Transcript P-value method

P-value method
2 means, both σ’s known
An economist is comparing credit card debt from two recent years.
She has gathered the following data:
Year 1
sample mean: $6618
sample size: 35
population standard deviation: $1928
Year 2
sample mean: $9205
sample size: 35
population standard deviation: $1928
Source: data is taken from problem 18, section 9-1 of Bluman, Elementary Statistics, eighth
edition
The economist
claims that
average credit
card debt
increased from
year 1 to year 2.
Evaluate her
claim using
the P-value
method with
α=.01.
If you want to try this
problem on your
own, click the kid to
the left.
Otherwise, click
away from the kid,
and we’ll work
through this
together.
Set-up
Summarizing the data using mathematical symbols, we get:
Year 1
population
𝜇1 =?
𝜎1 = 1928
sample
𝑋1 = 6618
𝑛1 =35
Year 2
population
𝜇2 =?
𝜎2 = 1928
sample
𝑋2 = 9205
𝑛2 = 35
These are what the
hypotheses will be about.
Step 1: State the hypotheses and
identify the claim.
The claim is that the average credit card debt increased
from year 1 to year 2. That is:
The debt from year 2 is bigger than the debt from year 1.
𝜇2
>
𝜇1
𝜇2 > 𝜇1
I don’t see an equals
sign. That should make
this the Alternate
Hypothesis, though I
suppose it could be the
cucumbers on my eyes.
𝐻0 :
𝐻1 :
𝜇2 = 𝜇1
𝜇2 > 𝜇1 (𝑐𝑙𝑎𝑖𝑚)
𝐻0 always
gets an equals
sign!
Yes. And it always
compares the same
quantities as 𝐻1 !
𝐻0 :
𝐻1 :
𝜇2 = 𝜇1
𝜇2 > 𝜇1
If we subtract, we’ll
be able to see what
number will be at the
center of our
distribution.
𝜇2 − 𝜇1 = 0
𝜇2 − 𝜇1 > 0
I hope she subtracts
“bigger minus
smaller” so we get a
positive number later,
when we work with
the sample values!
𝐻0 :
𝐻1 :
𝜇2 = 𝜇1
𝜇2 > 𝜇1
While the number 0 shows up in the
hypotheses, since we subtracted
“bigger minus smaller” (year 2 is
claimed to be bigger in the Alternate
hypothesis, and is actually bigger in the
sample data) we have set things up so
that we will have a right-tailed test and
our observed difference will be positive.
If you subtracted in the other order,
you’ll be doing a left-tailed test and your
observed difference will be negative.
𝜇2 − 𝜇1 = 0
𝜇2 − 𝜇1 > 0
Step (*)
Draw the picture and mark off the
observed value.
Do we know we have a normal distribution?
We do! Both sample sizes are
35, so they are big enough---they
are at least 30.
• First, draw the picture
Step (*):
Top level: Area
Middle Level: Standard Units (z)
We use z-values when we know both
population standard deviations.
Step (*):
• First, draw the picture
Top level: Area
Middle Level: Standard Units (z)
0
The center is always 0 in standard units.
Label this whenever you draw the picture.
Step (*):
• First, draw the picture
Top level: Area
Middle Level: Standard Units (z)
Bottom level: Actual Units ($)
0
In this case, the actual
units are dollars, since our
hypotheses are about
credit card debt.
Step (*):
• First, draw the picture
Top level: Area
Middle Level: Standard Units (z)
0
Bottom level: Actual Units ($)
0
The number from the Null Hypothesis always goes in the center
in standard units; that’s because we’re drawing the picture as if
the Null is true.
Then remember:
The P-value Method
is bottom-up
Step (*):
(continued)
Once you’ve drawn the picture,
start at the Bottom level and
mark off the observed value 𝑋2 −
𝑋1 = 9205 − 6618 = 2587.
Standard Units (z)
Bottom level
Actual units ($)
0
0
Mark off the right
tail, with its
boundary at 2587
2587
Step 2:
Middle level
Move up to the middle level. Convert
the observed value to standard units
and mark this off. (The value you
found is called the test value.)
Standard Units (z)
Actual units ($)
0
0
2587
Put your
answer here!
𝑧 =
𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟
(𝑋2 −𝑋1 )−(𝜇2 − 𝜇1 )
=
=
(𝜎1 )2 (𝜎2 )2
+𝑛
𝑛1
2
2587 − 0
(1928)2 (1928)2
+
35
35
Hypothesized
difference
= 5.613 … ≈ 5.61
Finishing up Step 2: Put the test value at the boundary of the
tail in standard units.
