1. OMGT1117 - LECTURE 10: Elements of Hypothesis Testing
Reading
Property Data Analysis – A Primer, Ch.10

2. Objectives
To demonstrate how to conduct tests of hypotheses for the population mean and population proportion .
To highlight the differences between Type 1 and Type 2 errors along with their associated cost consequences.
To define the meaning of the significance-level a and the appropriate level to choose in the light of the relative cost consequences of making Type 1 and Type 2 Errors.
Describe the difference between one and two tailed tests of hypotheses.
Describe various approaches to hypothesis testing including the: standardized approach, the un-standardized approach, the confidence interval approach and the P-Value approach

3. Tests of Hypotheses for the Population Mean m
An hypothesis test for the population mean m arises when one must decide which of 2 competing hypotheses concerning a population mean m is the more consistent with information emanating from a randomly selected sample.
How does one conduct such tests of hypotheses for m ?
The testing procedure depends on which one of 3 cases apply:
Case 1 s is known (and if n < 30, the population is assumed to be normal)
Case 2 s is unknown, n < 30 and the population is normal
Case 3 s is unknown and n ≥ 30
We begin with a discussion of Case 2 – arguably the most complex - leaving the less complicated cases to subsequent discussion.

4. Tests of Hypotheses for the Population Mean m Continued
Suppose supermarket management is willing to install - at great expense - a new check-out system only if there is strong plausible evidence that average takings per customer exceed $120.
The Null and Alternative Hypotheses
The two competing hypotheses might be written as:
Ho: m ≤ $120
Ha: m > $120
where: Ho (Ha) is called the null (alternative hypothesis)
Normally the hypotheses are written in such a way that acceptance (rejection) of Ho requires no action (action) that is, the new check out system is not (is) installed.

5. Tests of Hypotheses for the Population Mean m Continued
Types of Errors in Hypothesis Testing and their Cost Consequences:
In hypothesis testing there are 2 mutually exclusive errors: the Type I and Type II errors.
The Type I Error
This is the error involved in rejecting Ho when it is true. (eg failing to recognize that: m ≤ $120 when it is) The cost consequences of this type of error would be the unwarranted installation of the new check out system.
The Type II Error
This is the error involved in accepting Ho when it is false. (eg failing to recognize that : m > $120 when it is). The cost consequences of this type of error would be the forgone benefits associated with not having undertaken the warranted installation.

6. Tests of Hypotheses for the Population Mean m Continued

7. Tests of Hypotheses for the Population Mean m Continued

8. Tests of Hypotheses for the Population Mean m Continued

9. Tests of Hypotheses for the Population Mean m Continued

10. Tests of Hypotheses for the Population Mean m Continued

11. Tests of Hypotheses for the Population Mean m Continued

12. Tests of Hypotheses for the Population Mean m Continued
Caveat
In hypothesis testing we are never proving that H0 is true or false. We are simply using a mechanical rule to decide - on the basis of sample information - which of the two competing hypotheses is the more plausible. In other words, there is still the risk of making either a Type I or Type II Error.
If t* is set to (1.708) when H0 is true, then the probability of making a Type I error - prior to sample selection - is 1% (5%). This is because the decision rule rejects H0 when ttest exceeds t* and if H0 is true this will occur with a probability of .01 (.05).

13. Tests of Hypotheses for the Population Mean m Continued
Choice of Significance Level
The probability of rejecting H0 when it is true, is known as the significance level a. The larger is a, the greater is the chance of making a Type I error. On the other hand, if the alternative hypothesis Ha is true, the probability of making a Type II error is lower the higher is a.
The choice of significance level depends on which of the two errors, is the most critical to avoid. One selects a 1% (5%) significance level if one is more concerned about avoiding the cost consequences arising from a Type I (Type II) error. In cases where it is difficult to decide which error is the more severe, a suitable compromise is reached by selecting a significance-level between 1% and 5% (say 2% or 2.5%)

14. Tests of Hypotheses for the Population Mean m Continued
Two Tailed and One Tailed Tests
In our previous example, there was only one rejection region in the right hand tail of a Student t distribution. For this reason, such a test is referred to as a one tailed test.
A two tailed test arises when we wish to test the null hypothesis that m equals a particular value against the alternative that it does not equal this value. In such a test, the acceptance region for Ho is the region bounded by two critical t - values: t* and - t* both of which are equidistant from zero (see below).

