1. Business Statistics - QBM117 Testing hypotheses about a population mean

2. Objectives w To test hypotheses about a population mean when sigma is known and when sigma is unknown.

3. Testing a hypothesis about the population mean,  when  is known 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 The operations manager is concerned with determining whether the filling process for filling 100g boxes of smarties is working properly. He believes the standard deviation of the filling process is 10g.

4. Do we know the population standard deviation,  or do we only have the sample standard deviation s? we know  Does the question ask us to test a hypothesis about a mean or a proportion? a mean In order to test this hypothesis the manager selected a random sample of 25 boxes and found their average weight to be 95g. Is their sufficient evidence at  = 0.05 to conclude that the weight of the boxes differs from 100g?

5. Step 1 Step 2 Step 3

6. 0-1.961.96 Region of non-rejection Region of rejection  /2 = 0.025 Region of rejection  /2 = 0.025 0.95 Z

7. Step 1 Step 2 Step 3 Step 4

8. Step 5

9. 0-1.961.96 Region of non-rejection Region of rejection  /2 = 0.025 Region of rejection  /2 = 0.025 0.95 z -2.5

10. Since -2.5 < -1.96 we reject H A. Step 5 Step 6 There is sufficient evidence at  = 0.05 to conclude that the average fill is different from 100g.

11. Do we know the population standard deviation,  or do we only have the sample standard deviation s? Does the question ask us to test a hypothesis about a mean or a proportion? a mean Testing a hypothesis about the population mean,  when  is unknown Exercise 10.26 p346 (9.26 p312 abridged)

12. Step 1 Step 2 Step 3

13. 0 Critical value Region of rejection α = 0.01 Region of non-rejection 0.99 t -2.624

14. Step 1 Step 2 Step 3 Step 4

15. Step 5

16. 0 Region of rejection α = 0.01 Region of non-rejection 0.99 t -2.624 -3.87

17. Since -3.87 < -2.624 we reject H 0. Step 5 Step 6 There is sufficient evidence at  = 0.01 to conclude that the mean is less than 160.

18. Do we know the population standard deviation,  or do we only have the sample standard deviation s? Does the question ask us to test a hypothesis about a mean or a proportion? a mean Testing a hypothesis about the population mean,  when  is unknown Exercise 10.30 p347 (9.30 p313 abridged)

19. Step 1 Step 2 Step 3

20. 0-2.7762.776 Region of non-rejection α/2 = 0.025 0.95 t

21. Step 1 Step 2 Step 3 Step 4

22. Step 5

23. 0-2.7762.776 Region of non-rejection α/2 = 0.025 0.95 t -2.46

24. Since -2.46 > -2.776 we do not reject H 0. Step 5 Step 6 There is insufficient evidence at  = 0.05 to conclude that the mean is not equal to 32.

25. Strong and weak conclusions w Generally we will be presented with a null hypothesis, which we will try to reject. w Before carrying out the test, we know there is a possibility we may make a type I error. w This probability  is preset to a small number, say 0.05. w Knowing that we have a small probability of committing a type I error, ie rejecting a null hypothesis when it is true, makes our rejection of the null hypothesis a strong conclusion.

26. w Generally the same cannot be said about not rejecting the null hypothesis. w This is because the probability of , failing to reject a null hypothesis when it is should be rejected, is not preset to a known small number. w Therefore, failing to reject the null hypothesis is generally a fairly weak conclusion because we do not know the probability that we will fail to reject a null hypothesis when it should be rejected.

27. Reading for next lecture Chapter 10 Sections 10.7 (Chapter 9 Section 9.7 abridged) Exercises to be completed before next lecture S&S 10.2 10.3 10.5 10.11 10.29 10.33 (9.2 9.3 9.5 9.11 9.29 9.33 abridged)