Standard Units (z)
Actual units ($)
0
5.61
0
2587
Step3:
Move up to the top level and
calculate the area in the tail;
this is the P-value.
P
Top Level (area)
Standard Units (z)
Actual units ($)
0
5.61
0
2587
We can either use Table E or the
calculator to find the P-value. Click on
the option you prefer.
Table E
Calculator
Note: the calculator
used in this tutorial is
the Casio fx-115MS
plus.
We can use Table E to
find our P-value.
If our z-value is on
the table, table E will
give us the area to
the left of it, and
we’ll have to subtract
that area from 1 to
get the area in the
tail. But…
5.61 is so
BIG
, it’s off the chart!
Looks like we’re
supposed to use
0.9999.
Label .9999 as the area to the left
of the observed value.
P = 1-.9999 = .00001
P
.9999
Standard Units (z)
Actual units ($)
0
5.61
0
2587
Step 4: Decide whether or not to
reject 𝐻0
Please be
merciful!
𝐻𝑜
• Compare P to α.
• P = area in right tail
= probability of getting 2587 (or a
bigger value) if 𝐻0 is true
• α= maximum allowable probability of
making a Type I error (rejecting 𝐻𝑜 if it is
true)
P
.00001
α
<
.01
• The probability we would get the result
we did (if 𝐻0 is true) is small.
• But we did get that result.
• So the probability that 𝐻0 is true is small.
Reject 𝐻0.
Step 5: Answer the question.
• Talk about the claim.
• Since the claim is 𝐻1 , switch to the
language of “support.”
• We rejected the 𝐻0 , so we support
the claim.
There is enough evidence to support the
claim that credit card debt increased from
year 1 to year 2.
Let’s recap!
Each click will give
you one step. Step
(*) is broken up into
two clicks.
Step 1. 𝐻0 : 𝜇2 − 𝜇1 = 0
𝐻1 : 𝜇2 − 𝜇1 > 0 (𝑐𝑙𝑎𝑖𝑚)
P = .00001
Step 3
Standard Units (z)
Actual units ($)
0
5.61
Step 2
0
2587
Step (*)
Step 4. Reject 𝐻0 .
Step 5: There’s enough
evidence to support the claim.
And there was much rejoicing.
Press the escape key to exit the slide show.
With the calculator, there’s
no need to round the critical
value, so be sure you’ve still
got the calculated critical
value displayed on
your screen.
Then hit the “shift” key
followed by the “3” key.
LEFT
RIGHT
MIDDLE
You’ll see this menu.
RIGHT
Our test is right-tailed, so select the area to
the right.
You’ll see
To enter in the calculated test value after the
R( , just hit the “Ans” key and then hit the
equals key.
You should get 0; there is some area in the
right tail, but it is so small that it rounds to 0!
Add the P-value to the
picture.
P=0
P
Standard Units (z)
Actual units ($)
0
5.61
0
2587
Step 4: Decide whether or not to
reject 𝐻0
Please be
merciful!
𝐻𝑜
• Compare P to α.
• P = area in right tail
= probability of getting 2587 (or a
bigger value) if 𝐻0 is true
• α= maximum allowable probability of
making a Type I error (rejecting 𝐻𝑜 if it is
true)
P
0
α
<
.01
• The probability we would get the result
we did (if 𝐻0 is true) is small.
• But we did get that result.
• So the probability that 𝐻0 is true is small.
Reject 𝐻0.
Step 5: Answer the question.
• Talk about the claim.
• Since the claim is 𝐻1 , switch to the
language of “support.”
• We rejected the 𝐻0 , so we support
the claim.
There is enough evidence to support the
claim that credit card debt increased from
year 1 to year 2.
Let’s recap!
Each click will give
you one step. Step
(*) is broken up into
two clicks.
Step 1. 𝐻0 : 𝜇2 − 𝜇1 = 0
𝐻1 : 𝜇2 − 𝜇1 > 0 (𝑐𝑙𝑎𝑖𝑚)
P=0
Step 3
Standard Units (z)
Actual units ($)
0
5.61
Step 2
0
2587
Step (*)
Step 4. Reject 𝐻0 .
Step 5: There’s enough
evidence to support the claim.
And there was much rejoicing.