AP STATISTICS
LESSON 10 – 1
( DAY 3 )
CHOOSING THE SAMPLE SIZE
ESSENTIAL QUESTION:
How are sample sizes
determined?
Objectives:
• To find appropriate sample sizes
mathematically.
• To become aware of cautions involved in
creating confidence intervals.
How Confidence Intervals
Behave
Properties that are shared by all
confidence intervals:
• The user chooses the confidence level
and the margin of error..
• A small margin of error says that we have
pinned down the parameter quite
precisely.
Margin of error = z*•σ/√n
Making the Margin of Error
Smaller
z* and σ in the numerator and √n in the
denominator will make the margin of error
smaller when:
– z* gets smaller. Smaller z* is the same as
smaller confidence level C.
– There is a trade-off between the confidence
level and the margin of error.
Making Margin of Error Smaller
(continued…)
To obtain a smaller margin of error from the same
data you must be willing to accept lower confidence:
• σ gets smaller. The standard deviation σ measures the
variation in the population. You can think of the variation
among individuals in the population as noise that obscures
the average μ. It is easier to pin down μ when σ is small.
• n gets larger. Increasing the sample size n reduces the
margin of error for any fixed confidence level. Because n
appears under a square root sign, we must take four times
as many observations in order to cut the margin of error in
half.
Example 10.6
Page 550
Changing the Confidence Level
Suppose that the manufacturer in example
10.5 wants a 99% confidence level.
Find the new margin of error, confidence
level and compare this level to the 90%
confidence level found in Example 10.5.
Choosing the Sample Size
A wise user of statistics never plans
data collection without planning the
inference at the same time. You can
arrange to have both high confidence
and a small margin of error by taking
enough observations.
Example 10.7
Page 551
Determining Sample Size
Company management wants a report of
the mean screen tension for the day’s
production accurate to within ± 5 mV with
a 95% confidence.
How large a sample of video monitors
must be measured to comply with this
request?
Sample Size for
Desired Margin of Error
To determine the sample size n that will yield a confidence
interval for a population mean with a specified margin of
error m, set the expression for the margin of error to be less
than or equal to m and solve for n:
z*•σ/√n ≤ m
In practice, taking observations costs time and money. Do
notice once again that it is the size of the sample that
determines the margin of error. The size of the population
does not determine the size of the sample we need.
Some Cautions
• Any formula for inference is correct only in
specific circumstances.
• The data must be an SRS from the population.
• The formula is not correct for probability
sampling designs more complex than an SRS.
• There is no correct method for inference from
data haphazardly collected with bias of unknown
size.
• Because x is strongly influenced by a few
extreme observations, outliers can have a
large effect on the confidence interval. You
should search for outliers and try to correct them
or justify their removal before computing the
interval.
Cautions
(continued…)
• If the sample size is small and the population is
not normal, the true confidence level will be
different from the value C used in computing the
interval. Examine your data carefully for
shewness and other signs of nonnormality.
When n ≥15, the confidence level is not greatly
disturbed by nonnormal populations unless
extreme outliers or quite strong skewness are
present.
Cautions
(continued…)
• You must know the standard deviation σ of the
population. This unrealistic requirement renders
the interval x ±σ/√n of little use in statistical
practice.
• The margin of error in a confidence interval
covers only random sampling errors.
• Practical difficulties, such as undercoverage and
nonresponse in a sample survey, can cause
additional errors that may be larger than random
sampling error.