Transcript BIOSTAT 6 - Estimation

BIOSTAT 6 - Estimation
• There are two types of inference: estimation and
hypothesis testing; estimation is introduced
first.
• The objective of estimation is to determine the
approximate value of a population parameter
on the basis of a sample statistic.
• E.g., the sample mean ( ) is employed to
estimate the population mean ( ).
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Estimation
• The objective of estimation is to determine the
approximate value of a population parameter
on the basis of a sample statistic.
• There are two types of estimators:
• Point Estimator
• Interval Estimator
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Estimation
• For example, suppose we want to estimate the mean
weekly income for business students at the university.
For n=25 students,
•
is calculated to be 500 $/week.
• point estimate
interval estimate
• An alternative statement is:
• The mean income is between 480 and 520 $/week.
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Confidence Interval Estimator for
• If you know the value for σ
• What is Z/2 ? Traditionally this is the Zscore required in order to get /2 of the
area in the “RIGHT HAND TAIL” of the
normal distribution.
• However, this text will use Z(1-/2) for this
value.
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Notation Issue You will Just Have to Live With.
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Normal Z-Scores Used
• These Z-Scores will cover most problems you will need for
confidence intervals as well as hypothesis testing later.
•
Keep in mind the notation differences in this text.
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Example: Confidence Interval
• A nurse records the time in minutes it takes a patient to recover after
an operation before they are placed in a private room:
235
421
394
261
386
374
361
439
374
316
309
514
348
302
296
499
462
344
466
332
253
369
330
535
334
• Its is known that the standard deviation of recovery time is 75
minutes. We want to estimate the mean time with 95% confidence in
order to help determine the number of beds needed in recovery.
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Example:
• “We want to estimate the mean recovery
time with 95% confidence”
• The parameter to be estimated is the
population mean:
• The 95% confidence interval is
• Mechanically, all we need to do is fill in the
numbers for each symbol in this formula.
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Example:
• For this problem, we will need to calculate the
sample mean [later on we will also need to
calculate the sample standard deviation to use
for an unknown σ]
• The lower and upper confidence limits are
340.76 and 399.56
• Meaning: The odds are pretty high (95%) that
this interval actually contains the true mean.
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Confidence Intervals
• What affects the width of these confidence
intervals.
•
•
•
•
Effect of “n”
Effect of “σ”
Effect of Confident Level (Z-Score)
Effect of
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Sample Size to Achieve a Given Confidence Interval
• We can control the width of the confidence interval by determining
the sample size necessary to produce a desired confidence interval.
• Suppose we want to estimate the mean demand “to within 5 units”;
• i.e. we want to the interval estimate to be:
• From the formula for the confidence interval
• This implies that
• Solving for n results in
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Sample Size to Achieve a Given Confidence Interval
• The general formula for the sample size
needed to estimate a population mean
within + W with (1-)100% confidence is
• This author will use “d” instead of “W”
• Note that you need to know σ in order to
determine a sample size (n). Where do
you get a value for σ??
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Sample Size Example
• A lumber company wishes to estimate the mean diameter of trees in
order to determine the amount of money they will pay the land
owner. They need to estimate this to within 1 inch at a confidence
level of 99%. The tree diameters are normally distributed with a
standard deviation of 6 inches.
• How many trees need to be sampled?
• How would you take this ramdom sample????
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Confidence Interval for Unknown σ
• When the population standard deviation is
unknown and the population is normal, we
will use the sample standard deviation, s,
instead of the unknown value for σ. The
confidence interval now becomes
• Note that “t” replaces “Z” and “s” replaces
“σ” in the confidence interval formula
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The “t” Statistic
• The t-distribution looks very much like the
standard normal distribution but is spread out
wider (like someone sat on the normal
distribution and bulged it out). The t statistic has
one parameter called the degrees of freedom
(d.f., , or some other notation). In this case the
d.f. for a confidence interval is d.f. = n – 1
• See Table E in back of text. Note that the same
notation variation applies here as it did for the Z
statistics earlier.
• t/2 = t statistic to get /2 area to right (standard
notation used by most people)
• t1-/2 = t statistic to get 1- /2 to left (this text)
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Confidence Interval Unknown
Standard Deviation
• A nurse records the time in minutes it takes a
patient to recover after an operation before they
are placed in a private room: A sample of 25
times were recorded and the sample mean
calculated to be 370.12 and sample standard
deviation calculated to be 80.8. [same data as
previous example]
• We want to estimate the mean time with 95%
confidence in order to help determine the
number of beds needed in recovery.
• Students calculate this in class!!!!!!!!!!!!!!!!!
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HW: Estimation
• 6.2.1, 6.2.3, 6.3.1, 6.3.5, 6.5.1, 6.5.3,
6.7.1, 6.7.3, 6.8.1, 6.8.3
• Chap 6: Review questions and exercises
• 3, 6, 9
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