# Quantitative Methods

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```							Business Research Methods

An Overview of Hypothesis Testing
• Introduction
– The purpose of hypothesis testing is to determine
whether there is enough statistical evidence in favor
of a certain belief about a parameter.
– Examples
• Is there statistical evidence in a random sample of potential
customers, that support the hypothesis that more than
???% of the potential customers will purchase a new
product?
• Is a new drug effective in curing a certain disease? A
sample of patients is randomly selected. Half of them are
given the drug where half are given a placebo. The
improvement in the patients conditions is then measured
and compared.
• Concept of hypothesis testing
– The critical concepts of hypothesis testing.
• There are two hypotheses (about a population parameter(s))
– H0 - the null hypothesis        [ for example m = 5]
– H1 - the alternative hypothesis [m > 5] This is what you want to prove

• Assume the null hypothesis is
true.
– Build a statistic related to the
parameter hypothesized.
– Pose the question: How
probable is it to obtain a statistic
value at least as extreme as the
one observed from the sample?
m=5        x
– Continued
• Make one of the following two decisions (based on
the test):
– Reject the null hypothesis in favor of the alternative
hypothesis.
– Do not reject the null hypothesis in favor of the alternative
hypothesis.
• Two types of errors are possible when making
the decision whether to reject H0
– Type I error - reject H0 when it is true.
– Type II error - do not reject H0 when it is false.
– Describing the p-value
• If the p-value is less than 1%, there is
overwhelming evidence that support the
alternative hypothesis.
• If the p-value is between 1% and 5%, there is a
strong evidence that supports the alternative
hypothesis.
• If the p-value is between 5% and 10% there is a
weak evidence that supports the alternative
hypothesis.
• If the p-value exceeds 10%, there is no evidence
that supports of the alternative hypothesis.
• The p-value method (α-level)
– The p-value can be used when making
decisions based on rejection region
methods as follows:
• Define the hypotheses to test, and the required
significance level a.
• Perform the sampling procedure, calculate the
a = 0.05
test statistic and the p-value associated with it.
The p-value
• Compare the p-value to a. Reject the null
m x  170 hypothesis only if p <a; otherwise, do not reject
xL  175 .34 thenull hypothesis.
x 178
Conclusions of a test of Hypothesis
• If we reject the null hypothesis, we conclude that
there is enough evidence to infer that the
alternative hypothesis is true.
• If we do not reject the null hypothesis, we
conclude that there is not enough statistical
evidence to infer that the alternative hypothesis
is true.
The alternative hypothesis
is the more important
one. It represents what
we are investigating.
• As for Power…
Let’s Review:
– Judging the test

• A hypothesis test is effectively defined by the
significance level a and by the the sample size
n.
• If the probability of a type II error b is judged to
be too large, we can reduce it by
– increasing a, and/or
– increasing the sample size.
By increasing the sample size
the standard deviation of the
sampling distribution of the
a
mean decreases. Thus, x L
decreases.

xxLLLLLLL
xxxxx                          xL  m
za                , thus
    n
b1 > b2

As a result b decreases                                xL  m  z a
n
x xx
x LxLLLL
– In example 1, suppose n increases from 400 to
1000.                       65
xL  m  z a       170  1.645              173.38
n                    1000
173.38  180
b  P( Z                   )  P( Z  3.22)  0
65   1000
– In summary,
• By increasing the sample size, we reduce the
probability of type II error.
• Hence, we shall accept the null hypothesis
when it is false less frequently.
– Power of a test
• The power of a test is defined as 1 - b.
• It represents the probability to reject the null
hypothesis when it is false.

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