Estimation

Session 10 & 11
Using Sample Statistics to Estimate
Population Parameters

Source: Black, K. (2010), Business Statistics for Contemporary Decision Making, John Wiley & Sons, p. 252
Point & Interval Estimate

 A point estimate is a statistic taken from a sample that

is used to estimate a population parameter.
 A point estimate is only as good as the
representativeness of its sample.
 Because of variation in sample statistics, estimating a
population parameter with an interval estimate is often
preferable to using a point estimate.
 An interval estimate (confidence interval) is a range of
values within which the analyst can declare, with some
confidence, the population parameter lies.
Estimating the Population mean using
the z-statistics (σ known)
 Using the z formula for sample means, the
population mean formula can be written as

 Because a sample mean can be greater

than or less than the population mean, z
can be positive or negative, the preceding
expression takes the following form.
100(1 - α)% Confidence Interval to
estimate µ: σ known
μ=

Source: Black, K. (2010), Business Statistics for Contemporary Decision Making, John Wiley & Sons, p. 254
Confidence Interval to estimate µ: σ
known
 The more common levels of confidence used by
business researchers are 90%, 95%, 98%, and
99%.
 For 95% confidence, α = .05 and α/2 = .025. The
value of or zα/2 of z.025 is found by looking in the
standard normal table under 0.5000 - 0.0250 =
0.4750. This area in the table is associated with
a z value of 1.96.
What does a 95% confidence interval
indicate?
 It indicates that, if the company researcher
were to randomly select 100 samples of
size n and use the results of each sample
to construct a 95% confidence interval,
approximately 95 of the 100 intervals
would contain the population mean.
 It also indicates that 5% of the intervals
would not contain the population mean.
Exercise

 A survey was taken of U.S. companies that do business

with firms in India. One of the questions on the survey
was: Approximately how many years has your company
been trading with firms in India? A random sample of 44
responses to this question yielded a mean of 10.455
years. Suppose the population standard deviation for this
question is 7.7 years. Using this information, construct a
90% confidence interval for the mean number of years
that a company has been trading in India for the
population of U.S. companies trading with firms in India.
Exercise
 For a random sample of 36 items and a sample mean of
211, compute a 95% confidence interval for if the
population standard deviation is 23.
 A random sample of 81 items is taken, producing a
sample mean of 47. The population standard deviation is
5.89. Construct a 90% confidence interval to estimate
the population mean.
 A random sample of size 70 is taken from a population
that has a variance of 49. The sample mean is 90.4.
What is the point estimate of µ? Construct a 94%
confidence interval for µ
Exercise

 One of the properties of a good quality paper is

its bursting strength. Suppose a sample of 16
specimens yields mean bursting strength of 25
units, and it is known from the history of such
tests that the s.d. among specimens is 5 units.
Assuming the normality of test results, what are
the (i) 95 % and (ii) 98% confidence limits for the
mean bursting strength from this sample?
Confidence interval to estimate µ using
the finite correction factor

 Exercise: A study is conducted in a company that

employs 800 engineers. A random sample of 50
engineers reveals that the average sample age is 34.3
years. Historically, the population standard deviation of
the age of the company’s engineers is approximately 8
years. Construct a 98% confidence interval to estimate
the average age of all the engineers in this company.
Exercise
 A random sample of size 39 is taken from a
population of 200 members. The sample mean
is 66 and the population standard deviation is
11. Construct a 96% confidence interval to
estimate the population mean. What is the point
estimate of the population mean?
Estimating the Population Mean Using
the z Statistic when the Sample Size Is
Small
 The Central Limit theorem applies only when sample
size is large, but the distribution of sample means is
approximately normal even for small sizes if the
population is normally distributed.
 If it is known that the population from which the sample
is being drawn is normally distributed and if σ is known,
the z formulae can still be used to estimate a population
mean even if the sample size is small
Estimating the population mean using
the t-statistics (σ unknown)
 The z formulae are inappropriate for use
when the population standard deviation is
unknown
 Under these circumstances, we use the t
statistics (or Student’s t) developed by
William Gosset
The t-distribution
 t distribution is used instead of the z distribution for
doing inferential statistics on the population mean when
the population standard deviation is unknown and the
population is normally distributed.
 The formula of t-statistics

