MATH 1107 Elementary Statistics

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MATH 1107Elementary Statistics

• Lecture 8
• Confidence Intervals for proportions and
  parameters

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MATH 1107 Confidence Intervals

• All around Atlanta, we are increasingly seeing
  cameras at intersections a low cost way to
  ticket people for running lights. Its an
  election year. A local representative wants to
  know what people think about the new cameras.
• Lets say that 829 people were surveyed regarding
  the photo-cop and 423 said they hated it. How
  would you report the results? What is the
  representative figure?

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MATH 1107 Confidence Intervals
Definition A confidence interval (or interval
estimate) is a range (or an interval) of values
used to estimate the true value of a population
parameter. A confidence interval is sometimes
abbreviated as CI.
A confidence level is the probability 1? (often
expressed as the equivalent percentage value)
that is the proportion of times that the
confidence interval actually does contain the
population parameter, assuming that the
estimation process is repeated a large number of
times.
This is usually 90, 95, or 99. (? 10),
(? 5), (? 1)
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In the example, we noted that 829 adults were
surveyed, and 51 of them were opposed to the use
of the photo-cop for issuing traffic tickets.
Using these survey results, find the 95
confidence interval of the proportion of all
Atlantans opposed to photo-cop use.
Confidence Interval Estimation

• We are 95 confident that the interval from
  0.476 to 0.544 does contain the true value of p.

How was this calculated?
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MATH 1107 Confidence Intervals

• Lets take a look at the formula that we would use
  to answer this question (page 429)

p ? z ?/2 ? (pq)/n
Where p population proportion z ?/2 the
appropriate Z-score based upon the selected ?
value q 1-p n number of elements in sample
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MATH 1107 Confidence Intervals

• So, how did we calculate?We are 95 confident
  that the interval from 0.476 to 0.544 does
  contain the true value of p.?

So, .51 1.96(SQRT(.51.49)/829) .51 .034 or
.476 to .544 (.51-.034 .476 and .51.034
.544)
Here, p.51 q.49 z1.96 n829
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MATH 1107 Confidence Intervals
Where in the world did Z1.96 come from?
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MATH 1107 Confidence Intervals

• What if I wanted to be 90 confident?
• What if I wanted to be 99 confident?

Typical Z scores used in CI Estimation 90
confidence 1.645 95 confidence 1.96 99
confidence 2.575
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MATH 1107 Confidence Intervals
Go to http//www.gallup.com/ Lets replicate the
findings from the Gallup poll of the day.
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MATH 1107 Confidence Intervals
Question what is the relationship between the
Margin of Error and the confidence level? If we
need to be more confident, what happens to the
Margin of Error? What if we need to maintain the
MOE, AND increase the level of confidence?
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MATH 1107 Confidence Intervals

• Conceptually, we use the same process for
  estimating the confidence interval of a
  parameter

-
x ? z ?/2 (s/SQRT(n))
Where x sample mean z ?/2 the appropriate
Z-score s sample standard deviation n number
of elements in sample
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MATH 1107 Confidence Intervals
Lets say that we took a poll of 100 KSU students
and determined that they spent an average of 225
on books in a semester with a std dev of 50.
Report the 95 confidence interval for the
expenditure on books for ALL KSU students. Now,
assuming that you need to maintain this MOE, but
at a 99 confidence, what is the new sample size?
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MATH 1107 Confidence Intervals
Fun EXCEL exercise !