Download 1/24/2017 - Elizabeth School District

1/24/2017
Today
• We’re going to review Chapter 6
• Work on some practice problems
• Review what we have already covered from Chapter 7 also
• Regular schedule gets pushed back by a day
Test Scores
• The average ended up being a 49% (before the curve)
• Curved average of about 68%
• These are low enough that I still want to work on Chapter 6 a little bit
• And earn points back in the process
• See next slide
• On Thursday you will get a packet/take-home test
• I will grade it on accuracy (like a test)
• You can work together, you can ask for help, etc.
• Then, next class, I will give you a quiz that very much resembles the
problems you did for homework
• In order to get the points for the homework, you need to get these right
• You can NOT work together, and I will not help
• This will go into the gradebook as another test (probably between 30
and 50 points)
• Can only help you, not hurt you
So…let’s review
• Chapter 6 started us off by introducing random variables:
• Book definition: a random variable takes numerical values that
describe the outcomes of some chance process.
• Important point: numerical values
• BUT, it can also be a range of values
• The probability distribution of a random variable gives its possible
values and their probabilities
Practice Questions:
• I select a random card from a 52-card deck, and I look at the suit of
the card. Is this a random variable? Why or why not?
• I select a random card from a 52-card deck, and I look at the number
on the card (with face cards equal to a value of 10). Is this a random
variable?
Practice Questions:
• I select a random card from a 52-card deck, and I look at the suit of
the card. Is this a random variable? Why or why not?
• NO
• I select a random card from a 52-card deck, and I look at the number
on the card (with face cards equal to a value of 10). Is this a random
variable?
• YES
Discrete vs Continuous Random Variables
• Discrete have a finite number of potential outcomes—we can easily
list them all
• E.G. Rolling a die has 6 potential outcomes (1,2,3,4,5,6)
• Continuous random variables have an infinite number of potential
outcomes
• Usually occurs when we measure something
• E.G. If I measure how many minutes early I arrive to school
• It could be 20.643 minutes, or 30.983 minutes, or 9.805 minutes, etc.
Example Question:
• Is this a continuous or a discrete
random variable?
• Hypothetical test grades, where I
rounded everything up to the
nearest multiple of 10 (anything
below 60 rounded to 60)
100
90
80
70
60
.091
.182
.152
.303
.273
Example Question:
• Is this a continuous or a discrete
random variable?
• Discrete, because there are a
fixed number of outcomes
• Hypothetical test grades, where I
rounded everything up to the
nearest multiple of 10 (anything
below 60 rounded to 60)
100
90
80
70
60
.091
.182
.152
.303
.273
Finding mean and St dev. Of discrete
• Could use the formulas:
• Mean: For each value, (value times probability). Then add them up
• St dev:
•
•
•
•
1. for each value, do (value minus mean) squared
2. Multiply by the probability for that value
3. Add them up
4. Take the square root
• OR, use your calculator to calculate the mean and st dev
• Enter them in Stat (L1 and L2)
• Then do 1-var stats (using L2 as frequency list)
Example Question:
• Find the mean and standard
• Hypothetical test grades, where I
deviation of this random variable
rounded everything up to the
nearest multiple of 10 (anything
below 60 rounded to 60)
100
90
80
70
60
.091
.182
.152
.303
.273
Example Question:
• Find the mean and standard
• Hypothetical test grades, where I
deviation of this random variable
rounded everything up to the
nearest multiple of 10 (anything
below 60 rounded to 60)
• Mean: 75.2 (also known as the
expected value)
100
• St dev: 13.05
.091
90
80
70
60
.182
.152
.303
.273
A Continuous Example
• Calculate the probability of
getting above a 90%
• Note: with a continuous
variable, our probability
distribution is usually a normal
curve
• Ch 6 Test Scores (before curve):
• Mean: 49.09
• St dev: 18.21
A Continuous Example
• Calculate the probability of getting
above a 90%
• Normalcdf(90,8274020374,49.09,18.2
1 = .012
• If you did
normalcdf(90,100,49.09,18.2) you
would get a similar result:
• .01
• Either way, we would expect about 1%
of people get a 90 or above
• Ch 6 Test Scores (before curve):
• Mean: 49.09
• St dev: 18.21
6.2 Transforming Random Variables
• X is our random variable (discrete or continuous)
• If Y=a+bX, this means that we are adding a constant (a) to X and/or
multiplying X by a constant (b)
• The main thing that we are interested in here is how this changes the
mean and the standard deviation
6.2 Transforming Random Variables
• If Y=a+bX, this means that we are adding a constant (a) to X and/or
multiplying X by a constant (b)
• The main thing that we are interested in here is how this changes the
mean and the standard deviation
• New mean=a+b(old mean)
• Or, in other words, our new mean is our old mean times b, plus a
• New Stdev=|b|(Old stdev)
• Our new standard deviation is our old standard deviation times the absolute value of b
An Example
• Let’s use the raw Chapter 6 test scores again
• Mean: 49.09
• St dev: 18.21
• If I wanted to create a new curve, I could multiply each score by 1.2
and then add 20.
