Chapter%208%20Interval%20Estimation

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Chapter 8 Interval Estimation

• I. Basic Definitions
• II. Confidence Interval Estimation
• III. Sample Size and Precision
• I. Basic Definitions
• Point estimate a single value of sample
  statistic as an estimator of population
  parameter.
• Sampling error Sample Statistic Population
  Parameter
• Confidence interval An interval with the
  confidence level. It is constructed on the
  basis of a sample sample statistic margin of
  error.
• Confidence level of intervals formed by
  samples with sample size of n will contain the
  population parameter. (1-?)
• Margin of error half-width of the confidence
  interval.

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II. Confidence Interval Estimation Outline Proce
dure 1. Sample ? point estimator ( or )
2. Confidence level and Table ? Z or tn-1 3.
Formulas ? compute UCL and LCL point
estimator ? margin of error
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Confidence Interval Estimation of Population Mean
(s Known)
1-a.99 z2.575 1-a.95 z1.96 1-a.90
z1.645
1-a
?
UCL
LCL
--------------------- ---------------------
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1. Interval Estimation for Population
Mean Example 1 p.315 5 (? known case) In an
effort to estimate the mean amount spent per
customer for dinner at a major Atlanta
restaurant. Data were collected for a sample of
49 customers over a three-week period. Assume a
population standard deviation of 5. a. At
the 95 confidence, what is the margin error? b.
If the sample mean is 24.80, What is the 95
confidence interval for the population
mean? HWK p.315 8
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• Example 1 p.315 5
• Answer
• n 49
• ?5
• Z (1-?)/2 0.95/2 0.475 ? Table 1 Z
  1.96
• 1.
• 2.
• ? 23.4, 26.2

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• Example 2 p.324 16 (? unknown case)
• The mean flying time for pilots at Continental
  Airlines is 49 hours per month. This mean was
  based on a sample of 100 pilots and the sample
  standard deviation was 8.5 hours.
• At 95 confidence, what is the margin of error?
• What is the 95 confidence interval estimate of
  the population mean flying time?
• The mean flying time for pilots at United
  Airlines is 36 hours per month. Discuss
  difference between the flying times at two
  airlines.
• HWK p.324 16

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• Example 2 p.324 16
• Given n 100, 49, S 8.5, 1-? .95
• Think What to estimate? Use Z or t?
• Answer
• Sample info (given) n 100, 49, S 8.5
• t 1- ?0.95, so ?/20.025, d.f.n-199 ?
• Table 2 d.f.100, ?/20.025 ? t1.984
• d.f.80, ?/20.025 ? t1.990
• Interpolation
• a. m.o.e.
• b. UCL 491.69 50.69 LCL 49 1.69
  47.31
• ? 47.31, 50.69
• c. 36 lt LCL. The mean flying time is lower at
  United.

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• Aside
• Why t?
• t distribution
• flatter than Z - more spread out
• n ?, t ? Z
• Degree of freedom a measure of precision.
• The d.f. ? ? more precise.

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• 2. Interval Estimation for Population Proportion
• Example 3
• A survey asked 346 job seekers. The answer
  selected most (152 times) was higher
  compensation.
• What is the point estimate of the proportion of
  job seekers who would select higher
  compensation as the reason of changing jobs?
• What is the 95 confidence interval estimate of
  the population proportion?
• HWK p.332 35

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• Example 3
• Answer
• a. Point estimate of p
• b. Confidence interval
• Z (1-?)/20.475, Table 1 ? Z1.96
• Margin of error
  0.0523
• Confidence interval
• p .3870, .4916

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• III. Sample Size and Precision
• Quality of estimation
• Confidence level 1 - ?
• Precision margin of error
• Confidence level 1 - ? is guaranteed by
  procedure.
• P.312 Figure 8.3
• sampling distribution for sample mean.
• Probability that
• is 1 - ?. In general, any sample mean that is
  within this range will provide an interval that
  contains the population mean ?.
• Margin of error
• Given n, then (1 - ?)?? ? margin of error ??.
• Given 1 - ?, then n ?? ? margin of error ??.

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Determine sample size to meet requirements for
both confidence level and margin of error 1.
Determine sample size for estimation of ?
(p.326) E desired margin of error round
up (Be conservative) 2. Determine sample size
for estimation of p (p.330) p .5 (p.319
Table 8.5 Be conservative) E desired margin
of error round up (Be conservative) Homew
ork p.327 25, p.332 39
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• Example 4
• Brides magazine reported that the mean cost of
  a wedding is 19,000. Assume that the population
  standard deviation is 9,400. Use 95 confidence,
• What is the recommended sample size if the
  desired margin of error is 1,000?
• What is the recommended sample size if the
  desired margin of error is 500?
• Answer
• a.
• b.

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Example 6 The League of American Theatres and
Producers uses an ongoing audience tracking
survey that provides up-to-date information about
Broadway theater audiences. Every week, the
League distributes a one-page survey on random
theater seats at a rotation roster of Broadway
shows. a. How large a sample should be taken if
the desired margin of error on any proportion is
0.04? Use 95 confidence. Answer