| Overview
In this Authentic Learning Task (ALT), you use pennies to simulate
a population study. You use statistics to study the variability
of the years from this population of pennies and the variability
of averages taken from samples. You then speculate about what the
population distribution of years should look like and make a frequency
distribution. On the Discussion Board, you discuss what you think
a distribution of sample averages might look like and find averages
taken from samples of 30 until a good representation of the distribution
can be displayed.
After completing this ALT, you should be able to demonstrate the
following competencies:
- Use summary techniques such as box plots, stem and leaf plots,
and percentiles to analyze and interpret data sets (Comp. 1).
- Use the techniques of sampling and recognize the nature of variability
in sample data, including the use of computer analysis tools (Comp.
2).
Pre-Activity
Frequency Distributions/ Histograms
For the activity, review how
to construct a frequency distribution for a set of data.
If you need to Review how
to construct a histogram.
Sampling and Sampling Distributions
On page 24 of your Probability, Sampling, and Hypothesis Testing
Participant Journal, the histogram illustrates a skewed distribution.
Given any large sample of pennies, we would expect to see a similar
distribution, because older pennies are taken out of circulation
because they are lost, damaged or stored in someone’s penny
jar.
Visit the sampling
distribution simulator. Read the instructions and then perform
the following simulations, as outlined in Table 1 and illustrated
in Figure 1.
Table
1. Simulation Trials |
Trial/Settings |
Distribution |
Sample
Size |
Mean
N= |
Std.
Dev. |
| 1 |
Normal |
5 |
5 |
5 |
| 2 |
Normal |
10,000 |
5 |
5 |
| 3 |
Normal |
5 |
25 |
25 |
| 4 |
Normal |
10,000 |
25 |
25 |
| 5
thru 8 |
Skewed |
Repeat |
Repeat |
Repeat |
| 9
thru 12 |
Uniform |
Repeat |
Repeat |
Repeat |
What did you observe? Be prepared to discuss your findings with
your group or on the Discussion Board with the class.
Figure 1 - First simulation trial
Activity
In your team, perform the following step
to complete the activity:
Part A—Variability of the Distribution
- Your facilitator will place you and the other participants in
teams of two.
- Take a handful of pennies from the Pennyville bucket.
- With your team, construct one frequency distribution from the
pennies.
- Your facilitator will show the group the histogram obtained
on the 500+ pennies.
- In your team, discuss the distribution and compare it to the
distributions obtained from your samples and to the prediction
from the pre-activity discussion.
Part B—Variability in Sample Averages
- With your team, collect a few samples of 30
pennies each.
- Calculate and record in the notebook the average and standard
deviation of each sample.
- Return all pennies to the Pennyville bucket and mix thoroughly.
- Repeat this process to collect as many random samples as possible
in 20 to 30 minutes.
- With your team, go to the flipchart and put an X over the year
( rounded to the nearest year) that represents the sample average
in order to construct and display a histogram combining all averages
obtained from the teams.
- In your team, discuss this distribution and how it compares
to what was predicted.
- Complete Data Sheet: Sampling.
Note: Be sure to keep your results from this ALT
for ALT #3
Post-Activity
Your team should now post all of its results from the activity
to the Discussion Board.
Assignment
There are no instructions to prepare for ALT #3: Return to Pennyville.
Assessment
Your facilitator may use Assessment Sheet: A Trip to Pennyville
to evaluate your results from the activity and your posting to the
Discussion Board.
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