Transcript PPT Chapter 21 - McGraw Hill Higher Education

Introductory Mathematics
& Statistics for Business
Chapter 21
Hypothesis Testing
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Learning Objectives
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Understand the principles of statistical inference
Formulate null and alternative hypotheses
Understand one-tailed and two-tailed tests
Understand type I and type II errors
Understand test statistics
Understand the significance level of a test
Understand and calculate critical values
Understand the regions of acceptance and rejection
Calculate and interpret a one-sample z-test statistic
Calculate and interpret a one-sample t-test statistic
Calculate and interpret a paired t-test statistic
Calculate and interpret a two-sample t-test statistic
Understand and calculate p-values
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21.1 Statistical inference
• One of the major roles of statisticians in practice is to draw
conclusions from a set of data
• This process is known as statistical inference
• We can put a probability on whether a conclusion is
correct within reasonable doubt
• The major question to be answered is whether any
difference between samples, or between a sample and
a population, has occurred simply as a result of natural
variation or because of a real difference between the
two
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21.1 Statistical inference (cont…)
• In this decision-making process we usually undertake a
series of steps, which can include the following:
1. collecting the data
2. summarising the data
3. setting up an hypothesis (i.e. a claim or theory), which
is to be tested
4. calculating the probability of obtaining a sample such
as the one we have if the hypothesis is true
5. either accepting or rejecting the hypothesis
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21.1 Statistical inference (cont…)
• The null hypothesis
– A technique for dealing with these problems begins with the
formulation of an hypothesis
– The null hypothesis is a statement that nothing unusual has
occurred. The notation is Ho
– The alternative hypothesis states that something unusual
has occurred. The notation is H1 or HA
– Together they may be written in the form:
Ho: (statement) v H1(alternative statement)
(where ‘v’ stands for versus)
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21.1 Statistical inference (cont…)
• The null hypothesis (cont…)
– E.g. Question: Is the population mean, μ, equal to a specified
value?
Hypotheses:
H0: The population mean, μ, is equal to the specified value.
v.
H1: The population mean, μ, is not equal to the specified
value
– Another way of expressing the null and alternative
hypotheses is in the form of symbols, e.g.
H0 :   0
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21.1 Statistical inference (cont…)
• The alternative hypothesis
– The alternative hypothesis may be classified as two-tailed or
one-tailed
– Two-tailed test (two-sided alternative)
 we do the test with no preconceived notion that the
true value of μ is either above or below the
hypothesised value of μ0
 the alternative hypothesis is written:
H1: µ  µo
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21.1 Statistical inference (cont…)
• Significance level
– After the appropriate hypotheses have been formulated, we
must decide upon the significance level (or a –level) of the
test
– This level represents the borderline probability between
whether an event (or sample) has occurred by chance or
whether an unusual event has taken place
– The most common significance level used is 0.05,
commonly written as a = 0.05
– A 5% significance level says in effect that an event that
occurs less than 5% of the time is considered unusual
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21.2 Errors
• There are two possible errors in making a
conclusion about a null hypothesis
– Type I errors occur when you reject H0 (i.e. conclude that it
is false) when H0 is really true. The probability of making a
type I error is, in fact, equal to a, the significance level of the
test
– Type II errors occur when you accept H0 (i.e. conclude that it
is true) when H0 is really false. The probability of making a
type II error is denoted by the Greek letter b (beta)
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21.3 The one-sample z-test
• Deals with the case of a single sample being chosen
from a population and the question of whether that
particular sample might be consistent with the rest of
the population
• To answer this question, we construct a test statistic
according to a particular formula
• Is important that the correct test be used, since the use
of an incorrect test statistic can lead to an erroneous
conclusion
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21.3 The one-sample z-test (cont…)
• Information required in calculation:
– the size (n) of the sample
– the mean x  of the sample
– the standard deviation (s) of the sample
• Other information of interest might include:
1. Does the population have a normal distribution?
2. Is the population’s standard deviation   known?
3. Is the sample size (n) large?
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21.3 The one-sample z-test (cont…)
•
There are different cases for the one-sample z-test statistic
Case I is a situation where:
1. the population has a normal distribution and
2. the population standard deviation, , is known
Case II is a situation where:
1. the population has any distribution
2. the sample size, n, is large (i.e. at least 25), and
3. the value of  is known
In both these cases we can use a z-test statistic defined by:
z
x  0

