1. Hypothesis testing Chapter S12 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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

2. Hypothesis testing Chapter S12 Learning Objectives continued ...
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

3. Statistical inference
One of the major roles of statisticians is to draw conclusions from data
This is referred to as statistical inference
We can put a probability on whether a conclusion is correct within reasonable doubt
Conclusions can always be wrong
Statistical inference plays a major role in decision making
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

4. Statistical inference
Decision-making process:
1. Collect the data
2. Summarise the data (using either visual displays or descriptive statistics)
3.Set up an hypothesis (i.e. claim or theory) to be tested
4.Calculate the probability of obtaining a sample such as the one we have if the hypothesis is true
5. Either accept or reject the hypothesis
Conclusions will be made based on samples taken from the population
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

5. Ho:(statement) v H1(alternative statement)
The null hypothesis (questions dealing with differences between samples)
Technique for dealing with these problems begins with the formulation of an hypothesis.
Null hypothesis is a statement that nothing unusual has occurred. The notation is Ho.
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

6. Alternative hypothesis
May be classified as two-tailed test or one-tailed test
Two-tailed test (two sided alternative)
Test with no preconceived notion that the true value of µ is either above or below the hypothesised value of µ
H1: µ  µo
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

7. Errors
There are two possible errors in making a conclusion about a null hypothesis
1. Type I errors occur when you reject Ho as being false when Ho is really true
2. Type II errors occur when you accept Ho as being true when Ho is really false
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

8. Significance level
This level represents the borderline probability between whether an event has occurred by chance or whether an unusual event has taken place
Most common significance level used is 0.05, commonly written as = 0.05
5% significance level says in effect that an event that occurs less than 5% of the time is considered unusual
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

9. Test statistics one sample z-test one sample t-test
Test for determining if a single sample from a population is consistent with the rest of the population
one sample z-test
one sample t-test
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

10. Value of a test statistic
Information required in calculation:
1. the size (n) of the sample
2. the mean ( ) of the sample
3. 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?
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

11. Drawing a conclusion
The steps that should be undertaken to perform a one-sample test are:
1. Set up null and alternative hypotheses—including deciding whether to use a one-sided or two-sided test
2. Decide on significance level you are using
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
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

12. Test statistics (two sample problems)
Paired t-test
two samples that are to be compared with each other
often referred to as two-sample problems
have a structure such that the data are paired
samples must each contain the same number of observations
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

13. Test statistics (two sample problems)
Two-sample t-test
Samples are not paired, they are independent
Two samples need not contain the same number of observations
Most common test statistic used in this situation is a two-sample t-test
Also known as a pooled test
Calculation requires more work than for the paired t-test
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher

14. Test statistics (two sample problems)
p-values
Alternative to using critical values for testing hypotheses
Calculates the probability of obtaining a value as extreme as the value of the test statistic
© 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher