Transcript Document

Chapter 11
• Z Tests
• Type I and II errors
• Power
Pizza Delivery
• Pines Pizza claims that on average
they deliver in 20 min or less.
• Design an experiment to test their
claim.
• Remember your three principles of
experimental design( control,
replication, randomization.
• What would constitute sufficient
evidence that they are wrong in their
claim.
• Your group has 10 minutes.
Pizza Delivery
• FACTS….
• Claim is 20 minutes or less.
So the true mean is assumed to be u = 20
• Lets use 1.5 minutes as the true
standard deviation
• We need a sample mean from our
experiment that we can use as
evidence against the claim.
Pizza Delivery
• FACTS….
• Choose a sample mean that is greater
than 20 minutes but not too much
greater.
• Also choose a sample size, how many
deliveries do you think is necessary to
test this claim.
Pizza Delivery
• FACTS….
• Claim is 20 minutes or less.
Ho: u = 20
• If you were going to argue the claim
you are saying it is over 20 minutes
Ha: u > 20
Conclusions
• The whole idea is to make the
correct conclusion.
• There are 2 scenarios based on
your p-value.
• The sample mean drives your pvalue and conclusion…..use
your brain!!!
• This p-value is low enough to reject
Ho at the ____ level.
• This is evidence to suggest that
pines pizza may have a mean
delivery time of more than 20 min.
• This p-value is too high to reject Ho
at the ____ level.
• There is not enough evidence to
suggest that pines pizza may have a
mean delivery time of more than 20
min.
Check Your Pulse
• Most peoples pulse rate ranges
from 70 to 100 beats per minute.
• A previous study gave a mean of
73 and standard deviation 11.
• Let’s see how our pulse rates
compare to these.
• Let’s use the following Hypothesis
Ho: μ = 73 vs Ha: μ ≠ 73, where μ is
the true mean pulse rate for our
class today.
We will also use the standard
deviation from the previous study
σ=11
How to take your pulse
• Place 2 fingers on your other
wrist and count the beats for 15
seconds, then multiply by 4.
• You can also count the beats
for 60 seconds(takes more
concentration)
Ho: μ = 73 vs Ha: μ ≠ 73, where μ is the
true mean pulse rate for students at
Rancho.
Assumptions: We have an independent random sample of 39
pulse rates taken during fourth period in class. The
population of all students at school is more than 10x our
sample. Our normal condition is met by the CLT.
1-sample Z Test
Mean = 69.6
n = 39
Z = -1.92
P-value = .0547
This p-value is low enough to reject Ho at
the 10% level.
Based on this sample, there is evidence to suggest the the
pulse rates for Rancho students may be different than 73.
Type 1 and 2 errors
Type 1 error — Rejecting Ho when it’s true.
Type 2 error — Failing to reject Ho when it’s
false.
CH 11 Tests
Significance Tests
Use the HAN SOLO acronym
• H: hypotheses Ho and Ha
• A: assumptions/Conditions
• N: name the Test
•
•
•
•
S: stats from calc
O: obtain a p-value
L: low enough to reject Ho?
O: outcome in context
What is a P-Value?
• P-value assumes Ho is true.
• It is a percent.
• It is the probability of your sample
happening by chance if Ho is true.
What is a P-value?
When the p-value is very low like
p = .0123 it means that the data
you collected could only happen
1.23% of the time if Ho is true.
Since 1.23% is very rare, what does
this probably suggest?
It means that Ho is probably not true.
So we Reject Ho
What is a P-value?
Now, lets say a p-value is higher.
p = .3567. This means that the data
we collected could happen about
35.67% of the time, if Ho is true.
This percent is very likely to
happen.
There is no reason to doubt Ho
Do not reject Ho
Alpha Levels
The popular alpha levels are:
• α = 0.01 which is 1%
• α = 0.05 which is 5%
• α = 0.10 which is 10%
These levels are the cutoff points for rejecting Ho
Practice the Hard Parts
• The following slides will help
you write your alternative
hypotheses(Ha)
• Help with your p-values
• Help write your conclusions.
Jack claims he can throw a football 50 yards. He throws the
ball 75 times and averages 49.5 yards. Carry out a
significance test at the α = .05 level.
Ho: μ = 50
vs
Ha: μ < 50
Where μ is the true mean distance Jack can throw a football
After crunching the #’s on the calculator a p-value gives
P = .0766
What do we do with Ho?
Write your conclusion.
