Estimating a Population Mean:

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Section 7-4

• Estimating a Population Mean
• s Not Known

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• In general, we collect sample data and find the
  value of n, , and s.
• Recall, the sample mean is the best point
  estimate of the population mean ?, but doesnt
  reveal how good that estimate actually is.
• Therefore, we create confidence intervals to
  provide a range of appropriate values that we
  feel contains the true value of the mean.

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Assumptions

• Simple random sample.
• The original population is normally distributed
  and/or n gt 30.

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• Now, when s is not known, but the above
  conditions are satisfied, we use a distribution
  called the Student t distribution

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• Since we dont know s, we use s to estimate its
  value, but this is usually not the exact same
  value, so theres a chance that s is
  unreliable.
• To overcome this problem, we make the confidence
  interval wider by using the larger critical
  values of the Student t distribution (found in
  Table A-3).

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Student t Distributions for n 3 and n 12
Standard normal distribution
Student t distribution with n 12
Student t distribution with n 3
0
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• To use this table, we must know the number of
  degrees of freedom
• The number of sample values that can vary after
  certain restrictions have been imposed on all
  data values for the data set.
• For this section, the number of degrees of
  freedom is n 1

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Confidence Interval for the Estimate of µ(With s
Not Known)
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Example 1

• A publishing company has just published a new
  college textbook. Before the company decides the
  price at which to sell this textbook, it wants to
  know the average price of all such textbooks in
  the market. The research department at the
  company took a sample of 36 comparable textbooks,
  and collected information on their prices. This
  information produced a mean price of 70.50 and a
  standard deviation of 4.50 for this sample.
  Construct a 90 confidence interval.

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• 90 C.I. n 36 ? t
• Interval

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Example 2

• How many calories are there in three
  ounces of french fries?
  It depends on where you get them.
• Good Cholesterol Bad Cholesterol, by Roth and
  Streicher, sampled eight popular fast-food
  restaurants and found a mean of 244.5 calories
  and a standard deviation of 21.73 calories. Find
  a 99 confidence interval for the mean calorie
  count for 3 ounces of french fries obtained from
  fast-food restaurants. Assume that the population
  is normally distributed

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• In many cases we can find the minimum sample
  size needed to estimate some parameter (such as
  ?). To do so, we would use the following
  formula
• Notice, the sample size does not depend on the
  size of the population, but on the desired degree
  of confidence, the desired margin of error, and
  the value of ?.

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Example 3

• A Gallup poll conducted December 20-21,
  1999 asked Americans how many times they
  bathe during the week.
• How many subjects would be needed in order to
  estimate the number of times Americans bathed
  during the week in 1999 within 0.5 with 95
  confidence? Initial survey results indicate that
  s2.9

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• When using the formula for sample size, we may
• have one problem - ? is usually unknown. To
• work around this problem
• 1. Use the range rule of thumb to estimate the
• standard deviation ? range/4.
• 2. Conduct a pilot study to calculate s and use
  it
• in the place of ?.
• 3. Estimate ? by using the results of some
  other
• study previously done.