Download Var(B)

Adding 2 random variables
that can be described by the
normal model
AP Statistics B
1
Outline of lecture
• Review of Ch 16, pp.376-78 (Adding or
subtracting random variables that fit
the normal curve)—last of Ch 16
• Follow along in the text
• Remember that you can download this
PowerPoint and a smaller, un-narrated
one from the Garfield web site
• Write down slide number if you don’t
understand anything
2
First thing: make a picture, make
a picture, make a picture!
• Well, actually, a table, but a table IS
a kind of picture, right?
Mean
Standard
Deviation
Packing (P)
9 min
1.5 min
Boxing (B)
6 min
1 min
3
Preliminaries: set-up and
assumptions
Setting up:
NOTE: NOT time to get an entire box packed
for sending; rather, we are only packing,
not boxing!
1. P1=time for packing 1st stereo system
2. P2=time for packing NEXT stereo system
3. T=P1+P2
Assumptions:
• Normal models for each RV
• Both times independent of each other
4
Step one: calculate expected
value (aka find the mean)
• Remember that the expected value
(EV) is a fancy word for finding the
mean.
• And with the mean, the EV sum of
two random variables is the sum of
their means:
5
Application to the packing and
boxing problem
• Mean (EV) of packing 1ST system is 9
min
• Mean (EV) of packing 2nd system is
also 9 min
• Therefore:
6
Calculate standard deviation
just like we have before
• Nothing new, same old formula:
7
So what?
• (You should always ask yourself “so what?”
when somebody tells you to do something)
• Well, we know that we had two RVs
(random variables, not recreational
vehicles; this is statistics, after all, not an
auto show) that have a normal distribution.
• So we now have a normal distribution and
know the mean and standard deviation.
• In statistical terms: N(18, 2.12)
• Now we can evaluate this using what we
learned in Ch. 6! A seriously big deal!
8
Q: What is the probability that
packing 2 consecutive systems
takes over 20 min?
• This is the question we need to answer.
• Remember the z-formula from Ch 6?
• Write it down, and we’ll apply it on the
next slide.
9
Setting up the problem
• We already know the mean and
standard deviation from our earlier
calculations: 18 and 2.12,
respectively.
• The “y” value we are looking for is
20, so we set up the solution thusly:
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Are we done yet?
• Of course not.
• Our goal is to find the probability that it
will take MORE than 20 minutes to
package 2 consecutive stereo systems.
• The z-value of 0.94 simply means that
the area to the LEFT of that point on
the z-table (text, A79-A80) will be the
probability that packaging will take
LESS than 20 minutes.
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Draw a picture!
• The probability we
get for z=0.94 is
the dark blue area
on the left.
• However, we’re
interested in the
light blue area
that’s ABOVE
z=0.94
12
But first…..
• Remember that our table only
measures the cumulative probability
to the LEFT of the value.
• That’s all we have, so let’s find it,
and then answer the question more
directly.
13
How to use the table
• It’s been a while, but get the X.x value
on the inner column, and the 0.0x
value across the top. The intersection is
where the value lies, and looks like
this:
14
Finding the probability to the
left of 0.94
• Since 0.94>0, look on p. A-80
• Find 0.90 along the z-column on the
far left
• Read across the top row to 0.04
• The intersection of the 0.90 row and
the 0.04 is the percentage of the
normal curve to the LEFT of 0.94…..
• ……which should be 0.8294
15
What are we looking for?
• Not the blue area
to the LEFT, but
the clear area to
the RIGHT of 0.94
• Calculate by
subtracting the
area to LEFT from
1:
• 1.00000.8294=0.1706
16
Why the difference from the
textbook?
• Beats me.
• But 0.1706 isn’t all
THAT different
from 0.1736…3
parts in a
thousand.
17
Back to interpretation
• We get a z-value of just over 17%.
• That means, in everyday language,
that there’s only a 17% chance
(probability) that it will take more
than 20 minutes to package 2
consecutive stereo systems.
• NOTE: the AP exam expects you to
write out things like this
18
NEXT QUESTION
(BOTTOM OF PAGE 377)
What percentage of
stereo systems
take longer to back
than to box?
19
Set-up on questions like this is
crucial
• The key is to realize that you don’t
set it up as an inequality, exactly.
• That is, the question is NOT P>B
• Rather, the question is whether PB>0.
• We pick a different variable (D for
“difference”) and define it as
D=P-B
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Why do we do it like this?
• We now can ask a specific question,
namely what’s the expected value of
D?
• In statistical terms, we have
E(D)=E(P-B).
• We can now calculate these values
using what we’ve learned in Ch 16
and combining it with the normal
model from Ch 6.
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First the mean,
then the standard deviation
1. E(D)=E(P-B)=E(P)-E(B)
2. E(P) we get from reading the mean
for packing right off the table
3. We get E(B) the same way.
4. E(P)-E(B)=9 min – 3 min = 6 min
22
Calculating the standard
deviation
• I like to calculate the SD directly, but
you can start with the variance, and
then take the square root.
• Var(D)=Var(P-B)=Var(P)+Var(B)
• =1.52+ 1.02 (from the
table)=2.25+1=3.25
• σP-B=(3.25)½=1.8 min
(approximately)
23
Now we have the normal
model
• i.e., N(3, 1.80)
• We are interested only in the values
that are GREATER than 0, i.e., to the
RIGHT of the value, like the purple:
24
Now, to calculate
• Same as we did before (Slide 9):
• By table A-79, z=-1.67 has 0.0475
to its LEFT, but we want the area to
the RIGHT
• So subtract from 1 and get 0.9525.
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Interpreting the result
• We have determined the percent of
the time where P-B>0
• In other words, 95.25% of the time,
it takes longer to pack the boxes
than to box them.
• How do we know? Because P-B is
positive ONLY when P>B (otherwise,
the difference would be negative)
26
Exercise
• Do Exercise 33 on p. 383-84 of the
textbook
• Take 10-15 minutes to complete all
parts.
• Review your answers with Ms. Thien,
who has them all worked out.
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