n
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21.3 The one-sample z-test (cont…)
Case III is a situation where:
1. the population has any distribution
2. the sample size, n, is large (i.e. at least 25), and
3. the value of  is unknown (however, since n is large, the
value of  is approximated by the sample standard
deviation, s)
In this case we can use a z-test statistic defined by:
x  0
z
s
n
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21.3 The one-sample z-test (cont…)
• A critical value is one that represents the cut-off point for
z-test statistics in deciding whether we do not reject or do
reject H0. The particular critical value to use depends on two
things:
1. whether we are using a one-sided or two-sided test, and
2. the significance level used
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21.3 The one-sample z-test (cont…)
Rules for not rejecting or rejecting H0 in a onesample z-test
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21.4 The one-sample t-test
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Consider a test that has the same objective as a one-sample ztest, that is to determine whether a sample is significantly
different from the population as a whole
However, the one-sample t-test has a different set of
assumptions:
Case IV is a situation where:
1. the population is normally distributed
2. the population standard deviation,  , is unknown and
3. the sample size, n, is small
Then we can use a t-test statistic defined as:
x  0
t
s
n
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21.4 The one-sample t-test (cont…)
• Unlike the z-test statistic, the t-test statistic has associated
with it a quantity called degrees of freedom
• The degrees of freedom are denoted by the Greek letter υ
and are defined by υ = n – 1.
• Degrees of freedom relate to the fact that the sum of all
deviations in a sample of size n must add up to zero
• When using a t-test statistic, we must use special critical
values in a table
• This information is given in Table 7, and the critical values
are given to three decimal places
• After the value of the t-test statistic is found, the method of
either not rejecting or rejecting H0 is similar to Rules (1) to (6)
described for the z-test
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21.5 Drawing a conclusion
• Once a decision has been made to either not reject or
reject H0, a conclusion should be written in terms of
what the actual problem is
• To summarise, the steps that should be undertaken to
perform a one-sample test are:
1. Set up the null and alternative hypotheses. This includes
deciding whether you are using a one-sided or two-sided
test
2. Decide on the significance level
3. Write down the relevant data
4. Decide on the test statistic to be used
5. Calculate the value of the test statistic
6. Find the relevant critical value and decide whether H0 is to
be not rejected or rejected
7. Draw an appropriate conclusion
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21.6 The paired t-test
• Now we will consider problems in which there are two
samples that are to be compared with each other
• These are often referred to as two-sample problems
and there are a number of test statistics available
• In some instances, the two samples have a structure
such that the data are paired
• A comparison of the values in two samples requires the
calculation of a two-sample test statistic
• In such cases, the most commonly used test statistic is
a paired t-test, which takes into account this natural
pairing
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21.6 The paired t-test (cont…)
• The steps involved in the analysis are as follows:
1. Construct the null and alternative (two-tailed) hypotheses.
H0 always states that there is no difference between the two
samples. H1 always states that there is some difference
between the two samples.
In symbol form these can be written as:
H0: μd = 0
v.
H1: μd ≠ 0
2. Calculate the actual differences between the values in the
two samples. It is important to retain the minus sign if the
subtraction yields a negative number
3. Calculate the mean and standard deviation of the values of
the differences
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21.6 The paired t-test (cont…)
4. Calculate the value of the paired t-test statistic using:
x d  d
t
sd
n
where n = the number of pairs of differences
μd = 0 (according to the null hypothesis)
and with n = n – 1 degrees of freedom
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21.6 The paired t-test (cont…)
5. Look up the critical value in Table 7 for the desired
significance level
If | test statistic | > critical value, we reject H0 and the
conclusion is that H1 applies
If | test statistic | < critical value, we cannot reject H0
and the conclusion is that there is no evidence of a
significant difference between the two samples
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21.7 The two-sample t-test
• This time, suppose that the samples are not paired;
that is, they are independent
• The two samples need not contain the same number of
observations
• The most common test statistic used in this situation is
a two-sample t-test (also known as a pooled t-test)
• The steps for using a two-sample t-test are as follows:
1. Construct the null and alternative (two-tailed) hypotheses.
H0 always states that there is no difference between the two
samples, while H1 always states that there is some
difference between the two samples
H0: μ1 = μ2
v
H1: μ1 ≠ μ2
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21.7 The two-sample t-test (cont…)
2. Calculate the following statistics from the samples: the
number of observations in Sample 1 and 2, the mean of
Sample 1 and 2, the standard deviation of Sample 1 and 2
3. Calculate the value of the pooled standard deviation
sp 
n1  1s12  n2  1s22
n1  n2  2
4. Calculate the value of the two-sample t-test statistic
x1  x 2
t
sp
1
1

n1 n2
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21.7 The two-sample t-test (cont…)
5. Look up the critical value in Table 7 for the desired
significance level.
If | test statistic | > critical value, we reject H0 and the
conclusion is that H1 applies. Hence, there is a significant
difference between the two samples
If | test statistic | < critical value, we cannot reject H0 and
the conclusion is that there is no evidence of a significant
difference between the two samples
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21.8 p-values
• An alternative to using critical values for testing
hypotheses is to use a p-value approach
• The value of the test statistic is still calculated in the usual
way
• In this case we now calculate the probability of obtaining
a value as extreme as the value of the test statistic if H0
were true
• Compare the p-value with a. Then:
If p-value > a, we cannot reject H0 at that a-level
If p-value < a, we reject H0 at that a-level
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Summary
• In this chapter we looked at the principles of statistical
inference
• We formulated null and alternative hypotheses
• We understood
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one-tailed and two-tailed tests
type I and type II errors
test statistics
the significance level of a test
the regions of acceptance and rejection
• We understood and calculated critical values
• We calculated and interpreted
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a one-sample z-test statistic
a one-sample t-test statistic
a paired t-test statistic
a two-sample t-test statistic
• We understood and calculated p-values
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21-27