This project consists of 10 steps. You can either print this document (not including this page), complete
it by hand, scan it, and email that scanned file to me OR you can complete the document in Microsoft
Word. If you choose to complete the project using MS Word, you must make sure that the formatting
stays relatively similar to the original formatting.
When the project is complete, please e-mail cindy.langley (in eSchoolware) or
cindy.langley@sc.provostacademy.com. Please be sure to include your name and which project you are
submitting. For example, Cindy Langley _Class Name_ Part #_ Project #.
Please contact me with any questions or concerns.
eSchoolware: cindy.langley
Provost email: cindy.langley@sc.provostacademy.com
Cell: 803.465.7471
Have you ever wondered how long a coin stays in circulation? Do you know someone who
collects coins? There are a lot of people who are always on the watch for certain coins when they
get change because there are coins still in circulation that are more valuable than their face value.
Who knows? Maybe you should watch out for these coins as you are looking through your change.
Let’s talk about pennies…some pennies are worth more than face value already and have a
potential to increase in value over time as they become more rare. A good example of this are
pennies dated before 1982. In mid-1982, the composition was changed to 97.5% zinc and 2.5%
copper. Previous issues were 95% copper. The composition was changed because the 95% copper
coins contained more than a cents worth of copper making them too expensive to produce.
The Age of a Penny
To begin this project, you will need to have 30 pennies. Your first task is to form a distribution of
their ages.
1) Organize the data by using a frequency distribution chart.
a. Visit the following webpage for more information on frequency distribution charts:
http://www.ehow.com/how_6057641_create-distribution-chart.html
2) Describe the shape of the distribution? Why do you think it is this shape?
3) Did you find any outliers?
4) Do you think the distribution of all pennies in circulation is similar to your sample? Why or
why not?
5) Find the mean and standard deviation of the ages of the pennies in your sample.
6) Compute a 95% confidence interval for the mean ages of the pennies (pages 4-7 of this
document contain detailed steps for computing confidence intervals as well as an example).
7) What is the margin of error for your estimate?
8) The president of “COINS UNLIMITED” has just hired you as his chief statistician for his
research on the age of pennies. You are charged with the task of estimating the average age
of pennies in circulation within one year of age with 99% confidence. How large of a sample
would you need to obtain? Use the standard deviation from your sample as your best
estimate of the population standard deviation.
9) On the basis of your research with this project, how would you define the age of a “rare”
coin? Give a statistical definition for your choice.
10) What would be the age of the pennies that you would begin to save before they become hard
to find? Give a statistical reason for your choice. Consider “rare” as 2% or less of the
population.
Grading Rubric
Student Name:
CATEGORY
Mathematical
Concepts
________________________________________
4
Explanation
shows
complete
understanding
of the
mathematical
concepts used
to solve the
problem(s).
Uses complex
and refined
mathematical
reasoning.
3
Explanation
shows
substantial
understanding
of the
mathematical
concepts used
to solve the
problem(s).
Uses effective
mathematical
reasoning
2
Explanation shows
some
understanding of
the mathematical
concepts needed
to solve the
problem(s).
1
Explanation shows
very limited
understanding of
the underlying
concepts needed
to solve the
problem(s) OR is
not written.
Some evidence of
mathematical
reasoning.
Little evidence of
mathematical
reasoning.
Explanation
Explanation is
detailed and
clear.
Explanation is
clear.
Explanation is a
little difficult to
understand, but
includes critical
components.
Completion
All problems
are
completed.
All but one of
the problems
are completed.
All but two of the
problems are
completed.
Explanation is
difficult to
understand and is
missing several
components OR
was not included.
Several of the
problems are not
completed.
Mathematical
Reasoning
CONFIDENCE INTERVALS
Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample
estimate of a population parameter.
To express a confidence interval, you need three pieces of information.

Confidence level
o
In survey sampling, different samples can be randomly selected from the same
population; and each sample can often produce a different confidence interval. Some
o
confidence intervals include the true population parameter; others do not.
A confidence level refers to the percentage of all possible samples that can be expected
to include the true population parameter. For example, suppose all possible samples
were selected from the same population, and a confidence interval were computed for
each sample. A 95% confidence level implies that 95% of the confidence intervals would
include the true population parameter.

