1. Business Statistics - QBM117 Introduction to hypothesis testing

2. Objectives w To introduce the second type of statistical inference - hypothesis testing w To introduce the concept of hypothesis testing. w To gain a basic understanding of the methodology of hypothesis testing

3. Introduction to hypothesis testing w Hypothesis testing is another type of statistical inference where, once again, decisions are based on sample data. w 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. w Over the next few lectures we will develop a step-by-step methodology that will enable us to test these beliefs.

4. 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.

5. The null and alternative hypotheses w is the challenged hypothesis. It is the assertion we hold as true, until we have sufficient statistical evidence to conclude otherwise. w it always expresses a value of the population parameter which we intend to subject to scrutiny, based on sample data. The null hypothesis, denoted H 0, The purpose of scrutinizing the null hypothesis is to determine whether there is support for the alternative hypothesis, denoted H A,

6. 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

7. 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 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

8. The manager hopes to find the filling process is working properly, however, he may find the sampled boxes weigh too little or too much 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.

9. 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 The test statistic or its standardised value as long as the population is normal or

10. When testing hypotheses about the population mean , when the population variance is unknown, the test statistic will be or its standardised value as long as the population is normal.

11. When testing hypotheses about the population proportion p, the test statistic will be or its standardised value as long as

12. 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 H 0 is rejected. The non-rejection region consists of all values of the test statistic for which H 0 is not rejected. The value that separates the rejection region from the non- rejection region is the critical value.

13.  Critical value Region of non-rejection Region of rejection Two-tailed hypothesis test H A :   100

14.  Critical value Region of rejection Upper tailed hypothesis test H A :  > 100 Region of non-rejection

15.  Critical value Region of rejection Lower tailed hypothesis test H A :  < 100 Region of non-rejection

16. 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 H 0 ifis greater than 110. Reject H 0 ifis less than 90. or the rejection region in slide 3 might be described by

17. 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?

18. How do we determine this critical value? w The determination of the critical value depends on the size of the rejection region. w The size of the rejection region depends on the probability of making an error, when testing our hypotheses. w So, what are these errors?

19. Errors in hypothesis testing w rejecting the null hypothesis when it is true, a type I error; w 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.

20. Type I and type II errors w The probability of making a type I error is defined as . This probability is also referred to as the level of significance. w 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.

21. w The size of  is usually kept as small as possible, generally a value between 1% and 10%. w 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. w The size of this rejection region is .

22.  Critical value Region of non-rejection Region of rejection  /2 = 0.025 Region of rejection  /2 = 0.025 Two-tailed hypothesis test H A :   100 when  = 0.05 0.95

23.  Critical value Region of rejection  = 0.05 Upper tailed hypothesis test H A :  > 100 when  = 0.05 Region of non-rejection 0.95

24.  Critical value Region of rejection  = 0.05 Lower tailed hypothesis test H A :  < 100 when  = 0.05 Region of non-rejection 0.95

25. Reading for next lecture w Chapter 10, sections 10.3 and 10.5 w (Chapter 9, sections 9.3 and 9.5 abridged)