STEPS IN STATISTICAL HYPOTHESIS TESTING
Step 1: State the null hypothesis, H0, and the alternative hypothesis, Ha. The alternative hypothesis
represents what the researcher is trying to prove. The null hypothesis represents the negation of what the
researcher is trying to prove. (In a criminal trial in the American justice system, the null hypothesis is that
the defendant is innocent; the alternative is that the defendant is guilty; either the jury rejects the null
hypothesis if they find that the prosecution has presented convincing evidence, or the jury fails to reject
the null hypothesis if they find that the prosecution has not presented convincing evidence).
Step 2: State the size(s) of the sample(s). This represents the amount of evidence that is being used to
make a decision. State the significance level, , for the test. The significance level is the probability of
making a Type I error. A Type I error is a decision in favor of the alternative hypothesis when, in fact,
the null hypothesis is true. A Type II error is a decision to fail to reject the null hypothesis when, in fact,
the null hypothesis is false.
Step 3: State the test statistic that will be used to conduct the hypothesis test (the appropriate test
statistics for the different kinds of hypothesis tests are given in the tables on the reverse side of this
handout). The following statement must appear in this step: “The test statistic is _________, which
under H0 has a _____________ probability distribution (with _____ degrees of freedom).”
Step 4: Find the rejection region of the test, using the form of the alternative hypothesis from Step 1, the
value of  from Step 2, and the distribution of the test statistic from Step 3.
Step 5: Choose the random sample(s) from the population(s). Calculate the value of the test statistic, and
the p-value for the test, using the gathered data.
Step 6:
Compare the calculated p-value to the chosen level of significance. If the p-value is less than , then the
null hypothesis will be rejected, and the alternative hypothesis will be affirmed. If the p-value is greater
than , the null hypothesis will not be rejected.
If the decision is to reject H0, the statement of the conclusion should read as follows: “We reject H0 at the
_____ level of significance. There is sufficient evidence to conclude that (statement of the alternative
hypothesis).”
If the decision is to fail to reject H0, the statement of the conclusion should read as follows: “We fail to
reject H0 at the_____ level of significance. There is not sufficient evidence to conclude that (statement of
the alternative hypothesis).”1
1
Note: It is not correct to say that we accept the null hypothesis, or that we conclude that the null
hypothesis is true. We never attempt to prove a null hypothesis. We either disprove it, or we fail to
disprove it.
STATISTICAL INFERENCE FOR VALUES OF POPULATION PARAMETERS
Parameter
Population mean, 
Point Estimator
Sample mean,
Population proportion, p
Sample proportion,
X
The value 0 is the value specified in the null
hypothesis.
Inferential Statistic for
Hypothesis Testing
X  0
 s 


 n
X  t
2
The value p0 is the value specified in the null
hypothesis.
pˆ  p 0
p 0 1  p 0 
n
, when the population is normal, or when the
sample size is large. Under the null hypothesis, this
statistic has a t distribution with n – 1 degrees of
freedom.
Approximate
(1- )100%
Confidence Interval
for the Parameter
Value
p̂
s
, n 1
when n is large enough, and when np  5 and
n(p – 1)  5. Under the null hypothesis, this statistic has
an approximate standard normal distribution.
, when the population is normal or the
pˆ  z 
n
2
sample size is large..
Parameter
Difference between two independent population means,
1 - 2
Point Estimator
Difference between the sample means,
X1  X 2
pˆ 1  pˆ 
n
Difference between two independent population
proportions, p1 – p2
Difference between the sample proportions,
pˆ 1  pˆ 2
The value d0 in this formula is the difference specified in
the null hypothesis (here, d0  0).
X
1
Sp
Inferential Statistic
for Hypothesis
Testing
 X 2  d0
 pˆ 1  pˆ 2   d 0
pˆ 1 1  pˆ 1  pˆ 2 1  pˆ 2 

, when 1 and 2 are equal to
1
1

n1 n2
each other, and the populations are normal or the
sample sizes are large. Under the null hypothesis, this
statistic has a t distribution with n1 + n2 – 2 degrees of
freedom.
Here
Sp 
n1  1s12  n2  1s 22
n1  n2  2
n1
n2
when n1 and n2 are large enough, and when n1p1  5,
n2p2  5, n1(1-p1)  5, and n2(1-p2)  5. Under the null
hypothesis, this statistic has an approximate standard
normal distribution.
If the number appearing in the null hypothesis is 0, the
following test statistic is used:
 pˆ 1  pˆ 2   0
.
where
1
1 
pˆ 1  pˆ   
 n1 n 2 
n p  n2 p 2
. Under the null
pˆ  1 1
n1  n2
hypothesis, this statistic has an approximate standard
normal distribution.
Approximate
(1 - )100%
Confidence Interval
for the Parameter
Value
X
1
 X 2   t
2
,nn2
Sp
1 1

n1 n2
 pˆ 1  pˆ 2   z 
2
pˆ 1 1  pˆ 1  pˆ 2 1  pˆ 2 

n1
n2