1
Chapter 4 – Decision Making for a Single Sample
Definition: Inferential statistics is the branch of statistics concerned with inferring the characteristics
of populations (i.e., parameter values) based on the information contained in sample data sets.
Inferential statistics includes estimation of parameters and hypothesis testing.
There are two branches of statistical inference, 1) estimation of parameters and 2) testing hypotheses
about the values of parameters. We will consider estimation first.
Defn: A point estimate of a parameter  is a specific numerical value, ˆ , of a statistic, ̂ , based on
the data obtained from a sample. A particular statistic, such as ̂ , used to provide a point estimate of
a parameter is called an estimator.
Note: ̂ is a function of X1, X2, ..., Xn, the elements of a random sample, and is thus a random
variable, with its own sampling distribution.
Note: Nothing is said in the definition about the “goodness” of the estimate. Any statistic is an
estimate of a parameter. For a particular parameter , some statistics provide better estimates than
others. We need to examine the statistical properties of each estimator to decide which one gives the
best estimate for a parameter.
ˆ    . If
Defn: The point estimator ̂ is said to be an unbiased estimator of the parameter  if E 

ˆ    is called the bias of the estimator
the estimator is not unbiased, then the difference B  E 

̂ .
Example: Assume that we have a r.s. X1, X2, ..., Xn from a distribution with unknown mean . We
want to find an unbiased estimator of . From the linearity property of expectation, we have
1 n
1 n
 1 n
E  X   E   X i    E  X i       . Hence for any distribution, the sample mean is an
n i 1
 n i 1  n i 1
unbiased estimator of the distribution mean.
Example: Same situation, but now we want to find an unbiased estimator of the variance,  2 , of the
distribution. Also from the linearity property of expectation, we have
2
1
 1 n
 n

E  S 2   E 
Xi  X   
E    X i2  X 2  2 XX i 


 n  1 i 1
 n  1  i 1


1
1 n
 n

2
2 
E  X i2  nX 2  
 E  X i   nE  X   .
n  1  i 1
 n  1  i 1

2
2
2
But  2  E  X i2    2 , and  X  E  X    
E  S 2  
2
n
. Hence

 2
1 
 2
2
2 
2
             . Hence for any distribution, the sample variance is an
n 1 


 n

unbiased estimator of the distribution variance.
2
Sometimes there are several unbiased estimators of a given parameter. For example, each member of
the sample is also an unbiased estimator of the distribution mean. We want to decide which of them
provides the best estimate, based on the characteristics of the sampling distributions of the estimators.
Suppose that we have selected a r.s. X1, X2, ..., Xn from a distribution that is characterized by an
unknown parameter .
Suppose that we have two statistics
estimator should we use?
̂1 and ̂ 2 , both of which are unbiased estimators of . Which
Since we want our particular estimate to be close to the true value of , we want to use an estimator
whose sampling distribution has small variance.
Defn: In the class of all unbiased estimators of a parameter , that estimator whose sampling
distribution has the smallest variance is called the minimum variance unbiased estimator (MVUE) for
.
 

ˆ  E 
ˆ 
Defn: The mean square error of an estimator ̂ of a parameter  is MSE 

 
 
 
2
.
ˆ V 
ˆ  B 2 . If we are talking about the class of
Note: We can also write the MSE as MSE 
 
 
ˆ V 
ˆ .
unbiased estimators of , then B = 0, and MSE 
Defn: If we have two possible estimators,
ˆ
MSE 
1
given by
.
ˆ
MSE 
2
 
 
̂1 and ̂ 2 , of . The relative efficiency of ̂ 2 to ̂1 is
Example: We have a r.s. X1, X2, ..., Xn from a distribution with unknown mean  and standard
deviation . We could estimate the distribution mean using the sample mean, or using a single
member, say X1, of the sample. Which statistic has the better relative efficiency? We have
MSE  X   V  X   B 2 
2
, since the sample mean is unbiased. We also have
n
2
MSE  X1   V  X1   B   2 , since X1 is also unbiased. Hence the relative efficiency of X to
X1 is n.
Hence for n  2, the sample mean is a more efficient estimator of  than a single element of the
sample.
Defn: The standard error of an estimator ̂ of a parameter  is just the square root of the variance of
 
 
ˆ  V 
ˆ .
the sampling distribution of ̂ : S .E. 
3
Examples of parameters, together with best estimators and estimates, are given in the table below.
Unknown parameter

