Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Business Statistics - QBM117 Testing hypotheses about a population mean Objectives To test hypotheses about a population mean when sigma is known and when sigma is unknown. Testing a hypothesis about the population mean, when is known 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. 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 H 0 : 100 H A : 100 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? Does the question ask us to test a hypothesis about a mean or a proportion? a mean Do we know the population standard deviation, or do we only have the sample standard deviation s? we know x Z / n Step 1 H 0 : 100 H A : 100 Step 2 x Z / n Step 3 0.05 Region of non-rejection 0.95 3.00 1.96 2.50 2.00 1.50 1.00 0 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3 -1.96 Z Region of rejection Region of rejection /2 = 0.025 /2 = 0.025 Step 1 H 0 : 100 H A : 100 Step 2 x z / n Step 3 0.05 / 2 0.025 z0.025 1.96 Step 4 Reject H 0 if z sample 1.96 or z sample 1.96 Step 5 10 n 25 x z / n 95 100 10 / 25 2 .5 x 95 Region of non-rejection 0.95 3.00 1.96 2.50 2.00 1.50 1.00 0 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3 -1.96 z -2.5 Region of rejection Region of rejection /2 = 0.025 /2 = 0.025 Step 5 10 n 25 x z / n 95 100 10 / 25 2 .5 x 95 Step 6 Since -2.5 < -1.96 we reject HA. There is sufficient evidence at = 0.05 to conclude that the average fill is different from 100g. Testing a hypothesis about the population mean, when is unknown Exercise 10.26 p346 (9.26 p312 abridged) Does the question ask us to test a hypothesis about a mean or a proportion? a mean Do we know the population standard deviation, or do we only have the sample standard deviation s? x t s/ n Step 1 H 0 : 160 H A : 160 Step 2 x t s/ n Step 3 0.01 Region of non-rejection 0.99 3.00 2.50 2.00 1.50 1.00 0.50 0.00 α = 0.01 -0.50 -1.00 -1.50 -2.00 -2.50 -3 -2.624 Critical value Region of rejection 0 t Step 1 H 0 : 160 H A : 160 Step 2 x t s/ n Step 3 0.01 t ,n1 t0.01,14 2.624 Step 4 Reject H 0 if t sample 2.624 Step 5 s 10 n 15 x t sample s/ n 150 160 10 / 15 3.87 xx 150 Region of non-rejection 0.99 3.00 2.50 2.00 1.50 1.00 0.50 0.00 -0.50 α = 0.01 -1.00 -1.50 -2.00 -2.50 -3 -3.87 -2.624 Region of rejection 0 t Step 5 s 10 n 15 x t sample s/ n 150 160 10 / 15 3.87 x 150 Step 6 Since -3.87 < -2.624 we reject H0. There is sufficient evidence at = 0.01 to conclude that the mean is less than 160. Testing a hypothesis about the population mean, when is unknown Exercise 10.30 p347 (9.30 p313 abridged) Does the question ask us to test a hypothesis about a mean or a proportion? a mean Do we know the population standard deviation, or do we only have the sample standard deviation s? x t s/ n Step 1 H 0 : 32 H A : 32 Step 2 x t s/ n Step 3 0.05 Region of non-rejection 0.95 3.00 2.776 2.50 2.00 1.50 1.00 0 0.50 0.00 -0.50 α/2 = 0.025 -1.00 -1.50 -2.00 -2.50 -3 -2.776 t α/2 = 0.025 Step 1 H 0 : 32 H A : 32 Step 2 x t s/ n Step 3 0.05 t / 2,n1 t0.025, 4 2.776 Step 4 Reject H 0 if t sample 2.776 or t sample 2.776 Step 5 s 6.91 n5 x t sample s/ n 24.4 32 6.91 / 5 2.46 x 24.4 Region of non-rejection 0.95 3.00 2.776 2.50 2.00 1.50 1.00 0 0.50 0.00 -0.50 -1.00 -1.50 -2.00 -2.50 -3 -2.776 -2.46 α/2 = 0.025 t α/2 = 0.025 Step 5 s 6.91 n 5 x t sample s/ n 24.4 32 6.91 / 5 2.46 x 24.4 Step 6 Since -2.46 > -2.776 we do not reject H0. There is insufficient evidence at = 0.05 to conclude that the mean is not equal to 32. Strong and weak conclusions Generally we will be presented with a null hypothesis, which we will try to reject. Before carrying out the test, we know there is a possibility we may make a type I error. This probability is preset to a small number, say 0.05. 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. Generally the same cannot be said about not rejecting the null hypothesis. 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. 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. 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)