Week 8

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Statistical Inference
Week 8
To guess is cheap, To guess wrongly is
expensive. An old Chinese proverb
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OUTLINE
1. Statistical Inference 2. Point and Interval
Estimation 3. Confidence Intervals RATIONALE 4.
Assumptions and Conditions 5. A Six-Step Model 6.
Estimation of a population mean (? known) 7.
Choosing the Sample Size
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To recap Central Limit Theorem
When n is large
68 probability that our will be in this region
95 probability that our will be in this region
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STANDARD ERROR OF THE SAMPLE MEAN

• The standard error of the sample mean is computed
  by
• 
  
• is the symbol for the standard error of
  the
• sample mean.
• s is the standard deviation of the population.
• n is the size of the sample.

Shows how closely to population mean the sample
mean tends to be. It is a TYPICAL deviation of a
sample mean relative to the POPULATION mean.
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EXERCISE 1
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1. Statistical Inference
A population is a complete set of observations,
patients, entities, or measurements about which
we would like to draw conclusions.

• Statistical inference is a formal process that
  uses information from a sample to draw
  conclusions about a population.
• It also provides a statement of how much
  confidence can be placed in the conclusion.

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Statistical Inference

• Estimation
• Estimating unknown value of a population
  parameter.
• Hypothesis testing
• Making decisions about the value of a parameter
  by testing a pre-conceived hypothesis
• - Both types of of inference are based on the
  sampling distribution of statistics.
• - Both report probabilities that state what
  would happen if we used the inference method many
  times.

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Statistical Inference

• When do we use estimation?
• When we want to estimate the unknown population
  parameters and we do not have any previous
  knowledge about the population.

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2. POINT AND INTERVAL ESTIMATION

• Point Estimate
• is a single number (our best guess), calculated
  from available sample data, that is used to
  estimate the value of an unknown population
  parameter.
• Interval Estimate (Confidence Interval)
• is an interval that provides an upper and lower
  bound for a specific unknown population
  parameter.
• -gt arguably the most useful type of inference.

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3. Confidence Intervals RATIONALE
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Following are the data obtained from a study
involving 4 female rest home patients
Platelet counts (in
000, per mm3) 125 170
144 101 The sample mean is 135, and appears to
be a reasonable estimate of the population
mean. Can we conclude that the POPULATION mean
is 135? Why? How reliable is this estimate? A
second sample would definitely not give the same
mean again. A point estimate like this is of
little value unless we indicate its variability.
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Let us suppose that we know (unrealistically)
that the standard deviation, ? of the population
is 50. We know (CLT!) that in repeated sampling
the sample mean follows the normal distribution
centred at the unknown population mean. The
standard deviation of that distribution is
50/2 25
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• We say that we are 99.7 confident that the
  unknown population mean for all SA elderly women
  lies between 60 and 210.
• There are only two possibilities

The CI contains the true ?
The interval of numbers between the values
? 75 is called a 99.7 confidence interval for ? .
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Confidence Intervals

• Confidence Interval
• is an interval computed from a sample that is
  accompanied by a specific level of confidence
  (probability) of being correct (that is, of
  encompassing the true value of the parameter )
• The generic formula

Estimate - Margin of error lt Parameter lt Estimate
Margin of error
Margin of error shows how accurate we believe
our guess (estimate), is based on the variability
of the estimate
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5. A Six-Step Confidence Interval Model
Describe the population parameter of concern
STEP 1
STEP 2
Check the assumptions conditions
State the level of confidence
STEP 3
STEP 4
Find the point estimate
Obtain the critical values and calculate CI
STEP 5
Interpret the results
STEP 6
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6. Confidence Intervals for a Population Mean ?
s is known
Estimate - Margin of error ltParameter lt Estimate
Margin of error OR Estimate ? Margin of error
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Confidence Intervals for a Population Mean ?

• The two confidence intervals that are used
  extensively are the 95 and the 99.
• A 95confidence interval means that about 95 of
  the similarly constructed intervals will contain
  the parameter being estimated.

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4. Assumptions Conditions
(i) Observations are independent (this is
always the case when the sample is random) (ii)
The population has a normal distribution, OR the
sample size is large enough (say n gt 30) (iii)
Standard deviation of population, s, is known
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Confidence Intervals Example 1
Example Average potency(strength) of
tablets. Suppose that we want to determine the
average strength of certain tablet
product. Assume that the standard deviation of
the population is known and is equal 0.5 mg. A
random sample of twenty five tablets was taken
and the results are shown in Table 1.
Obtain the 95 confidence interval on the mean
tablet strength.
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Confidence Intervals Example 1
Table 1 Results of 25 Single-Tablet Assays
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Confidence Intervals Example 1
Step 1 Describe the population parameter of
concern ?
Step 2 Check the assumptions conditions
a) Observations are independent since the random
sample was taken b) Assume that the population is
normal! c) ? is known
Step 3 State the level of confidence
95
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Step 5 Obtain the critical values and calculate
the lower and upper confidence limits.
Margin of error
Confidence level Probability
z 90
0.05 1.645 95
0.025 1.96
99 0.005
2.578 z critical value
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Standard Error
C 95 z1.96 C 99 z2.58
For C 95
What is the margin of error? 0.196
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Step 6 Interpret the results. We are 95
confident that the mean tablet potency is
somewhere in the obtained interval.
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GRAPHICAL DESCRIPTION OF C.I. (simulation)
Hence we have calculated an interval with a
method that is guaranteed to catch the true mean
95 of the time in the long run.
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How CI Behave?

• HOW CONFIDENCE INTERVALS BEHAVE?
• What can the user control?
• - confidence level
• - sample size
• What are the desirable properties of the
  C.I.?
• High Confidence (this would imply that the
  method would
• give the correct answers almost always)
• Small margin of error (this would mean that his
  C.I . has
• pinned down the parameter quite precisely)

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7. CHOOSING THE SAMPLE SIZE
OBJECTIVE You want to have both a high
confidence level and a small margin of error How
can this be done? By taking a large enough
sample. So if m is your specified margin of
error and z is the critical value for your
desired confidence level the formula for the
sample size is
In our example 2 What is the sample size
required to decrease the margin of error to 0.1?
n ( (z s )/m )2 (1.96 0.5)/ 0.1)2 96
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Cautions When Computing Confidence Interval

• The data must be a SRS from the population.
• If the design used is more complex than the SRS
  the formula used might not be correct.
• If data is haphazardly collected then there is
  no way you can obtain valid inference.
• Since the sample mean, is largely affected by
  outliers,
• outliers can have a large effect on the
  Confidence interval as well.
• When the sample size is small and the population
  is roughly non-normal the true confidence level
  might be different from the value C used in
  computing the interval.