AP Stat
Random Sampling
Name
Period
A SAMPLING METHOD: BIASED OR UNBIASED?
Equipment:
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Random number generator
Graph paper
An airport has four public telephones. During peak hours these
phones are so busy that as soon as one is available, a new caller steps up
to use it. An airport official has asked us to estimate the mean length of
a phone call during "peak hours." Here is our sampling method:
We will get the phone company records showing the start and end
time of every call made on these four phones. Then we will randomly
select five different minutes from one "peak hour" period. We'll record the
phone call lengths from the four phones during each of those minutes.
This will produce 20 observations. Finally, we will take the mean of
these 20 phone calls as our estimate of the population mean.
Before we contact the telephone company and execute our plan,
however, let's do a simulation to see whether our method is biased. We
will randomly select phone calls from a population whose mean we know.
Then we'll see how far away our sample mean is from the correct mean.
The Simulation (on the TI-83 or TI-83 PLUS)
1} On a piece of graph paper, set up four rows of 40 squares to represent
the telephones. Label them 1 through 4, top to bottom. Label the minutes,
horizontally, 1 through 40, in the spaces. We will randomly select
integers between 1 and 10, inclusive, to represent phone call lengths .
. What is the pop. mean of random integers in this range?
_
Hit ON. Hit MATH. Arrow left to PRS. Hit 5. (randlnt). Input this
command: randlnt (1, 10, 50) ~
L1. (The arrow is produced by the STO
or store command above ON.) Here we are putting 50 random phone call
lengths, which is more than we'll need, into list L1.
2) Using arrows, diagram each call from list L1, into the four rows, or
telephones, just as if they were callers standing in a single line to use the
four. phones. For example, the fifth caller will use the first available
phone--it's the same phone as the shortest of the first four calls.
Continue until all 40 minutes for all four phones have been covered. Some
of the phone calls will go past the 40 minute mark.
Wes White • Active Math Instruction • Alhambra H.S., Alhambra, CA
50
3) Randomly select five minutes between 1 and 40.
Hit MATH. Arrow left to PRB. Hit 5. (randlnt). Input this
command: randlnt (1, 40). Hit ENTER repeatedly until it produces 5
unique numbers: Each of these is a given minute between 1 and 40.
4) Write down the phone call lengths of all four phones for each of the
five random minutes:
Minute#:
Phone 1:
Phone 2:
Phone 3:
Phone 4:
5) Calculate the mean of the 20 cal/s.
(
_
6) Now let's consider another method of sampling. We will randomly
select 20 phone call lengths directly from list L1. The easiest way to
do this is to first post 50 random decimals in list L2.
Hit 2nd LIST. Arrow right to OPS. Hit 5. (seq). Input this
command: seq(rand, X,-1:' 50, 1) STO
2nd L2. Next, we sort list L2
connected with list L1. Hit STAT. Hit 2. (SortA). Input this
command: SortA (L2, L1). Hit ENTER. Your calculator says "Done."
Our random sample of twenty phone call lengths is now the first
twenty numbers in list L1. Find the mean of these twenty calls. __
7) Compile the means for the whole class from Step 5, the sample of
randomly selected minutes, and from Step 6, the sample of randomly
selected phone calls.
a) Is there a big difference for the class in the two results?
_
b) Which method produced the greater mean length?
_
c) Would this difference in mean length be wiped out by repeated
trials-performing
the simulation, say, one thousand times?
_
d) Does this difference represent bias or high variation?
_
e) Explain the large difference.
_
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3) Randomly select five minutes between 1 and 40.
Hit MATH. Arrow left to PRB. Hit 5. (randlnt). Input this
command: randlnt (1, 40). Hit ENTER repeatedly until it produces 5
unique numbers: Each of these is a given minute between 1 and 40.
4) Write down the phone call lengths of all four phones for each of the
five random minutes:
Minute#:
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the mean of the 20 calls.
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6) Now let's consider another method of sampling. We will randomly
select 20 phone call lengths directly from list L 1. The easiest way to
do this is to first post 50 random decimals in list L2.
Hit 2nd LIST. Arrow right to OPS. Hit 5. (seq). Input this
command: seq(rand, X,~ 50, 1) STO
2nd L2. Next, we sort list L2
connected with list L 1. Hit STAT. Hit 2. (SortA). Input this
command: SortA (L2, L 1). Hit ENTER. Your calculator says "Done."
Our random sample of twenty phone call lengths is now the first
twenty numbers in list L 1. Find the mean of these twenty calls. S. bS
7) Compile the means for the whole class from Step 5, the sample of
randomly selected minutes, and from Step 6, the sample of randomly
selected phone calls.
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a) Is there a big difference for the class in the two results?
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c) Would this difference in mean length be wiped out by repeated
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d) Does this difference represent bias or h19h variation?
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