15. Tests of Hypotheses for the Population Mean m Continued
Two Tailed and One Tailed Tests Continued
If ttest falls in the interval: [-t* t*] one would accept H0. Were it to fall in either rejection region Ho would be rejected.
Note : In a 2-tailed test, the significance level a refers to the combined area of the two tails of the relevant Student t distribution.
Area in tails = a = Pr (t < -t*) + Pr (t > t*) = 2Pr (t > t*)

16. Tests of Hypotheses for the Population Mean m Continued

17. Tests of Hypotheses for the Population Mean m Continued
Worked Example Continued
Since df = n – 1 = 20 and 2.5% (not 5%) of the relevant t-distribution’s area lies to the right of t*, the acceptance range for H0 is now:
[-t* t*] = [ ]
Since ttest (i.e ) is less than we reject H0 that the mean expenditure is $120 in a two tailed test conducted at the 5% significance level.
Note: Had a been set to 1%, the acceptance region for H0 would now be given by:
[-t* t*] = [ ]
and Ho would be accepted not rejected.

18. Tests of Hypotheses for the Population Mean m Continued
Summary Of Case 2
A summary of the testing procedure is provided below for Case 2 where the population
standard deviation s is unknown, the sample size is less than 30 and the population is
normal.
Step 1: Set up the null and alternative hypotheses after deciding on a 1 or 2 tailed test
Step 2: Choose an appropriate significance level a that takes account of the cost consequences of making a Type 1 and Type II Error
Step 3: Construct the acceptance and rejection regions for H0
Step 4: Compute the standardized test statistic
Step 5: Accept (reject) H0 if the standardized test statistic falls within (beyond) the acceptance region in a 1 or 2 tailed test – as the case may be - at the designated significance level a.

19. Tests of Hypotheses for the Population Mean m Continued
Cases 1 and 3
How would one proceed for the remaining two cases:
Case 1 s is known (and if n < 30, the population is assumed to be normal)
Case 3 s is unknown and n ≥ 30
The steps outlined for Case 2 apply equally to Cases 1 and 3.
The critical t-values that are used in Case 2 to construct the acceptance and rejection
regions for Ho are replaced by critical z-values. For instance, if a is set to .05 in a two tailed
test the acceptance region for Ho is (-z* z*) = ( ).
On the other hand, if it is a one tailed test with the rejection region in the right (left) hand tail,
the critical z-value is 1.65 (-1.65).
Finally, the ttest value used in Case 2 is replaced by:

20. Tests of Hypotheses for the Population Proportion p
Hypotheses Testing for the population proportion p described below is only defensible if one is reasonably confident that np  5 and n(1 – p)  5
A Worked Example of the Steps Involved in Conducting an Hypothesis Test of p
At a general level, the steps involved in undertaking an hypothesis test for the population proportion are similar to the steps outlined earlier for the population mean.
Step 1: Choice of 1 or 2 tailed-test
Suppose a bank wishes to conduct a 2 tailed test of the hypothesis that the proportion of mortgage loan failures ( = p ) in its loan book equals 10% against the alternative hypothesis that it does not equal 10%. Then the competing hypotheses are:
H0 : p = .10  NB we assume action [no action] is required if Ho is rejected [un-rejected]
Ha: p  .10
Step 2: Choose an appropriate significance level a that takes account of the relative cost consequences of making a Type 1 and Type II Error
Suppose the Bank’s statistician chooses a significance-level of .02 (=a) because she makes the judgment that:
Cost Consequence of Type 1 error  Cost Consequence of Type 2 error

