1. Introductory Mathematics & Statistics for Business
Chapter 21
Hypothesis Testing
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

2. Learning Objectives 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

3. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

4. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

5. 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)
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

6. 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.
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

7. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

8. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

9. 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)
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

10. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

11. 21.3 The one-sample z-test (cont…)
Information required in calculation:
the size (n) of the sample
the mean 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?
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

12. 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, s, 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 s is known
In both these cases we can use a z-test statistic defined by:
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

13. 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 s is unknown (however, since n is large, the value of s is approximated by the sample standard deviation, s)
In this case we can use a z-test statistic defined by:
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

14. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

15. 21.3 The one-sample z-test (cont…)
Rules for not rejecting or rejecting H0 in a one-
sample z-test
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

16. 21.4 The one-sample t-test
Consider a test that has the same objective as a one-sample z-test, 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, s , is unknown and
3. the sample size, n, is small
Then we can use a t-test statistic defined as:
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

17. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

18. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

19. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

20. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

21. 21.6 The paired t-test (cont…)
4. Calculate the value of the paired t-test statistic using:
where n = the number of pairs of differences
μd = 0 (according to the null hypothesis)
and with n = n – 1 degrees of freedom
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

22. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

23. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

24. 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
4. Calculate the value of the two-sample t-test statistic
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

25. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

26. 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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e

27. Summary
In this chapter we looked at the principles of statistical inference
We formulated null and alternative hypotheses
We understood
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
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
Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e