P-VALUE ON
INFERENTIAL
STATISTICS
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Rule of p-value:
If the p-value is small,
then we reject the null
The p-value is NOT…
The p-value is not the probability that H0 is true.
A small p-value does not mean H0 is false.
Statistical significance
Sometimes, a threshold of evidence called a
“significance level” (denoted by  ) is set prior to
conducting the test.
Typically  =0.05,
 but sometimes people also use  =0.10 ,  =0.01 or
 =0.001.
If the p-value is
less than or equal to  we reject H0.
greater than  , we do not reject H0.
Hypothesis testing
A good analogy for hypothesis testing is in the American judicial
system: In a trial, there are two competing hypotheses: the
defendant is guilty or innocent. Also, the defendant is presumed
innocent until they are proven guilty.
For example, we would initially believe that Frito-Lay is telling the
truth until we have convincing evidence that the average weight is
< 14 oz.
NULL HYPOTHESIS: we initially assume that one of the
hypotheses is true
ALTERNATIVE HYPOTHESIS: We then consider the
evidence from the sample and reject the null hypothesis
(in favor of the)
Significance
Tests
Call the paramedics!
Vehicle accidents can result in serious injuries to
drivers and passengers. When they do, someone
calls 911 and paramedic response. Several cities
have begun to monitor paramedic response time
and they got a ℳ = 6.7 minutes with σ=2 minutes.
At the end of the following year, the city manager
selects simple random sample of 400 calls and the
mean response was x=6.48 minutes. Do these
data provide good evidence that response times
have decreased since last year?
Stating the hypothesis
We are seeking evidence of decrease in response time
this year, thus our null hypothesis says “no decrease” on
the average in the large population of all calls involving
life-threatening injury this year.
The alternative
hypothesis says “there is a decrease.” therefore:
Ho: ℳ = 6.7 minutes
Ha: ℳ < 6.7 minutes
Conditions for significance testing
SRS: the city manager took simple random sample of
400 calls for life-threatening injury this year.
Normality: the population distribution may not follow
Normal distribution but the sample size of 400 is large
enough to ensure Normality of x (by CLT)
Independence: Since the city manager is sampling
calls without replacement, we must assume that there
were at least (400)(10)=4000 calls involved lifethreatening injuries in the city this year.
Test Statistics
estimate - hypothesized value
test statistic =
standard deviation of the estimate
ℳ=6.7 minutes
x= 6.48 minutes
σ= 2 minutes
n=400
z= x-ℳ
σ/√n
z= 6.48-6.7
2/√400
z= -2.20
P-values
The Probability, computed assuming that Ho is true, that
the observed outcome would take a values as extreme as
or more extreme that that actually observed is called the
P-value of the test. The smaller the P-value is the
stronger the evidence against Ho provided by the data
z= -2.20
P=0.0139
x=6.48
z=-2.20
P=0.0139
ℳ=6.7
Conclusion
x=6.48
z=-2.20
P=0.0139
ℳ=6.7
There is about 1.4% chance that the city manager would obtain
a sample of 400 calls with a mean response of 6.48 minutes or
less. The small P-value provides strong evidence against Ho
and in favor the Ha where ℳ<6.7
Example 2
Coffee sales Weekly sales of regular ground coffee at a
supermarket have in the recent past varied according to
a Normal distribution with mean μ = 354 units per week
and standard deviation σ = 33 units. The store reduces
the price by 5%. Sales in the next three weeks are 405,
378, and 411 units. Is this good evidence that average
sales are now higher? The hypotheses are
H0: μ = 354
Ha: μ > 354
Assume that the standard deviation of the population of
weekly sales remains σ = 33.
Solution:
State the 3
conditions
z= x-ℳ
σ/√n
X-bar: 398
µ: 354
n:3
∂: 33
z= 398-354
33/√3
z= 2.31
p= 0.0104
The small P-value
provides strong
evidence against Ho
and in favor the Ha
where ℳ > 354 units.
We can say that there
is a pretty convincing
evidence that the
mean sales are
higher.
Classwork
At the bakery where you work, loaves of bread are
supposed to weigh 1 pound. From experience, the
weights of loaves produced at the bakery follow a
Normal distribution with standard deviation
 = 0.13 pounds. You believe that new personnel
are producing loaves that are heavier than 1 pound.
As supervisor of Quality Control, you want to test
your claim at the 5% significance level. You weigh
20 loaves and obtain a mean weight of 1.05 pounds.
Answer
Ho: Loaves of bread has an average weight of µ=1 lb.
Ha: Loaves of bread has an average weight of µ1 lb.
Since ∂ is given - use one-sample z-test
SRS: we are not told that the sample is randomly chosen so we
should proceed with caution
Normality: the distribution is said to be Normal
Independence: N ≥ 10(20)
Test-statistic:Z=1.72, p=.0427
With a p-value of .04247 which is less than .05 significant level, we
have strong evidence to reject the null hypothesis. Therefore, the
true average weight of the loaves of bread is greater than a pound.
However, we will proceed with caution since we can’t confirm
Normality.
Warm-up
Statistics can help decide the authorship of literary
works. Sonnets by an Elizabethan poet are known to
contain an average of µ= 6.9 new words (words not
used in the poet’s other works). The distribution of
new words in this poet’s sonnets is Normal with
standard deviation ∂= 2.7. Now a manuscript with
five new sonnets has come to light, and scholars are
debating whether it is the poet’s work. The new
sonnets contain an average of x= 9.2 words not used
in the poet’s known works. We expect poems by
another author to contain more new words than found
in the Elizabethan poet’s poems.
Car Pollution