Chapter 10
Statistical Inference:
Hypothesis Testing for
Single Populations
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Learning Objectives
Upon completion of this chapter, you will be able to:
 Understand hypothesis-testing procedure using one-tailed and
two- tailed tests
 Understand the concepts of Type I and Type II errors in
hypothesis testing
 Understand the concept of hypothesis testing for a single
population using the z statistic
 Understand the concepts of p-value approach and critical value
approach for hypothesis testing
 Understand the concept of hypothesis testing for a single
population using the t statistic
 Understand the procedure of hypothesis testing for population
proportion
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Introduction to Hypothesis Testing

A statistical hypothesis is an assumption about an
unknown population parameter.

Hypothesis testing is a well defined procedure which helps
us to decide objectively whether to accept or reject the
hypothesis based on the information available from the
sample.

In statistical analysis, we use the concept of probability to
specify a probability level at which a researcher concludes
that the observed difference between the sample statistic
and the population parameter is not due to chance.
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Hypothesis Testing Procedure
Figure 10.1: Seven steps of hypothesis testing
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Step 1: Set Null and Alternative Hypotheses

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The null hypothesis generally referred by H0 (H sub-zero), is the
hypothesis which is tested for possible rejection under the
assumption that is true. Theoretically, a null hypothesis is set as no
difference or status quo and considered true, until and unless it is
proved wrong by the collected sample data.
Symbolically, a null hypothesis is represented as:
The alternative hypothesis, generally referred by H1 (H sub-one),
is a logical opposite of the null hypothesis. In other words, when
null hypothesis is found to be true, the alternative hypothesis must
be false or when the null hypothesis is found to be false, the
alternative hypothesis must be true.
Symbolically, alternative hypothesis is represented as:
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Step 2: Determine the Appropriate Statistical
Test

Type, number, and the level of data may provide a
platform for deciding the statistical test.

Apart from these, the statistics used in the study (mean,
proportion, variance, etc.) must also be considered when a
researcher decides on appropriate statistical test, which
can be applied for hypothesis testing in order to obtain the
best results.
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Step 3: Set the Level of Significance

The level of significance generally denoted by α is the
probability, which is attached to a null hypothesis, which
may be rejected even when it is true.

The level of significance is also known as the size of the
rejection region or the size of the critical region.

The levels of significance which are generally applied by
researchers are: 0.01; 0.05; 0.10.
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Step 4: Set the Decision Rule
Figure 10.2: Acceptance and rejection regions of null hypothesis
(two-tailed test)
Critical region is the area under the normal curve, divided into two
mutually exclusive regions. These regions are termed as acceptance
region (when the null hypothesis is accepted) and the rejection region
or critical region (when the null hypothesis is rejected).
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Step 5: Collect the Sample Data

In this stage of sampling, data are collected and the
appropriate sample statistics are computed.

The first four steps should be completed before collecting
the data for the study.

It is not advisable to collect the data first and then decide
on the stages of hypothesis testing.
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Step 6: Analyse the data

In this step, the researcher has to compute the test
statistic. This involves selection of an appropriate
probability distribution for a particular test.

Some of the commonly used testing procedures are z, t, F,
and χ2.
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Step 7: Arrive at a Statistical Conclusion and
Business Implication

In this step, the researchers draw a statistical conclusion.
A statistical conclusion is a decision to accept or reject a
null hypothesis.

Statisticians present the information obtained using
hypothesis-testing procedure to the decision makers.
Decisions are made on the basis of this information.
Ultimately, a decision maker decides that a statistically
significant result is a substantive result and needs to be
implemented for meeting the organization’s goals.
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Two-Tailed Test of Hypothesis


