Transcript P-value

Testing a Claim
I’m a great free-throw shooter!
Significance Tests
A significance test is a formal procedure for
comparing observed data with a claim (also called
a hypothesis) whose truth we want to assess.
I’m a Great Free-Throw Shooter!
Our virtual basketball player in the previous
activity claimed to be an 80% free-throw
shooter. Suppose that he shoots 50 free
throws and makes 32 of them. His sample
proportion of made shots is
32
ˆp 
 0.64
50
H 0 : p  0.80
Null Hypothesis
H a : p  0.80
Alternate Hypothesis
One-sided
3/400= 0.0075
P-Value
DEFINITION: P-value
The probability, computed assuming H0 is true, that the
statistic (such as p̂ or X) would take a value as extreme as
or more extreme than the one actually observed is called
the P-value of the test. The smaller the P-value, the
stronger the evidence against H0 provided by the data.
Studying Job Satisfaction
Does the job satisfaction of assembly-line workers differ
when their work is machine-paced rather than self-paced?
One study chose 18 subjects at random from a company
with over 200 workers who assembled electronic devices.
Half of the workers were assigned at random to each of two
groups. Both groups did similar assembly work, but one
group was allowed to pace themselves while the other
group used an assembly line that moved at a fixed pace.
After two weeks, all the workers took a test of job
satisfaction. Then they switched work setups and took the
test again after two more weeks. The response variable is
the difference in satisfaction scores, self-paced minus
machine-paced.
(b)Describe
State appropriate
hypotheses
for performing
a significance test.
(a)
the parameter
of interest
in this setting.
: mean
 0 μ of the differences (self-paced
(a) The parameter of interestH
is0the
Two
sided
minus machine-paced) in jobH
satisfaction
scores
in
the
population
of all
:


0
a
assembly-line workers at this company.
For each of the following settings,
(a) describe the parameter of interest, and
(b) state appropriate hypotheses for a significance test.
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complete,
including the
instructions and answers.” We suspect that the actual time it takes
(a) p = proportion of students at Jannie’s high school who get
to complete the form may be longer than advertised.
less than 8 hours of sleep at night.
(a) μ = mean amount of time that it takes to complete the census
form.
(b) H0: p = 0.85 and Ha: p ≠ 0.85.
(b) H0: μ = 10 and Ha: μ > 10
For the job satisfaction study described, the hypotheses are
H0 :   0
Ha :   0
where μ is the mean difference in job satisfaction scores (self-paced
− machine-paced) in the population of assembly-line workers at the
company. Data from the 18 workers gave x  17 and sx = 60. That is,
these workers rated the self-paced environment, on average, 17
points higher.
Researchers performed a significance test using the sample data
and obtained a P-value of 0.2302
(b)
theprovide
P-valueconvincing
in context.
(c) Interpret
Do the data
anevidence
average difference
ofnull
17 orhypothesis?
more points
against the
(a) Explain
what
it means
for the null
between
the two
work
environments
Explain.
would
happento
23%
thein
time
by
hypothesis
beoftrue
thisjust
setting.
chance in random samples of 18 assemblyline workers when the true population
mean is μ = 0
In a nutshell, our conclusion in a significance test comes down to:
P-Value small
Reject H0
P-Value large
Fail to reject H0
How small of a P-value is small?
P-Value < α
Reject H0
P-Value ≥ α
Fail to reject H0
Conclude Ha in context
Cannot conclude Ha in context
Smaller than an α = 0.05 or 0.01
Conclude Ha in context
Cannot conclude Ha in context
Better Batteries
A company has developed a new deluxe AAA battery that is supposed
to last longer than its regular AAA battery. However, these new
batteries are more expensive to produce, so the company would like to
be convinced that they really do last longer. Based on years of
experience, the company knows that its regular AAA batteries last for
30 hours of continuous use, on average. The company selects an SRS of
15 new batteries and uses them continuously until they are completely
drained. A significance test is performed using the hypotheses
H 0 :   30 hours
H a :   30 hours
where μ is the true mean lifetime of the new
deluxe AAA batteries. The resulting P-value is
0.0276.
What conclusion would you make for α=0.05
significance levels? For α=0.01?
Answer:
(a) Since the P-value, 0.0276, is less than α = 0.05, the sample
result is statistically significant at the 5% level. We have sufficient
evidence to reject H0 and conclude that the company’s deluxe AAA
batteries last longer than 30 hours, on average.
(b) Since the P-value, 0.0276, is greater than α = 0.01, the sample
result is not statistically significant at the 1% level. We do not have
enough evidence to reject H0 in this case. Therefore, we cannot
conclude that the deluxe AAA batteries last longer than 30 hours, on
average.
Tasty Chips
For his second semester project in AP Statistics, Zenon decided to
investigate if students at his school prefer name-brand potato chips to
generic potato chips. He randomly selected 50 students and had each
student try both types of chips, in random order. Overall, 34 of the 50
students preferred the name-brand chips. Zenon performed a
significance test using the hypotheses:
H0 : p = 0.5
Ha: p > 0.5
where p = the true proportion of students at his school that prefer
name-brand chips. The resulting P-value was 0.0055.
Problem: What conclusion would you make
at each of the following significance levels?
(a) = 0.01
(b) = 0.001
Type I and Type II Errors
If we reject H0 when H0 is true, we have committed a Type I error
If we fail to reject H0 when H0 is false, we have committed a Type II
error.
Perfect Potatoes
A potato chip producer and its main supplier agree that each
shipment of potatoes must meet certain quality standards. If the
producer determines that more than 8% of the potatoes in the
shipment have “blemishes,” the truck will be sent away to get
another load of potatoes from the supplier. Otherwise, the entire
truckload will be used to make potato chips. To make the decision, a
supervisor will inspect a random sample of potatoes from the
shipment. The producer will then perform a significance test using
the hypotheses
H : p=0.08
0
Ha: p>0.08
Where p is the actual proportion of potatoes with blemishes in a
given truckload.
Describe a Type I and a Type II error in this setting,
and explain the consequences of each.
Faster fast food?
The manager of a fast-food restaurant want
to reduce the proportion of drive-through
customers who have to wait more than 2
minutes to receive their food once their
order is placed.
Based on store records, the proportion of customers who had to wait at
least 2 minutes was p = 0.63. To reduce this proportion, the manager
assigns an additional employee to assist with drive-through orders.
During the next month the manager will collect a random sample of
drive-through times and test the following hypotheses: H0: p=0.63
Ha: p<0.63
where p = the true proportion of drive-through customers who have to
wait more than 2 minutes after their order is placed to receive their
food.
Describe a Type I and a Type II error in this setting and explain the
consequences of each.
Significance and Type I Error
P(type I error) = Significance level (α)
Power and Type II Error
P(type II error) = β
DEFINITION: Power
The power of a test against a specific alternative is the probability
that the test will reject H0 at a chosen significance level α when the
specified alternative value of the parameter is true.
Power =1 - β
Lower α
Higher power
Higher n
Higher n
Read Section 9.2
Exercises on page 546,
# 1 – 15 odds,
# 19, 21,
23, 25