Business Analytics for Decision-Making
Module 4: Hypothesis Testing
M4 – Summary – Index
Null and Alternative Hypotheses
The Hypothesis Testing Process
Rejection Regions and Types of Tests
Decisions and Significance Levels
Hypothesis Testing Using the Rejection
Region for a Mean
The Probability Value (P-Value) of the Test
Statistic
Steps in Hypothesis Testing
Hypothesis Testing for the Mean
(Small Samples, n < 30)
Hypothesis Testing for a Population
Proportion
Two-Sample Hypothesis Testing
Two-Sample Hypothesis Testing:
Population Mean
Null and Alternative Hypotheses
Two Ways of Inferencing
Statistical Hypothesis
•
Without a preconceived idea
• Calculating a point estimate of a population
parameter and identifying a confidence interval
around this point estimate
Types of Hypothesis
The Null Hypothesis
With a preconceived idea
• Making a hypothesis by gathering data and
checking whether the data provides enough
evidence to support the hypothesis
Statements or a claims about the parameters
describing a population
•
•
Denoted by H0, is the
current thinking or the
status quo
Contains a statement of
equality such as ≤, =, ≥
The Alternative Hypothesis
•
Denoted by Ha , is the
complement of the null
hypothesis
Ha must be true if H0 is
false
•
Contains a statement of
inequality such as >, ≠, <
The Hypothesis Testing Process
Hypothesis Testing
• Uses sample statistics to test a claim about the
value of a population parameter
The Critical Value of a Test Statistic
• If the sample mean is close to the assumed population
mean, the null hypothesis is not rejected
• If the sample mean is far from the assumed population
mean, the null hypothesis is rejected
• The critical value of a test statistic determines whether to
reject or accept the null hypothesis
The Test Statistic
• The sample statistic which is compared with the parameter
in the null hypothesis is called the test statistic
Rejection Regions and Types of Tests
The Rejection Region
Refers to the range of values for which the null
hypothesis is not possible
The Rejection Region
Left-tailed test:
• H0: µ ≥ k
• H a: µ < k
Right-tailed test:
• H0: µ ≤ k
• H a: µ > k
Two-tailed test:
• H0: µ = k
• H a: µ ≠ k
Decisions and Significance Levels
Types of errors
The Significance Level of the Test
• The Significance level α: Maximum allowable probability
•
of making the Type I error
Typically used significance level: 𝛼 = 0.05
The Significance Level and Z-Score
Hypothesis Testing Using the Rejection Region for a
Mean
The Hypothesis Testing Process:
• Mentioning the claim and identifying the null and
alternative hypotheses
• Determining the critical value
• Determining the rejection regions
• Finding the standardised test statistic
• Decision-making and interpreting
The Probability Value (P-Value) of the Test Statistic
The P-Value
• The probability of obtaining a sample statistic
•
with a value as extreme or more extreme from
the determined sample data
The p-value of a hypothesis test depends on
the type of the test
The Decision Rule on the P-Value
If p ≤ α, then reject H0
If p > α, then fail to reject H0
P-Values for Test Types
Steps in Hypothesis Testing
• State the claim mathematically, find out the null and alternative hypotheses and
determine the hypotheses test type
• Specify the significance level
• Determine the standardised sampling distribution and draw the graph
• Calculate the test statistic and its standardised value
• Find the p-value
• Use the decision rule
• Write a statement to interpret the decision
Hypothesis Testing for the Mean (Small Samples, n < 30)
The T-Score Test Statistic
Critical Values in a T-Distribution
When the population standard deviation is given:
• The standard test statistic is the z-score
If the hypothesis test is left-tailed:
• T.INV with a negative sign (T.INV (alpha, d.f.) )
When the population standard deviation is not known:
• The standard test statistic is the t-score
If the hypothesis test is right-tailed:
• The standardised test statistic:
• T.INV with a positive sign (T.INV (alpha, d.f.))
t =x −μ
s n
• The test statistic has degrees of freedom: d.f. = n – 1
If the hypothesis test is two-tailed:
• T.INV with negative and positive signs (T.INV.2T (alpha,
d.f.))
Hypothesis Testing for a Population Proportion
The Test Statistic for the Population Proportion
The test statistic is the sample proportion 𝑝−ℎ𝑎𝑡;
the standardised test statistic is the z-test
pˆ − μ pˆ
p−p
z=
= ˆ
σ pˆ
pq n
Hypothesis Testing for a Proportion
•
Identify the null and alternative hypotheses
•
Verify that np  5 and nq  5
•
Specify the significance level and the sampling
distribution
•
Calculate the critical values
•
Interpret the decision in the context of the original
claim
•
Decide whether to reject or fail to reject the null
hypothesis
•
Find the standardised test statistic to check if z is in
the rejection region
•
Determine the rejection regions
Two-Sample Hypothesis Testing
Two-Sample Hypothesis Testing
The Two-Sample Z-Test
Null hypothesis H0 : States that there is no difference between
the parameters of two populations. Symbol: , =, 
Three conditions are necessary to perform a z-test:
Alternative hypothesis Ha: A statistical hypothesis that is true
when H0 is false. Symbol: >, , <
Null and alternative hypotheses can be written by translating the
claim of the population parameters from a verbal statement to a
mathematical statement
H0: μ1 = μ2
Ha : μ 1  μ 2
H0: μ1  μ2
Ha : μ 1 > μ 2
H 0: μ 1  μ 2
Ha : μ 1 < μ 2
Regardless of which hypothesis is used, μ1 = μ2 is always assumed
to be true
• Samples must be randomly selected
• Samples must be independent
• Each sample size must be at least 30, or each
population must have a normal distribution with a
known standard deviation
μ𝑥lj 1 −𝑥lj 2 = μ𝑥lj 1 − μ𝑥lj 2 = μ1 − μ2
σ𝑥lj 1 −𝑥lj 2 =
σ2𝑥lj 1 + σ2𝑥lj 2 =
𝑥lj 1 − 𝑥lj 2 − μ1 − μ2
z=
σ𝑥lj 1 −𝑥lj 2
σ12 σ22
+ .
n1 n2
where σ𝑥lj 1 −𝑥lj 2 =
σ12 σ22
+ .
n1 n2
Two-Sample Hypothesis Testing: Population Mean
A Two-Sample Z-Test for the Mean
• Identify the null and alternative hypotheses
• Specify the level of significance
• Sketch the sampling distribution
• Determine the critical value(s)
• Determine the rejection regions(s)
• Find the standardised test statistic
• Make a decision to reject or fail to reject the null hypothesis
• Interpret the decision in the context of the original claim
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