21. Tests of Hypotheses for the Population Proportion p
Worked Example of Steps Involved in Conducting an Hypothesis Test of p Continued
Step 3: Construct the acceptance and rejection regions for H0
For a 2 tailed test the acceptance region1 is: (-z* z*) where: z* = za/2
Since a = .02, the acceptance region is : (-z.01 z.01) = ( )
Step 4: Calculate the Standardized Test Statistic
For an hypothesis test of the population proportion, the standardized test statistic is given by:

22. Tests of Hypotheses for the Population Proportion p
Worked Example of Steps Involved in Conducting an Hypothesis Test of p Continued
Step 5: Accept (Reject) H0 if ztest falls within (beyond) the acceptance region for H0.
Since for the running example:
ztest = 1  ( ) = (-z* z*) = (-za/2 za/2) where a = .02
the bank’s statistician would be correct to conclude that:
H0: p = .10
is not rejected (or is accepted) in a two tailed test conducted at the 2% significance–level

23. Various Approaches to Conducting Tests of Hypotheses
The Standardised Approach to Conducting Tests of Hypotheses:
All discussion on previous slides relating to hypothesis testing has revolved around the
standardised approach to conducting hypotheses tests.
Other Approaches to Conducting Tests of Hypotheses:
Apart from the standardized approach there are 3 other approaches:
The un-standardized approach
The confidence-interval approach
The P-value approach
Each of these remaining approaches are illustrated in subsequent slides by continued
reference to the “mortgage loan failure” example

24. Other Approaches to Conducting Tests of Hypotheses Continued
The Un-standardised Approach to Conducting Tests of Hypotheses:
If with the preceding standardized approach the bank’s statistician were willing to accept H0
when:
Ztest  (-z* z*) or ( )+
+ This is because a = .02 and z* = za/2 = z.01 = 2.33

25. Other Approaches to Conducting Tests of Hypotheses Continued
The Confidence Interval Approach to Conducting Tests of Hypotheses:
With this approach, one constructs a (1 – a)100% confidence interval for p and if pHo falls in the confidence interval then H0 is accepted or un-rejected in a two-tailed test conducted at the a significance-level.
In our running example, if the bank’s statistician were to implement the confidence interval approach she would construct a 98% confidence interval. Since a = .02. this is given by:
Note: This approach can only be used for 2-tailed tests

26. Other Approaches to Conducting Tests of Hypotheses Continued
The P-Value Approach to Conducting Tests of Hypotheses:
The P-Value is the probability of obtaining a less likely sample statistic than the one obtained from the selected sample. If the P-Value is less (more) than a one rejects (accepts) Ho
In the running example the sample statistic was p = .112 (equivalent to z = 1) so from the z-distribution:
P-Value = 2Pr(z > 1) = 2( ) = 2(.1587) =.3174

27. Other Approaches to Conducting Tests of Hypotheses Continued
The P-Value Approach to Conducting Tests of Hypotheses Continued:
Furthermore, since the P-Value =.3714 > .02 = a the Bank’s statistician would accept Ho in a 2-tailed test conducted at the 2% significance - level

28. Other Approaches to Conducting Tests of Hypotheses Continued
Roundup of Alternative approaches to Conducting Tests of Hypotheses:
The last few slides (viz ) have been devoted to a discussion of alternative approaches to hypothesis testing in the context of two-tailed tests and hypotheses concerning the population proportion.
However, these slides have not covered hypotheses concerning the population mean and for that matter 1-tailed tests. For a more thorough general treatment that covers these situations, students are invited to read Lombardo (2006, pp. 390 – 393).
What is Important for the Exam in this Topic ?
Students are expected to apply the standardised approach (discussed on slides 1-22) when attempting any exam question related to this topic.
Nonetheless, it is important for students to know about the other approaches particularly the P-Value approach as the P-Value statistic is often generated by computerised statistical software.