Statistics 300: Elementary Statistics

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Statistics 300Elementary Statistics

• Sections 7-2, 7-3, 7-4, 7-5

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Parameter Estimation

• Point Estimate
• Best single value to use
• Question
• What is the probability this estimate is the
  correct value?

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Parameter Estimation

• Question
• What is the probability this estimate is the
  correct value?
• Answer
• zero assuming x is a continuous random
  variable
• Example for Uniform Distribution

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If X U100,500 then

• P(x 300) (300-300)/(500-100)
• 0

100 300 400
500
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Parameter Estimation

• Pop. mean
• Sample mean
• Pop. proportion
• Sample proportion
• Pop. standard deviation
• Sample standard deviation

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Problem with Point Estimates

• The unknown parameter (m, p, etc.) is not exactly
  equal to our sample-based point estimate.
• So, how far away might it be?
• An interval estimate answers this question.

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Confidence Interval

• A range of values that contains the true value of
  the population parameter with a ...
• Specified level of confidence.
• L(ower limit),U(pper limit)

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Terminology

• Confidence Level (a.k.a. Degree of Confidence)
• expressed as a percent ()
• Critical Values (a.k.a. Confidence Coefficients)

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Terminology

• alpha a 1-Confidence
• more about a in Chapter 7
• Critical values
• express the confidence level

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Confidence Interval for mlf s is known (this is
a rare situation)
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Confidence Interval for mlf s is known (this is
a rare situation) if x N(?,s)
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Why does the Confidence Interval for m look like
this ?
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Using the Empirical Rule
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Check out the Confidence z-scoreson the WEB
page. (In pdf format.)
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Use basic rules of algebra to rearrange the parts
of this z-score.
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Confidence 95a 1 - 95 5 a/2 2.5
0.025
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Confidence 95a 1 - 95 5 a/2 2.5
0.025
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Confidence Interval for mlf s is not known
(usual situation)
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Sample Size Neededto Estimate m within E,with
Confidence 1-a
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Components of Sample SizeFormula when Estimating
m

• Za/2 reflects confidence level
• standard normal distribution
• is an estimate of , the standard
  deviation of the pop.
• E is the acceptable margin of error when
  estimating m

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Confidence Interval for p

• The Binomial Distribution gives us a starting
  point for determining the distribution of the
  sample proportion

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For Binomial x
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For the Sample Proportionx is a random
variablen is a constant
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Time Out for a PrincipleIf is the mean of
X and a is a constant, what is the mean of
aX?Answer .
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Apply that Principle!

• Let a be equal to 1/n
• so
• and

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Time Out for another PrincipleIf is the
variance of X and a is a constant, what is the
variance of aX?Answer .
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Apply that Principle!

• Let x be the binomial x
• Its variance is npq np(1-p), which is the
  square of is standard deviation

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Apply that Principle!

• Let a be equal to 1/n
• so
• and

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Apply that Principle!
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When n is Large,
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What is a Large nin this situation?

• Large enough so np gt 5
• Large enough so n(1-p) gt 5
• Examples
• (100)(0.04) 4 (too small)
• (1000)(0.01) 10 (big enough)

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Now make a z-score
And rearrange for a CI(p)
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Using the Empirical Rule
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Use basic rules of algebra to rearrange the parts
of this z-score.
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Confidence Interval for p(but the unknown p is
in the formula. What can we do?)
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Confidence Interval for p(substitute sample
statistic for p)
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Sample Size Neededto Estimate p within E,with
Confid.1-a
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Components of Sample SizeFormula when Estimating
p

• Za/2 is based on a using the standard normal
  distribution
• p and q are estimates of the population
  proportions of successes and failures
• E is the acceptable margin of error when
  estimating m

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Components of Sample SizeFormula when Estimating
p

• p and q are estimates of the population
  proportions of successes and failures
• Use relevant information to estimate p and q if
  available
• Otherwise, use p q 0.5, so the product pq
  0.25

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Confidence Interval for sstarts with this
factthen
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What have we studied already that connects with
Chi-square random values?
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Confidence Interval for s