statistics - systematic collection
and arrangement of large numbers of observations and quantities
of numerical observations, and with ways of drawing useful
conclusions from such data
frequency table - a table showing a set
of values of a variable and the number of times each value
occurs
Age That 42 U.S.A. Presidents Began
Their Term
Age
Number of
Presidents Beginning Their Term at this Age
line of best fit (trend line)
- A line on a scatter plot which can be drawn near the points
to more clearly show the trend between two sets of data.
The line of best that rises quickly from left to right
is called a positive correlation.
The line of best that falls down quickly from left to
the right is called a negative correlation
Strong positve and negative correlations
have data points very close to the line of best fit..
Weak positve and negative correlations
have data points that are not clustered near or on the
line of best fit.
Data points that are not close to the line of best fit
are called outliers.
Directions: Use the
following interactive grid to create a scatter plot.
Study the changes in the line of best fit
as you add additional data points.
Create a scatter plot with a line of best fit showing
a positive correlation. Click "Reset"
to clear the data if you wish to make additional scatter
plots.
Create a scatter plot with a line of best fit showing
a negative correlation. Click "Reset"
to clear the data if you wish to make additional scatter
plots.
Create a scatter plot with a line of best fit showing
little or no correlation. Click "Reset"
to clear the data if you wish to make additional scatter
plots.
Notice a scatter plot with a strong correlation
has data points clustered very near to the line
of best fit. Weak correlations have data
points that are further from the line of best fit. Create
a scatter plot with a line of best fit showing weak positive
and weak negative correlation. Click "Reset"
to clear the data if you wish to make additional scatter
plots.
Interpolate is the process
one uses to determine a value on the line of best fit within
the cluster of scatter plot data.
Extrapolate is the process
one uses to determine a value on the line of best fit outside
the range of data values plotted. Extrapolated values on
a line of best fit are outside the cluster of scatter plot
data.
Directions: Use the following interactive
grid to create a scatter plot. Study the changes in
the line of best fit as you add additional
data points.
Click on the grid several times to plot data points.
Practice interpolating data. Pick a value
on the x-axis that is inside the cluster of points. Follow
the point upwards until your reach the line of best fit.
Estimate the y-axis value for this point on the line of
best fit. Click on the point to check your estimate. If
the line of best fit does not move - congratulations!
Click "Reset" to
clear the data.
Practice extrapolating data. Pick a
value on the x-axis that is outside the cluster of points
that determine the line of best fit. Follow the point
upwards until your reach the line of best fit. Estimate
the y-axis value for this point on the line of best fit.
Click on the point to check your estimate. If the line
of best fit does not move - your estimate is great! Click
"Reset" to clear
the data.
Enrichment: The
correlation coefficent (r) ranges between -1 and
1. For this course you do not have toremember
these value. The scale on the right of the grid
includes the correlation coefficient. Notice the
black regions are positive correlatioins and the
red are negative correlations.
My students will need to be
able to classify scatter plot patterns as strong/weak
positive, strong/weak negative or no correlation.
This next activity will help you see the mathematical
pattern of correlation that you may study in high
school.
Directions: Use this
interactive graph to study line of best fit and
correlation of scatter plots. Click "New
Sample" to generate another set of data.
Click and drag the red slider on the right to
change the correlation. Study the scatter plot
and line of best fit as you move the slider.
If you would like additional information for
one of the graph features, select the "Rollover
help" checkbox, then move your mouse
over the feature.
Review:
My students studied central
tendency in grade 7 and 8 mathematics. Central
tendency measures may be included in data one wishes
to analyze.
central tendency -
point within the range about which the rest of the
data is considered balanced. The three common
measures of central tendency are mean,
median and mode.
One of
the ways to display central tendency is a box and
whiskers plot.
50% of the data is between
the upper and lower quartiles.
The lower 75% of the data
is between the lower extreme and the upper
quartile.
probability - the likelihood
that some event will happen, measured or estimated on a
scale of 0 to 1. Zero probability means impossibility, a
probability of 1 means certainty and a 0.5 probability is
sometime called a fifty-fifty or an even chance.
theoretical probability -
facts (probability of a certain value by logic or theory)
expressed by the fraction
# ways the event can occur
total possible outcomes
experimental results - data
(obtained by a test/ survey) expressed by the fraction
# favourable outcomes
# trials
independent event - an event whose outcome
does not affect the occurrence of another event.
Interactive examples will be provided later. You
will be asked to calculate the probability of dependent
events.
draw two cards at random from a standard deck with
replacement
draw two marbles at random from a bag with
replacement
draw two candies at random from a dish with
replacement
draw two numbers at random from a hat with
replacement
select two keys at random from a keyring with
replacement
select two tools at random from a toolbox without
replacement
roll two die
spin two spinners
flip two coins
dependent event - an event
whose outcome affects the occurrence of another
event. For my course you will need to explain the difference
between independent and dependent events. You will not be
asked to calculate the probability of dependent events.
draw two cards at random from a standard deck without
replacement
draw two marbles at random from a bag without
replacement
draw two candies at random from a dish without
replacement
draw two numbers at random from a hat without
replacement
select two keys at random from a keyring without
replacement
select two tools at random from a toolbox without
replacement
Total possible outcomes are required for theoretical
probability calculations. A tree diagram (Method 1),
table (Method 2) or list (Method 3)
shows the possibilities. A tree diagram will include the possibilities
and the branches, while the chart generally illustrates the
same possibilities without the branches.
