Estimating a Population Mean: s Known

1
Section 7-3

• Estimating a Population Mean s Known

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ASSUMPTIONS s KNOWN

• The sample is a simple random sample.
• The value of the population standard deviation s
  is known.
• Either or both of the following conditions are
  satisfied
• The population is normally distributed.
• n gt 30

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SAMPLE MEANS

• For many populations, the distribution of sample
  means tends to be more consistent (with less
  variation) than the distributions of other sample
  statistics.
• For all populations, the sample mean is an
  unbiased estimator of the population mean µ,
  meaning that the distribution of sample means
  tends to center about the value of the population
  mean µ.

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MARGIN OF ERROR FOR THE MEAN
The margin of error for the mean is the maximum
likely difference observed between sample mean x
and population mean µ, and is denoted by E. When
the standard deviation, s, for the population is
known, the margin of error is given by
where 1 - a is the desired confidence level.
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CONFIDENCE INTERVAL ESTIMATE OF THE POPULATION
MEAN µ (WITH s KNOWN)
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CONSTRUCTING A CONFIDENCE INTERVAL FOR µ (s KNOWN)

• Verify that the required assumptions are met.
• Refer to Table A-2 and find the critical value
  za/2 that corresponds to the desired confidence
  interval.
• Evaluate the margin of error
• Find the values of x E and x E.
  Substitute these in the general format of the
  confidence interval x E lt µ lt x E.
• Round the result using the round-off rule on the
  next slide.

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ROUND-OFF RULE FOR CONFIDENCE INTERVALS USED TO
ESTIMATE µ

• When using the original set of data to construct
  the confidence interval, round the confidence
  interval limits to one more decimal place than is
  used for the original data set.
• When the original set of data is unknown and only
  the summary statistics (n, x, s) are used, round
  the confidence interval limits to the same number
  of places as used for the sample mean,

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FINDING A CONFIDENCE INTERVAL FOR µ WITH TI-83/84

• Select STAT.
• Arrow right to TESTS.
• Select 7ZInterval.
• Select input (Inpt) type Data or Stats. (Most
  of the time we will use Stats.)
• Enter the standard deviation, s.
• Enter the sample mean, x.
• Enter the size of the sample, n.
• Enter the confidence level (C-Level).
• Arrow down to Calculate and press ENTER.

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SAMPLE SIZE FORESTIMATING µ
where za/2 critical z score based on desired
confidence level E desired margin of
error s population standard deviation
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ROUND-OFF RULE FOR SAMPLE SIZE n
When finding the sample size n, if the use of the
formula on the previous slide does not result in
a whole number, always increase the value of n to
the next larger whole number.
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FINDING THE SAMPLE SIZE WHEN s IS UNKNOWN

• Use the range rule of thumb (see Section 3-3) to
  estimate the standard deviation as follows s
  range/4.
• Conduct a pilot study by starting the sampling
  process. Based on the first collection of at
  least 31 randomly selected sample values,
  calculate the sample standard deviation s and use
  it in place of s.
• Estimate the value of s by using the results of
  some other study that was done earlier.