ap statistics section 10.1 c determining necessary sample size
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AP Statistics Section 10.1 C Determining Necessary Sample Size. Consider the confidence interval for the mean of a population where is known. The user chooses the confidence level and the margin of error automatically follows from this choice. - PowerPoint PPT PresentationTRANSCRIPT
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AP Statistics Section 10.1 C
Determining Necessary Sample Size
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Consider the confidence interval for the mean of a population where is known. The
user chooses the confidence level and the margin of error automatically follows from this
choice.
nzx
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Ideally, we would like both high confidence and a small margin of error. High
confidence says that our method almost always gives correct answers. A small
margin of error says that we have pinned down the parameter quite nicely.
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An equivalent expression for the margin of error is . Since
the expression has z* and in the numerator and in the denominator, the margin of
error gets smaller when:
n
z
n
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z* gets smaller. This happens when _______________________
So there is a trade-off between the confidence level and the margin
of error. To obtain a smaller margin of error from the same data, you must be willing to
accept lower confidence.
smaller gets C
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gets smaller. Remember, is a fixed value in the population and
can’t be changed.
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n gets larger. Now this is something that we can control. For example, in order to cut the
margin of error in half, we need to take ___ times as many
observations.4
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A wise user of statistics never plans data collection without planning the inference
at the same time. 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.
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Example: Researchers would like to estimate the mean cholesterol level of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 mg/dcl of blood of the true value of at a 95% confidence level. A previous study involving
this variety of monkey suggests that the standard deviation of cholesterol level is about mg/dcl. What is the
minimum number of monkeys needed to generate a satisfactory estimate?
5
z 96.1
1
n
z 1E
1)5)(96.1(
n
n8.9
n8.9
n04.96
97n
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Always round up to the next whole number when finding n.
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It is the size of the sample that determines the margin of error.
The size of the population does not influence the sample size we need
- as long as the population is at least 10 times as large as the
sample.
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CAUTION! CAUTION!
The data must be an SRS from the population.
Nonresponse and other practical problems can frustrate choosing an
SRS.
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The margin of error in a confidence interval covers only random
sampling errors. The margin of error indicates how much error can
be expected because of chance variation in randomized data
production.
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There is no correct method for inference from data haphazardly
collected or biased.
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Different methods are needed for different designs. The CI formula
isn’t correct for probability samples more complex than an SRS. There are correct methods
for other designs.
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Outliers can distort results. Outliers can strongly influence ___,
which can have a large effect on the confidence interval.
x
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The shape of the population distribution matters. Examine your
data carefully for skewness and other signs of non-Normality.
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You must know the standard deviation, , of the population.
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Finally, you must understand what statistical confidence does not say. Recall our confidence interval of (107.8, 116.2) for the mean IQ score
for all BCU freshmen. We are 95% confident that the mean IQ score for all BCU freshmen lies
between 107.8 and 116.2. That is, these numbers were calculated by a method that gives
correct results in 95% of all possible samples.
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We cannot say that the probability is 95% that the true mean falls between 107.8
and 116.2. No randomness remains after we draw one particular sample and get from it one particular interval. The true
mean either _______________ between 107.8 and 116.2.
not isor is
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The probability calculations of standard statistical inference
describe how often the __________ gives correct answers.process