Is p < .05? No, we will not Reject Ho
There is not enough evidence at the 5% level to suggest that
Jack throws a football less than 50 yards.
A manufacturer claims that a new brand of air-conditioning
unit uses only 6.5 kilowatts of electricity per day. A
consumer agency believes the true figure is higher and runs
a test on a sample size of 50. Carry out a significance test at
the α = .10 level.
Ho: μ = 6.5
vs
Ha: μ > 6.5
Where μ is the true mean kilowatts of electricity used per day
After crunching the #’s on the calculator a p-value gives
P = .0590
What do we do with Ho?
Write your conclusion.
Is p < .10? Yes, we will Reject Ho
This evidence suggests that the true mean kilowatts of
electricity used per day might be more than 6.5 kilowatts.
Mr. Pines claims that the mean GPA for athletes at Rancho
Alamitos High School is 3.25. Carry out a significance test at
the α = .01 level.
Ho: μ = 3.25
vs
Ha: μ ≠ 3.25
Where μ is the true mean GPA of athletes at Rancho
After crunching the #’s on the calculator a p-value gives
P = .0083
What do we do with Ho?
Write your conclusion.
Is p < .01? Yes, we will Reject Ho
This evidence suggests that the true mean GPA of athletes at
Rancho might be different than 3.25.
No alpha(α) level given
Lets say our calculator gives a p-value of p = .0452
If there is no alpha level given, how do you know if you should
reject the Ho?
Typical alpha levels
.01
.05
.10
You need to reject at the level where it fits best, this p-value of
p =.0452 is low enough to reject at the 5% level but not at the
1% level.
Understand this…..
• It is NOT our job to find the true
mean or the true proportion in a
significance test.
• We just have to make a
decision on the claim.
• Our data helps us make a
decision
One and Two-sided tests
One sided tests:
Ha: μ < 20
Ha: μ > 20
Two sided test
Ha: μ ≠ 20
One-sided test
Ho: μ = 10
vs
Ha: μ > 10
Let’s say you are not given any data, all you have is a z-score
Z = 2.78
How do you get a p-value?
normalcdf(2.78,1000) = .0027
Two-sided test
Ho: μ = 10
vs
Ha: μ ≠ 10
Let’s say you are not given any data, all you have is a z-score
Z = 2.78
How do you get a p-value?
normalcdf(2.78,1000) = .0027
Because this is a two-sided
test(≠) you need to multiply
your p-value by 2
2(.0027) = .0054
P-value?
• Caution!!!
• You only need to multiply your
p-value by 2 if it is a two-sided
test and if you had to use
normalcdf to calculate it.
P-value
If you used one of the tests from the test menu
on your calculator, the p-value is already taken
care of.
This was done from the TEST MENU, do NOT multiply your
p-value by 2
Type 1 and 2 errors
• Type 1 error — Rejecting Ho when
it’s true.
• Type 2 error — Failing to reject Ho
when it’s false.
Here is the dilemna
• It might not be safe to launch
the space shuttle today.
• How is this a type I, type II error
situation?
Type 1 and 2 errors
Ho: The shuttle is safe, launch rockets
Ha: The shuttle is not safe, delay the launch
What is a type I error in this context?
What are its consequences?
What is a type II error in this context?
What are its consequences?
Which is worse?
Type I error: Reject Ho but it was true.
Assuming the shuttle is not safe, delaying the
launch, but the shuttle was safe.
Consequence: Not launching on time, have to
reschedule. But no one is in danger
Type II error: Fail to Reject Ho, when it is
false. Assuming shuttle is safe, when it
actually wasn’t safe.
Consequence: Shuttle blows up.
Reject Ho
Ho True
Ho False
Assume the Shuttle
is not safe, delay
launch
Assume the Shuttle
is not safe, delay
launch
TYPE I ERROR
Fail to Reject Ho
Correct Decision
Assume the Shuttle
is safe, launch
Assume the Shuttle
is safe, launch
Correct Decision
TYPE II ERROR
Which type of error is more
serious? …. You have to read
the problem!
Decreasing the chance of a type I
error increases the chance of a
type II error…and vice versa
Type 1 and 2 errors
• What is the probability of that error?
• Type 1 is the alpha level.
• Type 2 is β and complex and you do not
have to know.
Power
STUDY THIS PAGE
• What is power?
• Power = 1 - β
It is the probability of correctly
rejecting Ho when it is false.