Explore the total possible outcomes of the following pairs
of events.
Two Coins
Method
1
Complete a tree diagram to show all
of the possible outcomes of flipping two coins.
Compare it to the tree diagram below. It should
include branches connecting the 2
possibilities for Coin 1 with the
2 possibilities
for Coin 2. There are 4 possible
outcomes when you flip two coins.
Another way of drawing the ttree diagram
follows:
Event
A: Coin Flip
Event
B: Coin Flip
H
H
T
T
H
T
number of possible outcomes = 2
x 2
= 4
Method 2
Make a chart to show the 4 possibilities
for 2 coins. Compare your chart to the one given below.
H
T
H
H, H
H, T
T
T, H
T, T
Method 3
Make a list of all the possibile events
(lists for 3 and 4 coins are included).
Compare your list to the one given below.
#Coins
#Possibilities
List
of possibilites
2
2
x 2 = 4
HH, HT,
TH, TT
3
2
x 2 x 2 = 8
HHH, HHT,
HTH, THH, HTT, THT,
TTH, TTT
4
2
x 2 x 2 x 2 = 16
HHHH, HHHT,
HHTT, HTTTT,
TTTT, HHTH,
HTHH, THHH,
HTTH, HTHT,
HTTT, THTT,
TTHT, TTTH, THHT, THTH
Two
Six-sided Dice
Method 1
The following shows a tree diagram of the possible sums when
you toss two dice. To better understand the tree diagram, roll
Die 1 until you get 1. Find the corresponding
number in the chart below Die 1. Roll Die
2. It will display the value from 1 to 6.
The branches beside each event in the left cell shows there
are 6 possibilities
for the second event. Each possibility for Die 1
connects with the 6
possibilities for Die 2. There are 36
possibile outcomes when you roll two dice.
Event
A: Die Toss
Event
B: Die Toss
1
1
2
3
4
5
6
2
1
2
3
4
5
6
3
1
2
3
4
5
6
Event
A: Die Toss
Event
B: Die Toss
4
1
2
3
4
5
6
5
1
2
3
4
5
6
6
1
2
3
4
5
6
number of possible outcomes = 6
x 6 = 36
Method 2
Complete a chart to show the 36 possibilities
for 2 six-sided dice thrown. Compare your chart to the one given
below.
This is the standard probability table. Notice
that the possible outcomes of one die are placed across the
top of the chart and the possible outcomes for the second die
are placed down the left side of the chart. The sum of the two
die are placed in the appropriate table cell. For example if
the first and second die are both 1, then the sum of the dice
is 2.
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Method 3
Make a list of all the possibile events. Compare
your list to the one given below.
The following animation shows the pattern of the
possible sums when one tosses two. The 36 possible combinations
are also displayed. This information is used in the next table
to calculate the theoretical probability for the sum of two
dice tosses.
Theoretical probability for the sum of
two six-sided dice.
0
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Six-sided Die and Coin
Method 1
Complete a tree diagram to show all
of the possible outcomes of flipping a coin and tossing
a die. Compare your tree diagram to the one
given below. The table below shows the 6
possibilities for Die Toss followed
by the 2 possibilities
for Coin Flip. There are 12 possible
outcomes when you roll a die and flip a coin.
Event
A: Die Toss
Event B:
Coin Flip
1
H
T
2
H
T
3
H
T
4
H
T
5
H
T
6
H
T
number of possible outcomes = 6
x 2
= 12
Method 2
Make a chart to show the 12 possibilities
for a coin and six-sided die tossed. Compare
your chart to the one given below.
H
T
1
1, H
1, T
2
2, H
2, T
3
3, H
3, T
4
4, H
4, T
5
5, H
5, T
6
6, H
6, T
Method 3
Make a list of all the possibile events.
Compare your list to the one given below.
#Dice
#Possibilities
List
of possibilites
2
6
x 2 = 12
1H, 1T
2H, 2T
3H, 3T
4H, 4T
5H, 5T
6H, 6T
5 Sector Spinner and Coin
Complete
a tree diagram to show all of the possible outcomes
of flipping a coin and spinning a 5 sector spinner.
Compare it to the tree diagram below. Branches
connect the 5Spinner possibilities with the 2
possibilities for Coin. There are
10 possible outcomes when you spin a 5 sector spinner
and flip a coin.
Event
A: Spin Wheel
Event
B: Coin Flip
green
H
T
blue
H
T
white
H
T
grey
H
T
red
H
T
number of possible outcomes = 5
x 2
= 10
Remember when there are two independent events,
the number of possible outcomes equals the number
of possible outcomes of the first event times the
possible outcomes of the second event.
