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9.3: P-Value Approach to Hypothesis Testing
Steps to Hypothesis Testing
1. State the null and alternative hypothesis
2. Discuss the logic of this hypothesis test
3. Obtain a precise criterion for deciding whether to reject the null
hypothesis in favor of the alternative hypothesis
4. Apply the criterion in step 3 to the sample data and state the
conclusion
Example
Milton tells his friends that he averages 5 miles every time he runs. He
gets data from a random sample of 20 runs of his runs. The sample
mean of Milton’s runs is 4.8 miles, but he still maintains that the
variation and subsequent sample mean is due to chance. At the 10%
significance level, does the data provide sufficient evidence to
conclude that Milton’s running distance is less than 5 miles?
a. If  denotes the population mean of all of Milton’s running
distances then the null and alternative hypotheses are respectively,
H O :   5 miles (Milton's claim)
H a :   5 miles (his friend' s belief)
Note that this hypothesis test is left tailed
b. If the null hypothesis (Milton’s claim) is true, then the sample x of
Milton’s 20 runs should be approximately 5 miles. If the sample x is
significantly less than 5 miles, then we would reject the null
hypothesis (Milton’s claim) and in favor of the alternative hypothesis.
In this case we would believe that Milton runs on average less than
5 miles.
c. We use our information on sampling distributions for our normally
distributed to help us determine how much smaller is “too much
smaller”. Assuming that the null hypothesis true and key fact 7.4 on
pg 295 for samples of size 20, the sample mean of the running
distance x is normally distributed with mean and standard deviation
 x    5 miles
and
x 

n

1.2
 0.2683 miles
20
This means that the standardized version of x will become
z
x


x 5
0.2683
n
x 5
z

and has the standard normal distribution. The variable
0.2683 is
our test statistic.
Since the test is left tailed and we compute the probability of
observing a value of the test statistic z that is as small or smaller than
the value actually observed. This probability is called the P-value of the
hypothesis test and is denoted by the letter P. A picture of this will be
drawn. Our criterion for deciding whether to reject the null hypothesis is
as follows: if the P-value is less than or equal to the significance level,
we reject the null hypothesis; otherwise, we do not reject the null
hypothesis
d. Next, we compute our test statistic and obtain the P-value, the value
of the test statistic would be as follows:
z
x 5
4.8  5

 0.745
0.2683 0.2683
The p-value is 0.2281 and is greater than the significance level 0.05 so
we do not reject the null hypothesis.
P-value: of a hypothesis test is the probability of getting sample data
at least as inconsistent with the null hypothesis ( and supportive of the
alternative hypothesis) as the sample data actually obtained. The letter
P represents the P-value
Decision Criterion for a Hypothesis Test using theP-value
If the P-value is less than or equal to the specified significance level,
reject the null hypothesis; otherwise, we do not reject the null
hypothesis In other words, if P   , reject H O ; otherwise, do not
reject H O .
P-values as the Observed Significance Level
The P-value of a hypothesis test equals the smallest significance level
at which the null hypothesis can be rejected, that is, the smallest
significance level for which the observed sample data results in the
rejection of H O
Determining a P-value
To determine a P-value of a hypothesis test, we assume that the null
hypothesis is true and compute the probability of observing a value of
the test statistic as extreme as or more extreme than that observed.
Determine a P-value for a One Mean z-test
We need to set up our null and our alternative hypotheses,
H O :   0 where 0 is some specified value
Then we calculate our test statistic
z
x  0

n
We let z 0 be the observed value of our test statistic
P-value
Z0
Right tailed Test
Z0
P-value
P-value
 Z0
Z0
Z0
Two tailed Test
Z0
Left tailed Test
The P-value is determined as follows:
 For a right tailed test: The p-value is the area under the standard
normal curve that lies to the right of Z 0 . The p-value equals the
probability of observing a value of the test statistic that is as large
as or larger than the value actually observed.
 For a two tailed test: The p-value is the area under the standard
normal curve that lies outside of the interval of - Z 0 and Z 0 . The
p-value equals the probability of observing a value of the test
statistic that is as large as or larger than the value actually
observed
 For a left tailed test: The p-value is the area under the standard
normal curve that lies to the left of Z 0 . The p-value equals the
probability of observing a value of the test statistic that is as large
as or larger than the value actually observed.