Ways to increase power
Increase sample size
Increase alpha level.....ex.(use α = .10
instead of α = .05 or α = .01)
Criminal Trial QUIZ( yes, a quiz,
actually a group quiz, one paper
per group
• In a criminal trial, the defendant
is held to be innocent until
shown to be guilty beyond a
reasonable doubt.
• Ho: defendant is innocent
• Ha: defendant is guilty
Criminal Trial
1. Give the type I and type II
errors in this context.
2. Give the consequences for
each
3. Which is worse?
4. Based on your answer to #3, is
it better to use a significance
test with α = .05 or α = .01?
Explain your answer.
5. Explain what power would be
in the context of THIS setting.
1-sample Z-test
• Ho: u = 3.5 vs Ha: u ≠ 3.5 where
u is the true mean of this real
die.
• Assumptions: We have an
independent random sample of
50 rolls of a real die. Our normal
condition is met by the CLT.
z = -.33
X-bar = 3.42
σ = 1.70783
p=.7405
This p-value it too high to reject
Ho
There is not enough evidence to
suggest that the mean of this die
is different than 3.5
Back to the Real Die
Recall that a real die has μ = 3.5
and σ = 1.71….last week we
rolled the die 50 times.
What sample mean would we
need to get in order to reject
the null(μ = 3.5 ) at the 5%
level?
You have to use the z-test formula
x - 3.5
= 1.960
1.71
50
x - 3.5
= 1.960
1.71
50
x = 3.973987818
Plug the answer back in a Z-test
and see if your p-value gets
rejected at the 5% level.
Make sure you do a two-sided test.
Emergency Response Time
• Ho: The mean response time is
equal to 6.7 minutes
• Ha: The mean response time is
less than 6.7 minutes.
Give the type I and type II errors for this situation.
Give consequences for each
What should the alpha level be set at?
What is power in context of this situation
Type I Error:
Reject Ho, when it’s true
Assume the response time is less than 6.7
minutes when it’s not.
Consequence: Not trying to improve your
response time, people are at risk of not getting
helped as fast as they should be.
Type II Error: Fail to reject Ho, when it’s false
Assume the response time is equal 6.7 minutes
when it’s less.
Consequence: Wasting time and money on
improving your response time when you
already have.
Type I is worse because peoples lives are
at risk.
Because you want to protect the type I
error, you need to make Ho hard to
reject. Choose a low alpha level .01.
The power is correctly rejecting Ho when
it’s false.
Probability of correctly assuming
response time is less than 6.7 minutes.
Temperatures
Ho: u = 98.6 vs Ha: u ≠ 98.6
where u is the true mean
temperature for students in this
class
Conditions
We have an independent sample of
9 students who volunteered to
take their temperatures. Our
sample is not large enough to
assume Normality by the CLT. Our
box plot shows no outliers or
extreme skew. Ok to proceed.
1-sample t test
X-bar = 98.31
S = .86378
n=9
df = 8
t = -1.00
p = .3451
This p-value is too high to reject Ho
This evidence suggests that the true
mean temperature might not be
different than 98.6
Rejecting the Ho
• Determine what should be done
with the following P-Values
1.P(Z < -1.43)
= 0.0764, reject at the 10% level
1.P(Z > .34)
= 0.3669, do not reject Ho
1.P(Z ≠ 2.87)
=
0.0041, reject at the 1% level
Notice the small difference
(a) Mean = 536.7
P-Value = .0505 Do not reject Ho
(b) Mean = 536.8
P-Value = .0496 Reject at 5% level
(c) The sample mean only changed by a
tenth, slight variation completely
reversed our conclusion
Sample Size Matters
(a) n = 100 P-Value = .3628 not
even close to rejecting null
(b) n = 1000 P-Value = .1336 not
enough to reject null
(c) n = 10,000 P-Value = .0002
reject at any level
Conjecture
• Mr. Pines is considering whether
he should do his catching tic tacs
activity this year in AP Stats.
Ho: It is safe, do the activity.
Ha: It is not safe, don’t do it.
Fill out your type I and II error template completely.
Conjecture
• Mr. Pines got stung by a sting ray
over the break. He is trying to
decide whether he should go to the
doctor.
Ho: It will heal on its own.
Ha: It needs to be looked at by a doctor.
Fill out your type I and II error template completely.
Type I Error:
Reject Ho, when it’s true
Decide that it needs to be looked at by a doctor
when it would have healed on its own.
Consequence: Wasted time,money, and the
inconvenience of going to the doctor.
