ECON 2300 LEC

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ECON 2300 LEC 9

• 10/09/06

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Outline

• Interval Estimate
• Population Mean s known
• Margin of Error and Interval Estimate
• Population Mean s unknown
• Margin of Error and Interval Estimate
• Summary

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Interval Estimate

• Point estimator cannot be expected to provide the
  exact value of the population parameter
• Interval Estimate Provides information on how
  close the point estimate provided by the sample,
  is to the value of the population parameter.
• General Form Point estimate Margin of error.
• Population mean

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Sampling Distribution of x

• Example Director of personnel for Electronics
  Associates, Inc. (EAI), has been assigned the
  task of developing a profile of the companys
  2500 managers. The characteristics to be
  identified include the mean annual salary for the
  managers and the proportion of managers having
  completed the companys management training
  program.
• Using the population data, the population mean
  was obtained equal to 51,800 and population
  standard deviation equal to 4000 and population
  proportion equal to 0.6

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Sampling Distribution of x

• Using a sample of 30 EAI managers as shown in the
  table, the following results obtained
• Sample mean 51814
• Sample proportion 0.63
• Another sample yields
• Sample mean 52760
• Sample proportion 0.7
• If we repeat the same process over and over again
  and compute the statistical values, the resulting
  distribution sampling distribution

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Population Mean s known

• Population standard deviation known - When large
  amounts of historical data available
• - Quality control applications with processes
  operating under control

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Margin of Error and Interval Estimate

• St. Andrews College receives 900 applications
• annually from prospective students. The
  application
• forms contain a variety of information including
  the
• individuals scholastic aptitude test (SAT) score
  and
• whether or not the individual desires on-campus
• housing.

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Margin of Error and Interval Estimate

• The director of admissions would like to know
• the following information
• the average SAT score for the applicants, and
• the proportion of applicants that want to live on
  campus.

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Margin of Error and Interval Estimate

• Sampling Distribution of for the SAT Scores

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Margin of Error and Interval Estimate

• Sampling distribution of mean in example is
  normally distributed with a standard error of
  14.6 (n30 Central Limit Theorem)
• Using tables of areas for the standard normal
  distribution 95 of the values of any normally
  distributed random variable are within 1.96
  standard deviations of the mean.
• 95 of all the values lie within 28.616 of
  the population mean 990

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Sampling Distribution of x
95 of all values
µ
1.96sx
1.96sx
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Interval Estimate of a Population
MeanLarge-Sample Case

• ? ?Assumed Known
• where is the sample mean
• 1 -? is the confidence coefficient
• z?/2 is the z value providing an area of
• ?/2 in the upper tail of the
• standard normal
  probability
• distribution
• s is the population standard deviation
• n is the sample size

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Interval Estimate of a Population
MeanLarge-Sample Case

• Example Each week Lloyds Department Store
  selects a simple random sample of 100 customers
  in order to learn about the amount spent per
  shopping trip. Based on historical data, Lloyds
  now assumes a known value of s 20 for the
  population standard deviation. The historical
  data also indicate that the population follows a
  normal distribution. During the most recent week,
  Lloyds surveyed 100 customers (n100) and
  obtained a sample mean of 82.

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Interval Estimate of a Population
MeanLarge-Sample Case (n gt 30)

• To obtain a 95 confidence Interval

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Interval Estimate of a Population
MeanLarge-Sample Case (n gt 30)

• To obtain a 90 confidence Interval

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Population Mean s unknown

• Used when a good estimate of the population
  standard deviation is not available.
• Same sample to be used to estimate µ and s
• Margin of error and interval estimate are based
  on a probability distribution known as the t
  distribution.
• t distribution holds good for situations where
  population slightly deviates from normal
  distribution.

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Population Mean s unknown

• The t distribution is a family of distributions
  with a specific t distribution depending on a
  parameter known as the degrees of freedom.
• As number of degrees of freedom increases the
  difference between t and the standard normal
  distribution becomes smaller and smaller.
• A t distribution with more degrees of freedom
  has less dispersion.
• The mean of the t distribution is zero.

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t Distribution
Standard normal distribution
t distribution (20 degrees of freedom)
t distribution (10 degrees of freedom)
0
z, t
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t Distribution

• a/2 Area or Probability in the Upper Tail

?/2
0
ta/2
t
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Interval Estimation of a Population
MeanSmall-Sample Case (n lt 30) and ? Estimated
by s

• Interval Estimate
• where 1 -? the confidence coefficient
• t?/2 the t value providing an
  area
• of ?/2 in the upper
  tail of a t
• distribution with n -
  1 degrees
• of freedom
• s the sample standard deviation

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Degrees of freedom

• The reason the number of degrees of freedom
  associated with the t value in the expression is
  n-1 concerns the use of s as an estimate of the
  population standard deviation.
• Degrees of freedom Number of independent pieces
  of information that go into the computation of

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Degrees of freedom
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• Example Consider a study designed to estimate
  the mean credit card debt for the population of
  U.S. households. A sample of n45 households
  provided the credit card balances shown in the
  table. No previous estimate of the population
  standard deviation s is available. Thus sample
  data is used to estimate both the population mean
  and standard deviation.

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Credit Card Balances (n45)
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• Mean 5144
• Standard deviation 2927
• 95 confidence and n-1 44 degrees of freedom.
• Interval estimate51442.014x2927/sqrt(44)
• 5144889

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Using a Small Sample

• Scheer Industries is considering a new
  computer-assisted program to train maintenance
  employees to do machine repairs. In order to
  fully evaluate the program, the director of
  manufacturing requested an estimate of the
  population mean time required for maintenance
  employees to complete the computer-assisted
  training.

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Using a Small Sample

• Sample of 20 employees taken

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Using a Small Sample
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Statistical Precision

• It can be thought of as the narrowness of the
  margin of error.
• Larger sample size would result in a high
  precision
• Directly proportional to the square root of
  sample size -To cut a margin of error in half,
  increase the sample size by a factor of four
• The margin of error is also influenced by our
  level of significance or confidence level- A 99
  confidence interval will be wider than a 95
  confidence interval or less precise

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Summary of Interval Estimation Procedures for a
Population Mean
Yes
No
s known
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Examples

• Example The mean number of hours of flying times
  for pilots at Continental Airlines is 49 hours
  per month. Assume that this mean was based on
  actual flying times for a sample of 100
  Continental pilots and that the population
  standard deviation was 8.5 hours. Obtain a 95
  percent confidence Interval for the mean flying
  hours.

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Examples

• Example Sales personnel for Skillings
  Distributors submit weekly reports listing the
  customer contacts made during the week. A sample
  of 65 weekly reports showed a sample mean of 9.5
  customer contacts per week. The sample standard
  deviation was 5.2 hours. Obtain a 90 confidence
  interval for the mean customer contacts for the
  sales personnel.