 The t distribution actually is a series of distributions

because every sample size has a different distribution

Source: Black, K. (2010), Business Statistics for Contemporary Decision Making, John Wiley & Sons, p. 261
t Distribution
 Degrees of freedom refers to the number of
independent observations for a source of variation minus
the number of independent parameters estimated in
computing the variation.
 In this case, one independent parameter, the population
mean, µ, is being estimated by in computing s. Thus,
the degrees of freedom formula is n independent
observations minus one independent parameter being
estimated (n - 1).
 The degrees of freedom for the t statistic are computed
by n - 1
Looking up the value of t for 90%
confidence

Source: Black, K. (2010), Business Statistics for Contemporary Decision Making, John Wiley & Sons, p. 262
Confidence interval to estimate µ: σ
unknown and population normally
distributed
Exercise
 Suppose the following data are selected randomly from a
population of normally distributed values. Construct a
95% confidence interval to estimate the population
mean.
 40, 51, 43, 48, 44, 57, 54, 39, 42, 48, 45, 39, 43

 Assuming x is normally distributed, use the following

information to compute a 90% confidence interval to
estimate.
 313, 320, 319, 340, 325, 310, 321, 329, 317, 311, 307, 318
Exercise
 For assessing the number of monthly
transactions in credit card issued by a bank,
transactions in 25 cards were analyzed. The
analysis revealed an average of 7.4
transactions and sample standard deviation of
2.25 transactions. Find confidence limits for the
monthly number of transactions by all the credit
card holders of the bank.
Estimating the population proportion

 The central limit theorem for sample

proportions formula is given as

 However, for confidence interval purposes

only and for large sample sizes is
substituted for p in the denominator,
yielding
Confidence interval to estimate p
Exercise
 A clothing company produces men’s jeans. The jeans
are made and sold with either a regular cut or a boot cut.
In an effort to estimate the proportion of their men’s
jeans market in Pune City that prefers boot-cut jeans, the
analyst takes a random sample of 212 jeans sales from
the company’s two retail outlets. Only 34 of the sales
were for boot-cut jeans. Construct a 90% confidence
interval to estimate the proportion of the population in
Oklahoma City who prefer boot-cut jeans.
Exercise
 Suppose a random sample of 85 items has been taken
from a population and 40 of the items contain the
characteristic of interest. Use this information to
calculate a 90% confidence interval to estimate the
proportion of the population that has the characteristic
of interest. Calculate a 95% confidence interval.
Calculate a 99% confidence interval. As the level of
confidence changes and the other sample information
stays constant, what happens to the confidence
interval?
 Summary of formulae for
confidence limits
Finite Population Infinite Population
(n/N>0.05) (n/N < 0.05)

Estimating μ, when σ is known

When σ is not known ( and sample

size larger than 30

When σ is not known ( and sample

size < 30 and population is
approximately normal or normal
Estimating p (population proportion)
when n is larger than 30,
Estimating Sample Size when
estimating µ
 When μ is being estimated, the size of sample
can be determined by using the z formula for
sample means to solve for n.
 The difference between and µ is the error of
estimation resulting from the sampling process.
Let be the error of estimation.
 Sample size while estimating µ
Estimating Sample Size when
estimating µ
 Sometimes in estimating sample size the
population variance is known or can be
determined from past studies.
 Other times, the population variance is unknown
and must be estimated to determine the sample
size.
 Insuch cases, it is acceptable to use the following
estimate to represent .
Determine sample size when
estimating p
Exercise
 A bank officer wants to determine the amount of
the average total monthly deposits per customer
at the bank. He believes an estimate of this
average amount using a confidence interval is
sufficient. How large a sample should he take to
be within $200 of the actual average with 99%
confidence? He assumes the standard deviation
of total monthly deposits for all customers is
about $1,000.
Exercise
 A group of investors wants to develop a chain of fast-
food restaurants. In determining potential costs for each
facility, they must consider, among other expenses, the
average monthly electric bill. They decide to sample
some fast-food restaurants currently operating to
estimate the monthly cost of electricity. They want to be
90% confident of their results and want the error of the
interval estimate to be no more than $100. They
estimate that such bills range from $600 to $2,500. How
large a sample should they take?
Exercise
 Suppose a production facility purchases a
particular component part in large lots from a
supplier. The production manager wants to
estimate the proportion of defective parts
received from this supplier. She believes the
proportion defective is no more than .20 and
wants to be within .02 of the true proportion of
defective parts with a 90% level of confidence.
How large a sample should she take?