• What is the new mean?
• What is the new standard deviation?
6.2 Combining Random Variables
• When we combine 2 random variables, we can either add them or
subtract them
• When we add X+Y, the new mean is just mean(X)+ mean(Y)
• When we subtract X-Y, the new mean is just mean(X)-mean(Y)
• When we add X+Y or subtract X-Y, the new VARIANCE is equal to
variance(X)+variance(Y)
• WE MUST USE VARIANCES, NOT STANDARD DEVIATIONS
• Remember, variance is equal to standard deviation squared
An Example
• I drive 35 minutes to work every day, with a standard deviation of 2
minutes
• My neighbor drives 30 minutes to work every day, with a standard
deviation of 7 minutes
• We want to create a new variable (Z) that represents the combined
time that he and I drive to work…
• What is the mean of this variable?
• What is the standard deviation?
An Example
• I drive 35 minutes to work every day, with a standard deviation of 2
minutes
• My neighbor drives 30 minutes to work every day, with a standard
deviation of 7 minutes
• We want to create a new variable (Z) that represents the combined
time that he and I drive to work…
• What is the mean of this variable? 65 minutes
• What is the standard deviation?
7.28 minutes
Binomial and Geometric Random Variables
• Binomial
• Binary (each outcome is either a yes or a no—a success or a failure)
• Independent (Each trial is independent of the other trials)
• Number (you must know the number of trials)
• Geometric
• Binary (each outcome is either a yes or a no—a success or a failure)
• Independent (each trial is independent of the other trials)
• Trials (we are trying to figure out how many trials it takes to get a success)
Normalcdf, binompdf, binomcdf, geometpdf,
geometdf
• Normalcdf: used when we have a mean and a standard deviation, and we
want to know the probability of being in some range
• Binompdf: when we have a binomial setting, and we want to know the
probability of getting an exact number of successes in a certain number of
trials
• Binomcdf: when we have a binomial setting, and we want to know the
probability of getting a range of successes in a certain number of trials
• Geometpdf: when we have a gemoetric setting, and we want to know the
probability of it taking an exact number of tries to get a success
• Geometcdf: when we have a geometric setting, and we want to know the
probability of it taking a range of tries to get a success
1. What is the probability of scoring
between 60 and 70?
2. If I take a sample of 10 people,
what is the probability that
exactly 3 of them scored
between 60 and 70?
3. For the same sample, what is the
probability that less than 3 of
them scored between 60 and 70?
4. If I start randomly choosing
students, what is the probability
that it takes me exactly 3
selections to find someone who
scored between 60 and 70?
5. What is the probability that it
takes me more than 3 selections
to find someone who scored
between 60 and 70?
• The test scores (again) had a mean
of 49.09 and a standard deviation
of 18.21
1. What is the probability of scoring
between 60 and 70? .149
2. If I take a sample of 10 people, what
is the probability that exactly 3 of
them scored between 60 and 70?
.128
3. For the same sample, what is the
probability that less than 3 of them
scored between 60 and 70? .823
4. If I start randomly choosing
students, what is the probability
that it takes me exactly 3 selections
to find someone who scored
between 60 and 70? .108
5. What is the probability that it takes
me more than 3 selections to find
someone who scored between 60
and 70? .616
• The test scores (again) had a mean of
49.09 and a standard deviation of
18.21
Another Example
• What is the probability of having
between 4 and 5 pounds?
• If I greet random people in the
town and ask how many pounds of
sugar they have, what is the
probability of it taking me exactly 5
people to find someone with 4-5
pounds of sugar?
• If I randomly select 6 people, what
is the probability that at least 1 of
them has between 4 and 5
pounds?
• In a small town, people have a
mean of 3.4 pounds of sugar, with
a standard deviation of 1.2
Another Example
• What is the probability of having
between 4 and 5 pounds? .217
• If I greet random people in the
town and ask how many pounds of
sugar they have, what is the
probability of it taking me exactly 5
people to find someone with 4-5
pounds of sugar? .082
• If I randomly select 6 people, what
is the probability that at least 1 of
them has between 4 and 5
pounds? .770
• In a small town, people have a
mean of 3.4 pounds of sugar, with
a standard deviation of 1.2
Starting Chapter 7
• Population:
• Sample:
Starting Chapter 7
• Population: the entire group about whom we want to draw
conclusions
• Sample: the randomly selected subset of the population that we
survey/poll/gather information about
Starting Chapter 7
• Parameter:
• Statistic:
Starting Chapter 7
• Parameter: value in the population that we want to know
• Statistic: value in the sample that provides an estimate of the
parameter value