10.1 Confidence Intervals

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10.1 Confidence Intervals

• Part A

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UWG IQ- can it be used to market the university?

• Suppose you want to find the IQ of 50 students
  that attend UWG, where there are 5000 incoming
  freshmen. The mean IQ score for the sample is
  x-bar112.
• What likely is the mean of all 5000 students?
• Find the sd of the sample
• Use the 68-95-99.7 rule and describe a 95
  interval in which the mean will lie

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Some distributions
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Statistical Inference about UWG IQ

• The sample of 50 freshman gave an x-bar of 112.
  The resulting interval is 112 4.2, which can be
  written also as (107.8, 116.2). We say that we
  are 95 confident that the unknown mean IQ for
  ALL UWG freshmen is between 107.8 and 116.2.
• A confidence interval is written as
• x-bar margin of error
• 95 is the confidence level.
• This is a 95 confidence interval because it
  catches the unknown µ in 95 of all possible
  samples.

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Confidence Intervals and Levels
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A way to visualize 95 confidence
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CIs for a Population Mean (when s is known)

• Your calculation of your CI depends on 3
  conditions
• SRS- we assume that the data came from an SRS.
• The sampling distribution of x-bars is at least
  approximately Normal.
• Independence-observations must independent to
  find the sample SD-
• Be sure to check that these conditions for
  constructing a CI for µ are satisfied before you
  perform any calculations.

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Conditions for Constructing C.I
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80 Confidence

• What happens when its not a 68-95-99.7 CI?
• You must use Table A (and work backwards).
• Suppose I want an 80 CI on UWG IQ. DRAW A
  PICTURE OF THIS!!! 10 left on each side, which
  means Im actually looking for a 90 probability
  in Table A.

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80 confidence
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80 Confidence
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Our z-score is now a z

• z means that this is the number of standard
  deviations we must use to catch the correct
  confidence level, C , under our curve.
• z is still a z-score, its just a way of
  writing z, so that we know we are finding and
  interpreting CIs.
• z is called a critical value, and is only used
  when the SD is known.

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Critical Values
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Critical Values
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Critical Values- some useful zs
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CI for a Pop. Mean(when s is known)
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Example 10.5 (pg. 630)

• A manufacturer of high-res video terminals must
  control the tension on the mesh of fine wires
  that lies behind the surface of the screen. The
  SD of the tension readings is s43 mV. Here are
  the readings from a SRS of 20 from one day.
• Construct and Interpret a 90 CI for the mean
  tension of ALL the screens produced that day.

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Example 10.5

• Step 1 Parameter- what is our population of
  interest?
• Step 2 Check Conditions-
• 1) SRS
• 2) Normality (look at box plot)
• 3) Independence (population N is at least 10
  times the size of the sample)
• Step 3 Complete Calculation using
• Be sure to use Sample SD and not population
  SD!!!
• Step 4 Interpret Results
• We are 90 confident that the true mean tension
  in the batch produced that day is between ___ and
  ___.

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Toolbox for completing a CI problem

• Use the Steps in the Inference Toolbox on Pg. 631
  to help complete HW problems!

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About Margin of Error and Variability

• The LARGER your sample, the smaller the margin of
  error will be.
• LARGER samples give shorter intervals, which
  means less variability!

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10.1 Part A Homework

• 10.4
• 10.5
• 10.8
• 10.9
• 10.11

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Confidence Intervals

• As a statistician, you choose the confidence
  level, C
• The margin of error follows, after you choose
  your confidence level, C
• The BEST is to have a high C and a low M of E.

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Margin of Error get smaller when

• Z gets smaller. This is not always good to have
  a smaller Z because it means you have a smaller
  confidence level, C. So, its a trade off- which
  is your employer hoping for more?
• s gets smaller- not easy to do when the book is
  giving you this value. You can reduce the SD of
  x-bar by increasing your sample size, n.
• n gets larger- since we take the sq rt of n, we
  must take 4 times the observations to cut the
  margin of error in half.

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Determining a Sample Size

• Lets say that your employer only want a specific
  size (very small) M of E. Then you will have to
  solve backwards for your sample size, n-value.

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Example 10.7 Monkeys (pg. 633)

• We would like to estimate the mean cholesterol
  for a particular type of monkey.
• We want the estimate to be within 1 mg/dl of the
  true value of µ at a 95 confidence level.
• Previous studies show the SD for the cholesterol
  level is about s5mg/dl.
• What is the minimum of monkeys you will need
  for this estimate?

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Example 10.7 Monkeys

• Margin of Error must be less than or equal to
  1.Solve for n.
• Remember, you must have a whole number answer,
  since we want whole monkeys!

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Sample Size for a Desired M of E
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CAUTIONS! READ pg. 636-637

• The data must be an SRS from the population.
• Different methods are needed for different
  designs
• There is no correct method for inference from
  data haphazardly collected with bias
• Outliers can distort results
• The shape of the population matters, but if n
  15, then the Confidence Interval is not greatly
  affected.
• You must know the SD of the population!

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CIs in the TI-84s

• STAT- TESTS- Z-interval
• Adjust settings
• Choose Calculate

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Homework

• 10.13
• 10.15
• 10.20