A Guide to Statistical Inference Question Answer 2022

Running Head: A GUIDE TO STATISTICAL INFERENCE
A GUIDE TO STATISTICAL INFERENCE
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A GUIDE TO STATISTICAL INFERENCE1
Answer 1
Hypothesis Testing
Introduction
A statistical hypothesis is a conjecture about the population distribution. As the term
suggests, one wishes to decide whether or not some hypothesis that has been formulated is
correct. There are two choices- accepting or rejecting hypothesis. A decision procedure for
such problem is called testing o hypothesis. There are many real-life applications of
hypothesis testing such as-
To test whether more men suffer than women from obesity
 Whether the defendant is guilty or innocent.
 Whether male and female births are equally likely and so on.
 A formal definition is provided below.
Definition of Hypothesis:
A statistical hypothesis H, also known as confirmatory data analysis, is an assertion or
conjecture about the characteristics of a population. If the hypothesis specifies the population
distribution completely, then it is called simple hypothesis. If the statistical hypothesis does
not specify the population distribution completely, then it is called composite hypothesis.
Suppose weights of 1000 randomly chosen people are recorded. Now an assumption
like “the mean weight is less than 150 lbs.” does not specify a particular value. Hence, it is a
composite hypothesis. On the other hand, if it is assumed that the mean weight is 150 lbs.
then it specifies the distribution completely. Thus, it is called simple hypothesis.
Test of Statistical Hypothesis
A test of statistical hypothesis H is a rule or a procedure to decide whether to reject H
or accept H based on a given sample from the population.
There are four stages of performing a statistical hypothesis-
 The first step is to state the null and alternative hypothesis.
 The next step is to formulate how the data will be evaluated.
 Analyze the sample data.
 The final stage is to analyze the results and to decide whether reject or retain the null
hypothesis.
The above steps show that some basic terminologies should be known before performing
a hypothesis testing. These are described below.
Null and alternative hypothesis:
The null hypothesis is a. assumption about the population distribution which is to be
tested. In general, the null hypothesis assumes that there is no difference between certain
characteristics of a population. The alternative hypothesis is the one which assumes
differences.

A GUIDE TO STATISTICAL INFERENCE2
The choice of null and alternative hypothesis is a very crucial part of a test. Suppose that
the bulbs manufactured under a standard process have average lifetime 1 hours and that for
bulbs manufactured by a new process is 2 hours. If the experimenter wants to compare the
effectiveness of the two processes, then three possible statements can be drawn-
 Mean life of bulbs in standard process is greater than that for a new process (1>2)
 Mean life of bulbs in standard process is smaller than the bulbs manufactured by new
process (1<2)
 Mean life of bulbs in both the processes are equal (1=2)
While choosing the null hypothesis, the researcher should be completely impartial and
should not allow his personal views to influence the decision. The first two statements appear
to be biased as they reflect a preferential attitude. Thus, the best course is to adopt the
hypothesis with no difference as stated in the third one. This suggests that the statistician
should take up the neutral or null attitude regarding the outcome of the test. This neutral
attitude of the experimenter before the sample values are taken is the key of the choice of the
null hypothesis. Keeping in mind the potential loses due to the wrong decision, the decision
maker is somewhere conservative in holding the null hypothesis as true unless there is strong
evidence against it and to him, the consequences of wrongly rejecting a null hypothesis seems
to be more serious than wrong acceptations. The null hypothesis is denoted as H0 and the
alternative hypothesis as H1.
Critical Region of a test
Let  denotes the collection of all possible samples of size n from a population,
= {(x1,x2,..., xn): (x1,x2,..., xn)=x is an observed value of X=(X1, X2,..., Xn)}
Hence,  is called the sample space or potential data set. A test procedure assigns to
each possible value of X of  one of the two decisions: accept H or reject H and thereby
divided the sample space into two complementary regions. The regions denoted as W and -
W=W are such that if X falls in W then the hypothesis is rejected otherwise it is accepted.
The set W is called the rejection region or critical region and W is called the region of
acceptance.
Performance of a test
While performing a test, one may arrive at the correct decision or may commit one of the
two errors:
 Rejecting a null hypothesis H0 when it is actually true.
 Accepting H0 when it is actually false.
True state Decision based on the sample
Reject H0 Accept H0
H0 is true Type-I error Correct
H0 is false or H1 is true Correct Type-II error
The rejection of null hypothesis H0 based on a test when it is really true is called
Type-I error of the test. On the other hand, The acceptance of H0 when it is actually false is
called the Type-II error of the test.
It is desirable to carry out the test in such a manner that keeps the probabilities of two
types of error o a minimum level. Unfortunately, or a given sample of size n, the both

A GUIDE TO STATISTICAL INFERENCE3
probabilities cannot be controlled simultaneously. Type-I error is found to be more serious
than type-II error since in first case, the null hypothesis gets rejected although it is true.
A suitable method of finding a test of H0 against H1 is to restrict the error probability
which is more serious to the investigator that is the probability of type-I error and then to
minimize the probability of type-II error.
Level of Significance
The level of significance is defined as the probability of rejecting a null hypothesis
when it is actually true. It is denoted by α. The choice of significance level depends on the
experimenter himself. If he thinks that rejection of a true null hypothesis is a serious matter,
he will choose rather a small value of α, say 0.01 or 0.001. On the other way, if he thinks that
this error is not so serious, he will not mind taking a value as high as 0.05,0.1.
Test Statistic
A test statistic is a numerical summary of the dataset that reduces the data to a value
which can be used to perform the hypothesis test. In a testing problem, the test or the critical
region is specified in terms of test statistic.
Left, Right and Both tailed test
In a testing problem, depending on the nature of the alternative hypothesis, if a test
uses the left tail/ right tail / both tails of the curve of the sampling distribution of the test
statistic as the critical region then the test is known as the left tailed/ right tailed/ both tailed
test.
p-value or probability value
The choice of a specific value of α is completely arbitrary and is determined by non-
statistical consideration such as the possible consequences of rejecting H0 falsely. There is
another value associated with a statistical test called p-value. The p-value associated with a
test is the probability that the observed value of the test statistic that is more extreme in the
direction given by the alternative hypothesis, when H0 is true. A smaller p-value indicates
strong evidence in favor of the alternative hypothesis.
Power and Size
The power of a test is defined as the probability of rejecting a null hypothesis when it
is false. The size of a test is nothing but the probability of committing type-I error that is
rejection of true null hypothesis.
Confidence Interval
The confidence interval gives the range within which the value of the true parameter
lies. A 95% confidence interval means that if 100 repeated samples are taken and the
population parameter is estimated each time, then 95 out of them will contain the true value
of the parameter.
These are some preliminary concepts which should be kept in mind before performing
a statistical test. There are mainly two types of test- Parametric and Non-parametric tests.
Here some of the parametric tests are described with examples.