Coin and 4 Sector
Spinner
Make a chart to show the 8 possibilities for a
coin and four sector spinner (blue, green, cyan, magenta)? Check
your table with the one below.
Theoretical probability is displayed
in the opening screen of the virtual coin below. The green line
shows the theoretical probability for heads and tails is 50%.
Experiment probability:
Toss a coin 20 times using the virtual coin provided by clicking
the "?" button. The experimental
probability does not have to match the theoretical probability.
Click the "Reset" button if you wish
to complete additional experimental probability trials.
The
theoretical probability of the event is the fraction:
# ways the event can occur
total possible outcomes
The
experimental probability for equally likely events is the
fraction:
# favourable outcomes
# trials
In the Experiment, the theoretical probability
of the event that heads comes up is:
P(H)
=
1/2
= 0.5 =
50%
P(T)
=
1/2
= 0.5 =
50%
In the Experiment,
if heads comes up 13 of 20 times, the experimental probability
of the event that heads comes up is:
Independent
Events: Rolling Dice Probability (Equally Likely
Events)
Theoretical probability
is:
# ways
the event can occur
total possible
outcomes
Experimental probability
is:
#
times the event occurred
# trials
Theoretical probability is
displayed in the opening screen of the virtual
coin below. The green line shows the of the
theoretical probability for each sum. The numbers
displayed above the green line are the probabilities
of each event. The number of ways each sum can
occur is displayed in an animation below. The
total outcomes is 36.
Experiment probability:
The theoretical probability numbers above the
green line will change to experimental probability
when the "?" button is selected. Click
the "?" button 36 times to toss the
virtual dice. The experimental probabilities
will be recalculated each time you do so. Compare
the results with the theoretical probability
represented by the green line. Click the "Reset"
button if you wish to complete another trial.
If you wish to record your own results make
a table like the one below the virtual dice.
Click on "?" 36 times and record the
results in the Experimental
column.
Independent Events: Probability of Tossing
Die and Spinning a Colour
Event
A: Toss Die
Event
B: Spin Wheel
Toss one six-sided dice
and spin a five sector color spinner.
What is the theoretical probability
of a particular number and color being
selected?
Solution:
You can use a real die
and spinner or use the virtual ones
above. The theoretical
probability of of any number
being tossed is 1/6, while the theoretical
probability of the spinner
stopping on each color is 1/5. Examples:
P(5, green)
means the probability of tossing
5 and spinning green
P(5,
green) = P(5)
x P(green) = 1/6
x 1/5 = 1/30 or 3.333...%
P(3, red) means
the probability of tossing
3 and spinning red
P(3, red)
= P(3) x P(red)
= 1/6 x 1/5 = 1/30 or 3.333...%
Complete 30 trials and complete
the experimental probability.
Two
Events
Theoretical
Probability
Experimental
Probability
P(5,
green) =
1/6
x 1/5 = 1/30 = 0.0333...3.3%
/30
P(3,
blue) =
1/6
x 1/5 = 1/30 = 0.0333...3.3%
/30
Independent
Events: Probability of Rolling Dice and Spinning
a Colour
Event
A: Toss Dice
Event
B: Spin Wheel
Toss two six-sided die and spin a five sector color
spinner. What is the probability of a particular number and color being selected?
Solution:
You can use real dice and spinner
or use the virt ual ones. The probability
of of any number being tossed varies, while the
probability of the spinner stopping on each color
is 1/5. Examples:
P(5,
green) means the probability of tossing
5 and spinning green
P(5,
green) = P(5) x
P(green) = 4/36 x 1/5 = 4/180
or 2.222...%
P(3, red)
means the probability of tossing 3
and spinning red
P(3,
red) = P(3) x P(red)
= 2/36 x 1/5 = 2/180 or 1.111...%
Complete 180 trials. Record the
number of times the selected two events occur
in the experiment, then complete the experimental
probability. Will the theoretical and
experimental probability always be the same? answer:
No
Two
Events
Theoretical
Probability
Experimental
Probability
P(5,
green) =
4/36
x 1/5 = 4/180 = 0.0222...2.2%
/180
P(3,
red) =
2/36
x 1/5 = 2/180 = 0.0111...1.1%
/180
Independent
Events: Probability of Drawing Two Cards
To be independent, you must put the first card back,
before drawing the second. This is equivalent
to drawing from two different decks of cards.
Complete the appropriate number
of trials needed for the experimental
probability. Will the theoretical and
experimental probability always be the same? answer:
No
Complete the appropriate number
of trials needed for the
experimental probability. Will
the theoretical and experimental probability always
be the same? answer:
No
Theoretical probability
is displayed in the opening screen of
each probability explorer selected. The green
line shows the theoretical probability for each
event.
Experiment probability:
Clicking the "?" button for Event
A and Event B starts
a probability experiment. The experimental
probability will likely approach the theoretical
probability for a large sample. Click
the "Reset" button
for Event A and Event
B if you wish to complete additional
experimental probability trials.
Note: the sectors for the
spinners included are the same size. Do not assume
that other resources will do the same. If you
are using a new spinner take a few moments to
examine the sector size.
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