Estimating Population Values

1
Chapter 7

• Estimating Population Values

2
Chapter 7 - Chapter Outcomes

• After studying the material in this chapter, you
  should be able to
• Distinguish the difference between a point
  estimate and a confidence interval estimate.
• Construct and interpret a confidence interval
  estimate for a single population mean using both
  the z and t distributions.

3
Chapter 7 - Chapter Outcomes(continued)

• After studying the material in this chapter, you
  should be able to
• Determine the required sample size for an
  estimation application involving a single
  population mean.
• Establish and interpret a confidence interval
  estimate for a single population proportion.

4
Point Estimates

• A point estimate is a single number determined
  from a sample that is used to estimate the
  corresponding population parameter.

5
Sampling Error

• Sampling error refers to the difference between a
  value (a statistic) computed from a sample and
  the corresponding value ( a parameter) computed
  from a population.

6
Confidence Intervals

• A confidence interval refers to an interval
  developed from sample values such that if all
  possible intervals of a given width were
  constructed, a percentage of these intervals,
  known as the confidence level, would include the
  true population parameter.

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Confidence Intervals
Lower Confidence Limit
Upper Confidence Limit
Point Estimate
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95 Confidence Intervals(Figure 7-3)
0.95
z.025 -1.96
z.025 1.96
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Confidence Interval- General Format -
Point Estimate ? (Critical Value)(Standard Error)
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Confidence Intervals

• The confidence level refers to a percentage
  greater than 50 and less than 100 that
  corresponds to the percentage of all possible
  confidence intervals, based on a given size
  sample, that will contain the true population
  value.

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Confidence Intervals

• The confidence coefficient refers to the
  confidence level divided by 100 -- i.e., the
  decimal equivalent of a confidence level.

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Confidence Interval- General Format ? known -
Point Estimate ? z (Standard Error)
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Confidence Interval Estimates

• CONFIDENCE INTERVAL ESTIMATE FOR ? (? KNOWN)
• where
• z Critical value from standard normal table
• ? Population standard deviation
• n Sample size

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Example of a Confidence Interval Estimate for ?

• A sample of 100 cans, from a population with ?
  0.20, produced a sample mean equal to 12.09. A
  95 confidence interval would be

12.051 ounces
12.129 ounces
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Special Message about Interpreting Confidence
Intervals

• Once a confidence interval has been constructed,
  it will either contain the population mean or it
  will not. For a 95 confidence interval, if you
  were to produce all the possible confidence
  intervals using each possible sample mean from
  the population, 95 of these intervals would
  contain the population mean.

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Margin of Error

• The margin of error is the largest possible
  sampling error at the specified level of
  confidence.

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Margin of Error

• MARGIN OF ERROR (ESTIMATE FOR ? WITH ? KNOWN)
• where
• e Margin of error
• z Critical value
• Standard error of the sampling
  distribution

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Example of Impact of Sample Size on Confidence
Intervals

• If instead of sample of 100 cans, suppose a
  sample of 400 cans, from a population with ?
  0.20, produced a sample mean equal to 12.09. A
  95 confidence interval would be

12.0704 ounces
12.1096 ounces
n400
n100
12.051 ounces
12.129 ounces
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Students t-Distribution

• The t-distribution is a family of distributions
  that is bell-shaped and symmetric like the
  standard normal distribution but with greater
  area in the tails. Each distribution in the
  t-family is defined by its degrees of freedom.
  As the degrees of freedom increase, the
  t-distribution approaches the normal distribution.

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Degrees of freedom

• Degrees of freedom refers to the number of
  independent data values available to estimate the
  populations standard deviation. If k parameters
  must be estimated before the populations
  standard deviation can be calculated from a
  sample of size n, the degrees of freedom are
  equal to n - k.

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t-Values

• t-VALUE
• where
• Sample mean
• Population mean
• s Sample standard deviation
• n Sample size

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Confidence Interval Estimates

• CONFIDENCE INTERVAL
• (? UNKNOWN)
• where
• t Critical value from t-distribution with
  n-1 degrees of freedom
• Sample mean
• s Sample standard deviation
• n Sample size

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Confidence Interval Estimates

• CONFIDENCE INTERVAL-LARGE SAMPLE WITH ? UNKNOWN
• where
• z Value from the standard normal
  distribution
• Sample mean
• s Sample standard deviation
• n Sample size

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Determining the Appropriate Sample Size

• SAMPLE SIZE REQUIREMENT - ESTIMATING ? WITH ?
  KNOWN
• where
• z Critical value for the specified
  confidence interval
• e Desired margin of error
• ? Population standard deviation

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Pilot Samples

• A pilot sample is a sample taken from the
  population of interest of a size smaller than the
  anticipated sample size that is used to provide
  and estimate for the population standard
  deviation.

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Example of Determining Required Sample
Size(Example 7-7)

• The manager of the Georgia Timber Mill wishes to
  construct a 90 confidence interval with a margin
  of error of 0.50 inches in estimating the mean
  diameter of logs. A pilot sample of 100 logs
  yield a sample standard deviation of 4.8 inches.

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Estimating A Population Proportion

• SAMPLE PROPORTION
• where
• x Number of occurrences
• n Sample size

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Estimating a Population Proportion

• STANDARD ERROR FOR p
• where
• ? Population proportion
• n Sample size

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Confidence Interval Estimates for Proportions

• CONFIDENCE INTERVAL FOR ?
• where
• p Sample proportion
• n Sample size
• z Critical value from the standard normal
  distribution

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Example of Confidence Interval for
Proportion(Example 7-8)

• 62 out of a sample of 100 individuals who were
  surveyed by Quick-Lube returned within one month
  to have their oil changed. To find a 90
  confidence interval for the true proportion of
  customers who actually returned

0.70
0.54
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Determining the Required Sample Size

• MARGIN OF ERROR FOR ESTIMATING ?
• where
• ? Population proportion
• z Critical values from standard normal
  distribution
• n Sample size

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Determining the Required Sample Size

• SAMPLE SIZE FOR ESTIMATING ?
• where
• ? Value used to represent the population
  proportion
• e Desired margin of error
• z Critical value from the standard normal
  table

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Key Terms

• Confidence Coefficient
• Confidence Interval
• Confidence Level
• Degrees of Freedom
• Margin of Error

• Pilot Sample
• Point Estimate
• Sampling Error
• Students t-distribution