Download What do you think will happen to the data if we roll a single 6

September 06, 2012
What do you think will happen to the data if we roll a single 6sided die over and over? Do you think one number will come
up more over another?
What do you think will happen to the data if we roll two 6-sided
dice over and over? Do you think one sum will come up more
over another?
September 06, 2012
To Answer the first question, we'll generate our own data set using the TI-nSpire.
To come up with data that's random
from everyone else's, type in the
RANDSEED command followed by
any 1 to 4 digit number
Create a Lists/Spreadsheets page with
the following for columns. Enter the
formulas exactly how they're seen
here. This will generate 10 rolls of the
dice. Notice in column D the two die
values are added up.
September 06, 2012
Create a Data/Statistics page...
...and click on "roll1" on the
horizontal axis to create
your dot plot.
Then change the data
display to a histogram.
So much for answering question #1 correctly! The data doesn't look even at all. Is there
anything we can do to our data to make it look more even???
YES!! More Data!!
Go ahead and change the "no.rolls" column from 10 to 100 and then to 1000. How does
the histogram look now? Better?
September 06, 2012
Results after 100 rolls of the die
While better than 10 rolls it's not as even as it should
be, especially since each number has an equal chance
of being rolled.
Results after 1000 rolls of the die!
Much better, don't you think? Imagine what the
histogram would look like after 10,000 rolls! You get
the idea.
September 06, 2012
To answer the second question, all we need to do is
change the histogram from "roll1" to "sumroll".
10 rolls of the dice
100 rolls of the dice
1000 rolls of the dice
The results are quite a bit different from our first question. That's because
there are more ways to roll a 6, 7, or 8, for example, than a 2 or 12. The other
thing to notice is that as the number of rolls increase, the closer our histogram
is approaching a normal density curve (or "bell" curve).
September 06, 2012
In fact, there is a way to verify that our data set is in-fact approaching a normal density curve
mathematically. If we calculate the percent of the data that's within 1, 2, and 3 standard deviations of
the population mean, mu, we should get our 68%, 95%, and 99.7% (or very close to them). But first,
we need to know what our population mean, mu, and our population standard deviation, sigma, are.
To do this, just calculate the one variable statistics in the column next to "sumroll" in your list and
spreadsheet page.
The TI-nSpire only calculates x, but
they assume that the data you've
collected is a sample, not the
population. For all intense and
purposes they are one and the same, as
long as you know it's one or the other
ahead of time. Since we know that x is
actually mu, we know to look at sigma
instead of s for our standard deviation
when calculating our percent under the
density curve in our next step.
September 06, 2012
To calculate the percent under the density curve, we could use Table A in
the back of the book, but hey, it's 2012, not 1986 any more! So let's have
the TI-nSpire generate these percentages for us!
Step 1: Figure out the range
of values 1 standard deviation
is away from our mean, mu.
Step 2: Use the normCdf
command on the TI-nSpire
to determine percent under
the curve.
BTW, normCdf stands for
"normal Cumulative
density function".
So 1 standard deviation below the mean is
4.81 and 1 standard deviation above the mean
is 9.77. Is this 68% of the data in the density
curve??
n
mi
ma
x
mu
ma
sig
Wow! Incredible, right? This only happened
because we rolled the dice 1000 times. We
would not have achieved this kind of
accuracy for 10 rolls or even 100 rolls. So
number of trials makes a difference!
September 06, 2012
Likewise, we see that the percent under the density curve for 2
and 3 standard deviations falls right in line with the rest of the
68-95-99.7 Rule.
Percent under the density curve 2
standard deviations away from mu
Percent under the density curve 3
standard deviations away from mu
September 06, 2012