Let us consider the null and alternative hypotheses as
below:
Two-tailed tests contain the rejection region on both the
tails of the sampling distribution of a test statistic. This
means a researcher will reject the null hypothesis if the
computed sample statistic is significantly higher than or
lower than the hypothesized population parameter
(considering both the tails, right as well as left).
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Figure 10.3: Acceptance and rejection regions (alpha = 0.05)
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One-Tailed Test of Hypothesis
Let us consider a null and alternative hypotheses as below:
One-tailed test contains the rejection region on one tail of the
sampling distribution of a test statistic. In case of a left-tailed
test, a researcher rejects the null hypothesis if the computed
sample statistic is significantly lower than the hypothesized
population parameter (considering the left side of the curve in
Figure 10.5).
In case of a right-tailed test, a researcher rejects the null
hypothesis if the computed sample statistic is significantly higher
than the hypothesized population parameter (considering the
right side of the curve in Figure 10.6).
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Figure 10.5: Acceptance and rejection regions for one-tailed
(left) test (alpha = 0.05)
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Figure 10.6: Acceptance and rejection regions for one-tailed
(right) test (alpha = 0.05)
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Type I and Type II Errors
When a researcher tests statistical hypotheses, there
can be four possible outcomes as follows:
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1.
2.
3.
4.
Rejecting a true null hypothesis (Type I error)
Accepting a false null hypothesis (Type II error)
Accepting a true null hypothesis (Correct decision)
Rejecting a false null hypothesis (Correct decision)
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Table 10.3 : Errors in hypothesis testing and power of the test
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Hypothesis Testing for a Single Population
Mean Using the Z Statistic
Example 10.1: A marketing research firm conducted a
survey 10 years ago and found that the average household
income of a particular geographic region is Rs 10,000. Mr
Gupta, who has recently joined the firm as a vice president
has expressed doubts about the accuracy of the data. For
verifying the data, the firm has decided to take a random
sample of 200 households that yield a sample mean (for
household income) of Rs 11,000. Assume that the
population standard deviation of the household income is
Rs 1200.
Verify Mr Gupta’s doubts using the seven steps of
hypothesis testing. Let α = 0.05.
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Example 10.1 (Solution)
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p-Value Approach for Hypothesis Testing

The p-value approach of hypothesis testing for large
samples is some times referred to as the observed level of
significance. The p-value defines the smallest value of α
for which the null hypothesis can be rejected.

Example 10.2: For Example 10.1, use the p-value
method to test the hypothesis using alpha = 0.01 as the
level of significance. Assume that the sample mean is
10,200.
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Example 10.2 (Solution)
The observed test statistic is computed
as 2.36. From the normal table, the
corresponding probability area for z
value 2.36 is 0.4909. So, the
probability of obtaining a z value
greater than or equal to 2.36 is 0.5000
– 0.4909 = 0.0091 (shown in Figure
10.9). For a two-tailed test, this value
is multiplied by 2 (as discussed above).
Thus, for a two-tailed test, this value is
(0.0091 × 2 = 0.0182). So, the null
hypothesis is accepted because (0.01 <
0.0182). It has to be noted that for α =
0.05 and α = 0.1, the null hypothesis is
rejected because 0.0182 < 0.05 and
0.0182 < 0.1.
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Critical Value Approach for Hypothesis Testing
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Example 10.3
A cable TV network company wants to provide modern facilities
to its consumers. The company has five-year old data which
reveals that the average household income is Rs 120,000.
Company officials believe that due to the fast development in the
region, the average household income might have increased. The
company takes a random sample of 40 households to verify this
assumption. From the sample the average income of the
households is calculated as 125,000. From historical data,
population standard deviation is obtained as 1200. Use alpha =
0.05 to verify the finding.
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Example 10.3 (Solution)
The null and alternative
hypotheses can be set as
below:
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Figure 10.10: Critical value method for testing a
hypothesis about the population mean for Example 10.3
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Solved Examples\Excel\Ex 10.1.xls
Solved Examples\Excel\Ex 10.2.xls
Solved Examples\Excel\Ex 10.3.xls
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Solved Examples\Minitab\Ex 10.1.MPJ
Solved Examples\Minitab\Ex 10.2.MPJ
Solved Examples\Minitab\Ex 10.3.MPJ
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Hypothesis Testing for a Single Population Mean
Using the T Statistic (Case of a Small Random
Sample When N < 30)
When a researcher draw a small random
sample (n < 30) to estimate the population
mean μ and when the population standard
deviation is unknown and population is
normally distributed, t-test can be applied.
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Example 10.4: Royal Tyres has launched a new brand of
tyres for tractors and claims that under normal
circumstances the average life of the tyres is 40,000 km. A
retailer wants to test this claim and has taken a random
sample of 8 tyres.He tests the life of the tyres under
normal circumstance. The results obtained are presented
in Table 10.4.
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Example 10.4 (Solution)
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Figure 10.18: Computed and critical t values for Example
10.4
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Solved
Solved
Solved
Solved
Examples\Excel\Ex 10.4.xls
Examples\Minitab\Ex 10.4.MPJ
Examples\SPSS\Ex10.4.sav
Examples\SPSS\Output Ex 10.4.spo
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Hypothesis Testing for a Population
Proportion
Example 10.5: The production manager of a company that
manufacturers electric heaters believes that at least 10% of the
heaters are defective. For testing his belief, he takes a random
sample of 100 heaters and finds that 12 heaters are defective.
He takes the level of significance as 5% for testing the
hypothesis. Applying the seven steps of hypothesis testing, test
his belief.
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Example 10.5 (Solution)
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Using Minitab for Hypothesis Testing for a
Population Proportion

Solved Examples\Excel\Ex 10.5.xls

Solved Examples\Minitab\Ex 10.5.MPJ
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