business statistics - qbm117 introduction to hypothesis testing
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Business Statistics - QBM117
Introduction to hypothesis testing
Objectives
To introduce the second type of statistical inference - hypothesis testing
To introduce the concept of hypothesis testing.
To gain a basic understanding of the methodology of hypothesis testing
Introduction to hypothesis testing
Hypothesis testing is another type of statistical inference where, once again, decisions are based on sample data.
The purpose of hypothesis testing is to determine whether the sample results provide sufficient statistical evidence to support (or fail to support) a particular belief about a population parameter.
Over the next few lectures we will develop a step-by-step methodology that will enable us to test these beliefs.
The objective of hypothesis testing is captured by this question:
Is the sample evidence consistent with a particular hypothesized population parameter, or does the sample evidence contradict the hypothesized value?
By rejecting the plausibility of the initially hypothesized value, we indirectly establish the plausibility of an alternative hypothesized value or range of values.
The null and alternative hypotheses
is the challenged hypothesis. It is the assertion we hold as true, until we have sufficient statistical evidence to conclude otherwise.
it always expresses a value of the population parameter which we intend to subject to scrutiny, based on sample data.
The null hypothesis, denoted H0 ,
The purpose of scrutinizing the null hypothesis is to determine whether there is support for the alternative hypothesis, denoted HA ,
Examples of null and alternative hypotheses
The operations manager is concerned with determining whether the filling process for filling 100g boxes of smarties is working properly.
If the manager wants to know whether the average fill of the boxes is less than 100g, he would specify the null and alternative hypotheses to be
100:
100:0
AH
H
If the manager wants to know whether the average fill of the boxes is more than 100g, he would specify the null and alternative hypotheses to be
100:
100:0
AH
H
If the manager wants to know whether the average fill of the boxes differs from 100g, he would specify the null and alternative hypotheses to be
100:
100:0
AH
H
The manager hopes to find the filling process is working properly, however, he may find the sampled boxes weigh too little
100: AHor too much
100: AH
As a result he may decide to halt the production process until the reason for the failure to fill to the required weight of 100g is determined.
By analysing the difference between the weights obtained from the sample and the 100g expected weight, he can reach a decision based on this sample information, and one of the two conclusions can be drawn.
The test statistic is a sample statistic calculated from the data. Its value is used in determining whether to reject or not reject the null hypothesis.
When testing hypotheses about the population mean , when the population variance is known, the test statistic will be x
The test statistic
or its standardised value
n
xz
/
as long as the population is normal or 30n
When testing hypotheses about the population mean , when the population variance is unknown, the test statistic will be x or its standardised value
ns
xt
/
as long as the population is normal.
When testing hypotheses about the population proportion p, the test statistic will be p̂ or its standardised value
npq
ppz
ˆ
as long as 5 and both nqnp
The rejection region
The sampling distribution of the test statistic is divided into regions, a region of rejection (critical region) and a region of non-rejection.
The rejection region consists of all values of the test statistic for which H0 is rejected.
The non-rejection region consists of all values of the test statistic for which H0 is not rejected.
The value that separates the rejection region from the non-rejection region is the critical value.
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Critical value Critical value
Region of non-rejection
Region of rejection Region of rejection
Two-tailed hypothesis test HA: 100
Critical value
Region of rejection
Upper tailed hypothesis test HA: > 100
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Region of non-rejection
Critical value
Region of rejection
Lower tailed hypothesis test HA: < 100
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Region of non-rejection
The decision rule
This is a rule that specifies the conditions under which the null hypothesis will be rejected.
It is a mathematical representation of the region of rejection as seen in the previous three slides.
For example the rejection region in slide 2 might be described by
Reject H0 if x is greater than 110.
Reject H0 if x is less than 90.
or the rejection region in slide 3 might be described by
The critical value
Therefore, in order to illustrate these rejection regions and describe them mathematically, we need to know the critical value(s), ie that value or values which separate the rejection region(s) from the non-rejection region.
How do we determine the critical value?
How do we determine this critical value?
The determination of the critical value depends on the size of the rejection region.
The size of the rejection region depends on the probability of making an error, when testing our hypotheses.
So, what are these errors?
Errors in hypothesis testing
rejecting the null hypothesis when it is true, a type I error;
not rejecting the null hypothesis when it is false, a type II error.
A hypothesis test concludes with a decision to either reject or not reject the null hypothesis. This decision, together with whether the hypothesis is true or not, results in two possible errors:
Because the decision we make and the conclusion we reach is based on sample data, there is always a possibility of making an error.
Type I and type II errors
The probability of making a type I error is defined as . This probability is also referred to as the level of significance.
The probability of making a type II error is defined as .
Ideally we would like to keep both errors and as small as possible. Unfortunately however, as decreases increases and vice versa therefore generally the size of is decided by the cost of making a type I error.
The size of is usually kept as small as possible, generally a value between 1% and 10%.
Once the value of is specified by the decision maker, the size of the rejection region is known because is the probability of rejecting the null hypothesis when it is true.
The size of this rejection region is .
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Critical value Critical value
Region of non-rejection
Region of rejection
/2 = 0.025
Region of rejection
/2 = 0.025
Two-tailed hypothesis test HA: 100 when = 0.05
0.95
Critical valueRegion of rejection
= 0.05
Upper tailed hypothesis test HA: > 100 when = 0.05
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Region of non-rejection
0.95
Critical value
Region of rejection
= 0.05
Lower tailed hypothesis test HA: < 100 when = 0.05
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Region of non-rejection
0.95
Reading for next lecture
Chapter 10, sections 10.3 and 10.5 (Chapter 9, sections 9.3 and 9.5 abridged)
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