1.96
n
n = 0.70(1 0.70)
0.70(1  .70)
P 0.70 1.96
75
75 0.5963 8037
P 0. = 4) Confidence Interval for Population
MeanSmall Samplet Distribution 4) Confidence Interval for Population
MeanSmall Sample
The 1confidence interval for the population
mean is given by X n
t1 t n
1 S
S n
μX t1
n
n is the value of the t distribution with n 1
degrees of freedom for an area of in both
/2
the tails of the distribution. 4) Confidence Interval for Population
MeanSmall SampleExample
The average travel time taken based on a random
sample of 10 people working in a company to
reach the office is 40 minutes with a standard
deviation of 10 minutes.
Establish the 95% confidence interval for the
mean travel time of everyone in the company?
This will help the company redesign the working
hours. 4) Confidence Interval for Population
MeanSmall SampleSolution
Substituting in the formula below the values of
X 40 S
10 .16
3
n
10 X n
t1 S
S n
μX t1
n
n t9 is the value of the t distribution with 9
degrees of freedom for an area of 0.025 in both
the tails of the distribution. It is = 2.26 4) Confidence Interval for Population
MeanSmall SampleSolution
On simplification, we have 32.86 .14
μ 47
The probability that the true mean travel time of
everyone in the office will fall in this interval of 32.86 .14 is 95%
μ 47 5) How to Determine Sample Size
using Confidence Interval Sample Size Population Mean
2
Z2 σ
n 2
E E
Sampling Error Xμ Sample Size –Population Proportion Z 2 p(1 )
p
n
E2 E
Sampling Error  P
p 5) How to Determine Sample Size
using Confidence Interval
Example 1
A marketing manager of a fast food restaurant in a
city wishes to estimate the average yearly amount
that families spend on fast food restaurants. He
wants the estimate to be within Rs 100 with a
confidence level of 99%. It is known from an
earlier pilot study that the standard deviation of
the family expenditure on fast food restaurant is
Rs 500. How many families must be chosen for
this problem? 5) How to Determine Sample Size
using Confidence Interval
Example 1Solution
2
Z2 σ , we have
Applying the formula, n E2 2.58 2 (5002 )
n
100 2 . Simplifying we find n = 166 5) How to Determine Sample Size
using Confidence Interval
Example 2
A company manufacturing sports goods wants to
estimate the proportion of cricket players among
high school students in India. The company wants
the estimate to be within 0.03 with a confidence
level of 99%. A pilot study done earlier reveals
that out of 80 high school students, 36 students
play cricket. What should be the sample size for
this study? 5) How to Determine Sample Size
using Confidence Interval
Example 2Solution
p=36/80 =0.45
Applying the formula for calculating the sample size
Z 2 p(1 )
p
n
. Substituting, we have
2
E 2.582 (0.45)(1 )
0.45
n
0.032
n = 1831 . Simplifying, we have ...
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