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.

nzx

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.

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

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

gets smaller. Remember, is a fixed value in the population and

can’t be changed.

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

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.

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

Always round up to the next whole number when finding n.

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.

CAUTION! CAUTION! 

The data must be an SRS from the population.

Nonresponse and other practical problems can frustrate choosing an

SRS.

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.

There is no correct method for inference from data haphazardly

collected or biased.

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.

Outliers can distort results. Outliers can strongly influence ___,

which can have a large effect on the confidence interval.

x

The shape of the population distribution matters. Examine your

data carefully for skewness and other signs of non-Normality.

You must know the standard deviation,     , of the population.

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.

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

The probability calculations of standard statistical inference

describe how often the __________ gives correct answers.process