Statistic
o
A statistic is a characteristic of a sample. Generally, a statistic is used to estimate the
value of a population parameter (a parameter is a measurable characteristic of a
population, such as a mean or a standard deviation).
o
For instance, suppose we selected a random sample of 100 students from a school with
1000 students. The average height of the sampled students would be an example of a
statistic. So would the average grade point average. In fact, any measurable
characteristic of the sample would be an example of a statistic.

Margin of error
o
The margin of error expresses the maximum expected difference between the true
population parameter and a sample estimate of that parameter. To be meaningful, the
margin of error should be qualified by a probability statement (often expressed in the form
of a confidence level).
o
For example, a pollster might report that 50% of voters will choose the Democratic
candidate. To indicate the quality of the survey result, the pollster might add that the
margin of error is +5%, with a confidence level of 90%. This means that if the survey
were repeated many times with different samples, the true percentage of Democratic
voters would fall within the margin of error 90% of the time.
o
To compute the margin of error use the following equation:


Margin of error = Critical value x Standard deviation of the statistic
The critical value is a factor used to compute the margin of error.

This section describes how to find the critical value, when the sampling
distribution of the statistic is normal or nearly normal.

The central limit theorem states that the sampling distribution of a statistic will
be normal or nearly normal, if any of the following conditions apply:

The population distribution is normal.

The sampling distribution is symmetric, unimodal, without outliers, and
the sample size is 15 or less.

The sampling distribution is moderately skewed, unimodal, without
outliers, and the sample size is between 16 and 40.


The sample size is greater than 40, without outliers.
When one of these conditions is satisfied, the critical value can be expressed
as a z score. To find the critical value, follow these steps.

Compute alpha (α): α = 1 - (confidence level / 100)

Find the critical probability (p*): p* = 1 - α/2

To express the critical value as a t score, follow these steps.

Find the degrees of freedom (DF). When estimating a mean score or a
proportion from a single sample, DF is equal to the sample size minus
one. For other applications, the degrees of freedom may be calculated
differently. We will describe those computations as they come up.

The critical t score (t*) is the t score having degrees of freedom equal to
DF and a cumulative probability equal to the critical probability (p*).
How to Construct a Confidence Interval
There are four steps to constructing a confidence interval.

Identify a sample statistic. Choose the statistic (e.g, sample mean, sample proportion) that you
will use to estimate a population parameter.

Select a confidence level. As we noted in the previous section, the confidence level describes the
uncertainty of a sampling method. Often, researchers choose 90%, 95%, or 99% confidence
levels; but any percentage can be used.

Find the margin of error.

Specify the confidence interval. The uncertainty is denoted by the confidence level. And the range
of the confidence interval is defined by the following equation.
Confidence interval = sample statistic + Margin of error
The sample problem in the next section applies the above four steps to construct a 95% confidence
interval for a mean score.
Example of Finding a Confidence Interval
Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a
random sample of 1,000 men from a population of 1,000,000 men and weigh them. We find that the
average man in our sample weighs 180 pounds, and the standard deviation of the sample is 30 pounds.
What is the 95% confidence interval.

Identify a sample statistic. Since we are trying to estimate the mean weight in the population, we
choose the mean weight in our sample (180) as the sample statistic.

Select a confidence level. In this case, the confidence level is defined for us in the problem. We
are working with a 95% confidence level.

Find the margin of error. Previously, we described how to compute the margin of error. The key
steps are shown below.

Find standard error. The standard error (SE) of the mean is:
SE = standard deviation / sqrt( n ) = 30 / sqrt(1000) = 30/31.62 = 0.95

Find critical value. The critical value is a factor used to compute the margin of error. To
express the critical value as a t score (t*), follow these steps.
o
Compute alpha (α): α = 1 - (confidence level / 100) = 0.05
o
Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
o
Find the degrees of freedom (df): df = n - 1 = 1000 - 1 = 999
o
The critical value is the t score having 999 degrees of freedom and a cumulative
probability equal to 0.975. From the t Distribution Calculator
(http://danielsoper.com/statcalc3/calc.aspx?id=10), we find that the critical value
is 1.96.

Compute margin of error (ME): ME = critical value * standard error = 1.96 * 0.95 = 1.86
Specify the confidence interval.
The range of the confidence interval is defined by the sample statistic + margin of error. And the
uncertainty is denoted by the confidence level.
Therefore, this 95% confidence interval says that the population mean falls within the interval 180 + 1.86.