Hypothesis Testing (Mean)
with a Single Population
By the end of the lecture you should be able to:
Know the steps in testing hypotheses and define H0 and H1.
Define Type I error, Type II error, and power.
Formulate a null and alternative hypothesis for μ or π
Explain decision rules, critical values, and rejection regions.
Perform a hypothesis test for a mean with known σ using z.
Perform a hypothesis test for a mean with unknown σ using t.
Perform a hypothesis test for a proportion.
Hypothesis testing
In statistics, a hypothesis is a claim or a statement about the property of an
underlying population.
In undertaking hypothesis testing, you want to determine: Is the hypothesis
true or false
In other words: are the data consistent with a particular hypothesis
e.g. given our data, can we reject the claim that women are paid less, on
average, than men
Examples of daily uses of data in business
To support marketing claims
Help managers make decisions
Measure business improvement
The use of data allows businesses to find the best answers to their questions, e.g.:
Should a manufacturing company change suppliers of an input
Did product defects decrease after a new manufacturing process was introduced
Has the average service time at Starbucks decreased since last year
Has the NHS ambulance service decreased its average response time to accidents
Hypothesis testing uses
Hypothesis testing is used in science and business to test assumptions
and theories and guide managers when facing decisions.
First we will explain the logic behind hypothesis testing and then show
how statistical hypothesis testing helps businesses make decisions.
Logic of hypothesis testing
The business analysts asks questions, makes assumptions, and proposes
testable theories about the values of key parameters of the business
operating environment.
Each assumption is tested against observed data.
If an assumption has been disproved, in spite of rigorous efforts to do so,
the business may operate under the belief that the statement is true.
The analyst states the assumption, called a hypothesis, in a format that
can be tested using well-known statistical procedures. The hypothesis is
compared with sample data to determine if the da
ta are consistent or
inconsistent with the hypothesis
When the data are found to be inconsistent (i.e. in conflict) with the
hypothesis, the hypothesis is either discarded or reformulated.
Estimation
Estimation is the process of using sample data to draw inferences about
the population
In order to estimate a parameter (such as the population mean) a rule
which describes how to derive that estimate is required ; such a rule is
known as an estimator.
Sample
information
Population
parameters
2, sx 2,σμ
inferences
Principles of hypothesis testing
The approach involves making a tentative assumption about the population
parameter, which is usually referred to as the null hypothesis and denoted by Ho.
Ho is what we wish to test.
Another hypothesis referred to as the alternative hypothesis (and denoted by
Ha) is then defined.
The hypothesis-testing procedure involves using data from a sample to test two
competing statements – one denoted by Ho and the other by Ha.
Principles of hypothesis testing
A random sample of data is chosen.
If the sample data are consistent with the null hypothesis, then we do not reject the
null hypothesis.
Never ‘accept’ a null hypothesis – just because a hypothesis is consistent with our
data does not mean it is true!
If the sample data are inconsistent with the null hypothesis, then we reject the null
hypothesis in favour of the alternative.
Choosing the Null Hypothesis
The first step in setting up a hypothesis test is to decide on the null and the alternative
hypothesis.
The parameter we will focus on will be the population mean (μ).
The null hypothesis will always specify a single value for the parameter in question.
The null is expressed as:
Ho: μ = μ0
μ0 is some numeric value assumed for the population mean μ.
Choosing the Alternative Hypothesis
Three choices are possible for the alternative hypothesis:
(a) Ha: μ ≠μ0
A hypothesis test of this kind is known as a two-tailed test.
(b) Ha: μ < μ0
A hypothesis test of this kind is known as a left-tailed test but is also known as a one-tailed test.
(c) Ha: μ > μ0
A hypothesis test of this kind is known as a right-tailed test but is also known as a one-tailed test.
Type I & Type II Errors in Hypothesis Testing
State of Nature