G/Acc. Math 3 – Summary Statistics
Name: _____________________________
GPS Accelerated Math 3 Unit 3 Standards
MA3D1. Using simulation, students will develop the idea of the central limit theorem.
MA3D2. Using student-generated data from random samples of at least 30 members, students will
determine the margin of error and confidence interval for a specified level of confidence.
MA3D3. Students will use confidence intervals and margin of error to make inferences from data about
a population. Technology is used to evaluate confidence intervals, but students will be aware of the ideas
involved.
Vocabulary and Formulas
Central Limit Theorem:

Choose a simple random sample of size n from any population with mean  and standard
deviation . When n is large (at least 30), the sampling distribution of the sample mean x is
approximately normal with mean  and standard deviation


n
.
Choose a simple random sample of size n from a large population with population parameter p
having some characteristic of interest. Then the sampling distribution of the sample proportion
p̂ is approximately normal with mean p and standard deviation
p 1  p 
n
. This approximation
becomes more and more accurate as the sample size n increases, and it is generally considered
valid if the population is much larger than the sample, i.e. np  10 and n(1 – p)  10.

The CLT allows us to use normal calculations to determine probabilities about sample
proportions and sample means obtained from populations that are not normally distributed.
Confidence Interval is an interval for a parameter, calculated from the data, usually in the form
estimate  margin of error. The confidence level gives the probability that the interval will capture the
true parameter value in repeated samples.
Margin of Error is the value in the confidence interval that says how accurate we believe our estimate
of the parameter to be. The margin of error is comprised of the product of the z-score and the standard
deviation (or standard error of the estimate). The margin of error can be decreased by increasing the
sample size or decreasing the confidence level.
Parameter is a number that describes the population. A parameter is a fixed number, but in practice we
do not know its value because we cannot examine the entire population.
Sample Mean is a statistic measuring the average of the observations in the sample. It is written as x .
The mean of the population, a parameter, is written as .
Sample Proportion is a statistic indicating the proportion of successes in a particular sample. It is
written as p̂ . The population proportion, a parameter, is written as p.
Sampling Distribution of a statistics is the distribution of values taken by the statistic in all possible
samples of the same size from the same population.
Sampling Variability refers to the fact that the value of a statistic varies in repeated random sampling.
Statistic is a number that describes a sample. The value of the statistics is known when we have taken a
sample, but it can change from sample to sample. We often use a statistic to estimate an unknown
parameter.
ENDURING UNDERSTANDINGS

As the sample size increases, the value of a statistic approaches the true value of the population
parameter, the standard deviation of the sample means is
sample proportions is
p 1  p 
n

n
, and the standard deviation of the
. These formulas for the standard deviation are valid as long as
the population is at least 10 times the size of the sample.

The Central Limit Theorem allows us to use normal probability calculations, given certain
conditions are met, for sample means and proportions even if the original distributions are nonnormal.

Confidence intervals provide a range of values that estimate the population parameter.

The margin of error, often cited in media articles, tells how accurate we believe our estimate of
the parameter to be. To decrease the margin of error, we could increase the sample size or
decrease how confident we need the result to be.