Estimator
Point Estimate
x
2
X
2
1 n
S 
Xi  X 


n  1 i 1
s2
p
p̂
1  2
X
Pˆ 
n
X1  X 2
x1  x2
p1  p2
Pˆ1  Pˆ2
pˆ1  pˆ 2
2
Examples: p. 135, Exercises 4-2, 4-10
Hypothesis Testing
Often, instead of estimating the value of a parameter based on sample data, we simply want to decide
whether we believe a specific assertion about the value of the parameter.
Definition: An hypothesis is a statement about the value of a population parameter.
Examples:
1) Nothing outlasts the Energizer.
2) More doctors recommend Tylenol for the relief of headache pain than any other pain reliever.
Hypotheses are tested in pairs, to decide which of the two statements is more believable.
The Null Hypothesis, H0
This hypothesis usually represents the state of no change or no difference, from the researcher’s point
of view. Often the null hypothesis is a statement of current belief about the value of the parameter; the
researcher doubts the null hypothesis and wants to disprove it. Symbolically, this hypothesis is never a
strict inequality.
The Alternative Hypothesis, Ha
This statement is what the researcher is attempting to prove. It usually is the negation of the null
hypothesis. Symbolically, this hypothesis is always a strict inequality. This hypothesis can take one of
three forms for a parameter  and a given number 0:
1) Ha:  >  0
2) Ha:  <  0
3) Ha:    0
4
Examples: Let pT be the proportion of all doctors who recommend Tylenol, and let pA be the
proportion of all doctors who recommend alternatives to Tylenol. We want to test the following two
hypotheses against each other:
H0: pT  pA
vs.
Ha: pT > pA
Whenever incomplete information, such as that from a sample, is used to make an inference about the
value of a population parameter, there is the risk of making a mistake. In a hypothesis testing
situation, there are two possible mistakes that could be made.
Type I Error
This type of error occurs when our test leads us to reject the null hypothesis when, in fact, the null
hypothesis is true. The probability of making a Type I error is denoted by the Greek letter , and is
called the significance level of the test. In scientific research, a Type I error is usually considered to be
more serious. This error is made when the researcher concludes that she has proved what she wanted
to prove, but this conclusion is mistaken.
Type II Error
This type of error occurs when our test leads us to fail to reject the null hypothesis when, in fact, the
null hypothesis is false. The probability of making a Type II error is denoted by the Greek letter .
This error is made when the researcher concludes that the data do not give sufficient evidence to
support the researchers original conjecture, but in fact the conjecture is true. Later research may then
provide sufficient evidence to validate the researcher’s conjecture.
Possible Results of a Hypothesis Test
Reject H0
Fail to Reject H0
____H0 True________Ha True___
|__Type I Error__|___________|
|____________|_Type II Error_|
Before conducting a hypothesis test, the researcher decides on an acceptable level of risk for
committing a Type I error (i.e., chooses a value for ). Most commonly,  = 0.05. If a Type I error is
deemed to have more serious consequences, then the researcher may choose a smaller value for , such
as 0.01 or 0.001. The researcher also chooses the amount of evidence to collect (sample size or sizes).
Test Statistic
The researcher summarizes the information contained in the simple random sample(s) in the form of a
test statistic, a random variable whose value is calculated from the sample data. The value of this
statistic will tell the researcher whether to reject H0 or to fail to reject H0. The test statistic must be
chosen so that its probability distribution under the null hypothesis is known.
Rejection Region( or Critical Region)
The rejection region is that range of possible values of the test statistic such that, if the actual value
falls in this region, the researcher will reject H0 and conclude that Ha is true. The boundary point(s) of
this region is(are) called the critical value(s) of the test.
5
The form of the rejection region depends on the form of the alternative hypothesis:
1) If the alternative hypothesis has the form Ha:  >  0, then the rejection region is a right-hand tail
of the distribution of the test statistic, with area .
2) If the alternative hypothesis has the form Ha:  <  0, then the rejection region is a left-hand tail of
the distribution of the test statistic, with area .
3) If the alternative hypothesis has the form Ha:    0, then the rejection region is the union of a
right-hand tail and a left-hand tail of the distribution of the test statistic, each having area /2.
Note: You may be wondering why we don’t simply always choose a very small value for the
significance level of the test, so that we will have a very slim chance of making a Type I error. The
reason is that, for a given level of evidence (sample size(s)), if we make the probability of a Type I
error smaller, we will automatically increase the probability of making a Type II error. Our goal is to
make both probabilities as small as possible. Usually, the consequences of making a Type I error are
more serious than the consequences of making a Type II error. Hence, we want to control . It is
important, however, to make both  and  as small as possible. We do this by choosing an appropriate
sample size(s). For a given chosen value of , if we make our sample size(s) larger, we will decrease
the probability of making a Type II error.
Steps in Statistical Hypothesis Testing
The following steps must appear in each statistical hypothesis test. The first four steps are the set-up
of the test. These steps are completed before the researcher chooses a sample(s) and collects data.
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 following statement
should appear in this step: “The test statistic is _________ , which under H0 has a _____________
probability distribution (with ____ degrees of freedom).”
Step 4: Find the critical value for the test. This value represents the cutoff point for the test statistic.
If the value of the test statistic computed from the sample data is beyond the critical value, the decision
will be made to reject the null hypothesis in favor of the alternative hypothesis.
Step 5: Calculate the value of the test statistic, using the sample data. (We may also compute a pvalue, and make our decision based on a comparison of the p-value with our chosen level of
significance.)
6
Step 6: Decide, based on a comparison of the calculated value of the test statistic and the critical value
of the test, whether to reject the null hypothesis in favor of the alternative.
If the decision is to reject H0, the statement of the conclusion should read as follows: “We reject H0 at
the (value of ) 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 (value of ) level of significance. There is not sufficient evidence to conclude that
(statement of the alternative hypothesis).”
Power and Sample Sizes
We want to minimize the probability of making an error in our conclusion. The only way to reduce
both  and  simultaneously is to increase the sample size(s). We want to choose a sample size that
will optimize the power of the test.
Defn: The power, 1 - , of a hypothesis test is the probability of rejecting the null hypothesis when in
fact the null hypothesis is false. I.e., the power of the test is the probability of correctly rejecting a
false null hypothesis.
Power is a measure of the sensitivity of a test. By sensitivity we mean the ability to detect effects, or
differences, when they are actually present. If a difference d 
Powerd   1  P(Type II error | d ) .
   0 exists, then
We decide what size difference d we wish to be able to detect, in units of the population standard
deviation, and we decide on a value of the Power(d); this number is between 0 and 1, preferably
greater than 0.5. We also decide on the type I error rate, . Knowing these three quantities, we can
find the necessary sample size from a chart of Operating Characteristic Curves for the test.