Type II Error: Fail to reject Ho, when it’s false
Assume that it would heal on its own when it
needed to be looked at by a doctor
Consequence: It takes a long time to heal, could
get infected, and may possibly result in loss of
a limb.
Type II error is worse because you’re
putting your body in serious risk
Because you want to protect against the
type II error, you need to make Ho easy
to reject. Choose a high alpha level .10.
The power is correctly rejecting Ho when
it’s false.
So in this situation the power is the
probability of correctly choosing to go
to the doctor when you need to go.
• That was a true story.
• Mr. Pines made a type II error.
• Luckily the doctor was able to
heal him in time.
Conjecture
• Mr. Pines is wondering whether the
shark swimming near him is a
friendly shark or a mean shark.
Ho: It is a friendly shark.
Ha: It is a mean shark.
Fill out your type I and II error template completely.
Type I Error:
Reject Ho, when it’s true
Decide that the shark is mean when it actually
was friendly
Consequence: Miss out on an opportunity to
swim with a shark and have a great experience
to share with all your friends and family.
Type II Error: Fail to reject Ho, when it’s false
Assume that the shark is friendly when it is
actually a mean shark.
Consequence: Get eaten.
Type II error is worse because you could
get eaten by a shark.
Because you want to protect against the
type II error, you need to make Ho easy
to reject. Choose a high alpha level .10.
The power is correctly rejecting Ho when
it’s false.
So in this situation the power is the
probability of correctly identifying a
mean shark when it is actually mean.
Conjecture
• Mr. Pines is wondering if he should
quit is his job and try to join the
circus.
Ho: Keep teaching.
Ha: Circus.
Fill out your type I and II error template completely.
Conjecture
• Mr. Pines is wondering if he should
still eat the last half of his bagel
after it fell on the carpet in this room
so that the cream cheese side was
touching the carpet.
Ho: Eat it.
Ha: Throw it away.
Fill out your type I and II error template completely.
4 Boys Ditch
It was the day before a vacation and four boys, who are all
friends, were absent from Mr. Pines’ 2nd period class. They were
generally healthy boys. After 60 days of class, each had missed
only one day. There was not a flu going around at the time and
the day was typical except for the fact that it was the day before
a long break. All boys were healthy and present the day before.
Is there mathematical evidence to suggest that these boys were
ditching? Justify your response.
Ho: The boys were just absent coincidentally.
Ha: The boys organized a ditch day.
How would you calculate the P-value? (The probability that
those boys were all absent by chance on the same day of
school)
4 Boys Ditch
It was the day before a vacation and four boys, who are all
friends, were absent from Mr. Pines’ 2nd period class. They were
generally healthy boys. After 60 days of class, each had missed
only one day. There was not a flu going around at the time and
the day was typical except for the fact that it was the day before
a long break. All boys were healthy and present the day before.
Is there mathematical evidence to suggest that these boys were
ditching? Justify your response.
What would your conclusion be?
Which error type might you have committed? Looking at
the error type, what do you think Mr. Pines should do with
this situation?
Conjecture
• Mr. Pines is wondering whether to
accuse these 4 boys of ditching.
Ho: The boys were just absent coincidentally.
Ha: The boys organized a ditch day.
We need a p-value to base our decision on.
The probability that one boy was absent is based on his
attendance so far this year…(1/60)
So for these 4 boys absent on the same day is (1/60)^4
P-value = .00000008
Type I Error:
Reject Ho, when it’s true
Decide that the boys ditched when they really
were just absent coincidently
Consequence: Boys may get in trouble for
something they didn’t do, Mr. Pines may get
parent complaints and possible law suit?.
Type II Error: Fail to reject Ho, when it’s false
Assume that the boys were just absent
coincidently when they did organize a ditch
day.
Consequence: Boys get away with missing a day
of school, they don’t learn their lesson and
continue to do more serious offenses which
leads to a life of crime.
For Mr. Pines a Type I error is worse
because it’s not worth the time and
hassle.
Because you want to protect against the
type I error, you need to make Ho hard
to reject. Choose a high low level .01.
The power is correctly rejecting Ho when
it’s false.
So in this situation the power is the
probability of correctly deciding that
they ditched when they really ditched.
You could have chosen that a Type II error
is worse because maybe you feel that they
shouldn’t get away with ditching
Because you want to protect against the
type II error, you need to make Ho easy to
reject. Choose a high high level .10.
The power is correctly rejecting Ho when
it’s false.
So in this situation the power is the
probability of correctly deciding that they
ditched when they really ditched.