ECON 3300 LEC

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ECON 3300 LEC 10

• 10/11/06

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Outline

• Sampling Distribution of
• Expected value of
• Standard deviation of
• Form of sampling distribution of
• Interval Estimate and Margin of Error of
• Sample size determination

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Sampling Distribution of

• Sample proportion - point estimator of
  population proportion p.

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Sampling Distribution of

• Sampling distribution of
• Probability distribution of all possible values
• of the sample proportion p.
• Expected value of
• E( )p
• Where
• E( ) the expected value of p
• p the population proportion

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Sampling Distribution of

• Standard Deviation of
• Finite Population Infinite
  Population
• A finite population is treated as being
  infinite if n/N lt .05.
• is the finite correction
  factor.
• is referred to as the standard error of the
  mean.

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Sampling Distribution of

• The sampling distribution of can be
  approximated by a normal probability distribution
  whenever the sample size is large.
• The sample size is considered large whenever
  these conditions are satisfied
• np gt 5
• and
• n(1 p) gt 5

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Sampling Distribution of

• For values of p near .50, sample sizes as small
  as 10 permit a normal approximation.
• With very small (approaching 0) or large
  (approaching 1) values of p, much larger samples
  are needed.

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Example St. Andrews

• Sampling Distribution of for instate residents

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Example St. Andrews

• Sampling Distribution of for instate
  residents
• What is the probability that a simple random
  sample of 100 students will provide an estimate
  of the population proportion of applicants
  desiring on-campus housing that is within plus or
  minus .05 of the actual population proportion?
• In other words, what is the probability that
• will be between .55 and .65?

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Example St. Andrews

• Sampling Distribution of for instate
  residents

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Example St. Andrews

• Sampling Distribution of for instate residents
  
• For z .05/.049 1.02, the area (.3461)(2)
  .6922.
• The probability is .6922 that the sample
  proportion will be within /-.05 of the actual
  population proportion.

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Example St. Andrews (n30)

• Sampling Distribution of
• For z .05/.0894 .559, the area
  (.2123)(2) .4246.
• The probability is .4246 that the sample
  proportion will be within /-.05 of the actual
  population proportion.

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Example

• Example The president of Doerman Distributors,
  Inc. believes that 30 of the firms orders come
  from first time customers. A simple random sample
  of 100 orders will be used to estimate the
  proportion of first-time customers.
• Assume that the president is correct and p.30.
  What is the sampling distribution of
• for this study
• What is the probability that sampling proportion
  will be between .20 and .40
• What is the probability that the sample
  proportion will be between .25 and .35

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Example

• Example Business week reported that 56 of the
  households in the United States have Internet
  access. Use a population proportion p.56 and
  assume that a sample of 300 households will be
  selected.
• Show the sampling distribution of where
• is the sample proportion of households
  that have internet access.
• b. What is the probability that the sample
  proportion will be within .03 of the population
  proportion?
• c. Answer part (b) for sample sizes of 600 and
  1000.

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Interval estimate of population proportion

• Interval estimate how close point estimate is
  to the population parameter
• General form point estimate margin of error.
• Whenever npgt5 and n(1-p)gt5, can be
  approximated by normal distribution.
• If is normally distributed, za/2sp if chosen
  as margin of error in developing interval
  estimates would result in intervals, 100(1-a) of
  which would contain the true population
  proportion.

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Interval estimate

• General expression

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• Example A national survey of 900 women golfers
  was conducted to learn how women golfers view
  their treatment at golf courses in the United
  States. The survey found that 396 f the women
  golfers were satisfied with the availability of
  the times. Construct a 95 confidence interval
  for the proportion of women golfers satisfied
  with times.

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Sample Size for an Interval Estimateof a
Population Mean

• Let E the maximum sampling error mentioned in
  the precision statement.
• E is the amount added to and subtracted from the
  point estimate to obtain an interval estimate.
• E is often referred to as the margin of error.

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Sample Size for an Interval Estimateof a
Population Mean

• Margin of Error
• Necessary Sample Size

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s unknown

• Preliminary or planning value of s
• Use the estimate of the population standard
  deviation computed from data of previous studies
  as the planning value for s
• Use a pilot study to select a preliminary sample.
  The sample standard deviation from the
  preliminary sample can be used as the planning
  value for s
• Use judgment as the best guess for s

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Example

• Example A previous study that investigated the
  cost of renting automobiles in the United States
  found a mean cost of approximately 55 per day
  for renting a midsize automobile. Suppose that
  the organization that conducted this study would
  like to conduct a new study in order to estimate
  the population mean daily rental cost for a
  midsize automobile in United States. In designing
  the new study the project director specifies that
  the population mean daily rental cost be
  estimated with a margin of error of 2 and a 95
  level of confidence

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• E2 and 95 confidence -gt z0.251.96
• From Historical data, s 9.65
• Therefore sample size

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Sample Size for an Interval Estimateof a
Population Proportion

• Let E the maximum sampling error mentioned in
  the precision statement.
• Margin of Error
• Necessary Sample Size

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p unknown

• Preliminary or planning value of p
• Use the sample proportion from a previous sample
  of the same or similar units.
• Use a pilot study to select a preliminary sample.
  The sample proportion from the preliminary sample
  can be used as the planning value for p
• Use judgment as the best guess for p
• If none of the preceding alternatives apply, use
  a planning value of p.50

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Example

• Example Company is interested to conduct a new
  survey to estimate the current proportion of the
  population of women golfers who are satisfied
  with the availability of the tee times. How large
  should the sample if the survey director wants to
  estimate the population proportion with a margin
  of erro of .025 at 95 confidence?

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• E2 and 95 confidence -gt z0.251.96
• From Historical data, p .44
• Therefore sample size

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Example

• Example Audience profile data collected at the
  ESPN SportsZone Web site showed that 26 of the
  users were women. Assume that this percentage was
  based on a sample of 400 users.
• At 95 confidence, what is the margin of error
  associated with the estimated proportion of users
  who are women?
• What is the 95 confidence interval for the
  population proportion of ESPN SportsZone Web site
  users who are women?
• How large a sample should be taken if the desired
  margin of error is .03?

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Example

• Example An American Express retail survey found
  that 16 of U.S. customers used the Internet to
  buy gifts during the holiday season. If 1285
  customers participated in the survey, what is the
  margin of error and what is the interval estimate
  of the population proportion of customers using
  the Internet to buy gifts? Use 95 confidence.