IEEM 3201 One and Two-Sample Tests of Hypotheses

IEEM 320IEEM151 Notes 20, Page 2 Statistical Hypotheses  Statistical Hypothesis: an assertion or conjecture concerning (parameters of) one or more populations ? e.g., the population mean is equal to a particular value:  =  0 ? Hypothesis testing: accept or reject a hypothesis based on the sample information

IEEM 320IEEM151 Notes 20, Page 3 Null and Alternative Hypotheses  H 0, null hypothesis: the hypothesis subject to testing ? H 1, alternative hypothesis: H 1 is rejected if H 0 is accepted, and vice versa ? condition to reject H 0 : if H 0 is true, it is highly unlikely to get the given set of sample values ? the rejection provides a firmer, clearer assertion ? tend to set the desirable conclusion as H 1

IEEM 320IEEM151 Notes 20, Page 4 Null and Alternative Hypotheses ? the form of H 1 affects the procedure of the test ? two-tailed test: H 0 :  =  0 vs. H 1 :    0 ? one-tailed test: H 0 :  =  0 vs. H 1 :  >  0 ? one-tailed test: H 0 :  =  0 vs. H 1 :  <  0

IEEM 320IEEM151 Notes 20, Page 5 Null and Alternative Hypotheses  Null Hypothesis: the hypothesis we wish to test and is denoted by H 0. ? Alternative Hypothesis: the rejection of the null hypothesis implies the acceptance of an Alternative hypothesis denoted by H 1. 1–   /2 (1-  )100% confidence interval Critical values Critical regions (reject H 0 if here) Acceptance region (accept H 0 if here) 00 ? e.g., H 0 :  =  0 H 1 :  ≠  0

IEEM 320IEEM151 Notes 20, Page 6 Type I and Type II Error  Rejection of the null hypothesis when it is true is called a type I error. ? Acceptance of the null hypothesis when it is false is called a type II error. Correct decisionType I error Reject H 0 Type II errorCorrect decision Accept H 0 H 0 is falseH 0 is true 1–   /2 11 Probability of committing a type I error. Probability of committing a type II error if  =  1  00

IEEM 320IEEM151 Notes 20, Page 7 Important Properties  Relationships among ,  and sample size ? type I error   type II error  ; type I error   type II error  ? type I error changes with the critical value(s) ? n    and  ?  if the difference between the true value and the hypothesized value increases

IEEM 320IEEM151 Notes 20, Page 8 The Power of A Test  The power of a test is the probability of rejecting H 0 given that a specific alternative is true. ? The power of a test = 1 – .

IEEM 320IEEM151 Notes 20, Page 9 One- and two-Tailed Tests ? One-tailed test: H 0 :  =  0 H 1 :  >  0 or H 1 :  <  0 ? Two-tailed test: H 0 :  =  0 H 1 :   0 ? e.g., a one-tailed test: H 0 :  =68 H 1 :  >68

IEEM 320IEEM151 Notes 20, Page 10 One- and two-Tailed Tests ? One-tailed test: H 0 :  =  0 H 1 :  >  0 or H 1 :  <  0 ? Two-tailed test: H 0 :  =  0 H 1 :   0 ? e.g., a one-tailed test: H 0 :  =68 H 1 :  >68

IEEM 320IEEM151 Notes 20, Page 11 Two-Tailed Test on Mean  H 0 :  =  0, H 1 :    0  X 1, …, X n ~ i.i.d. normal with variance  2 if the true mean is  0, it is unlikely for or

IEEM 320IEEM151 Notes 20, Page 12 Two-Tailed Test on Mean 1–   /2 00 Critical regions (reject H 0 if here) Acceptance region (accept H 0 if here) (1-  )100% confidence interval Critical values

IEEM 320IEEM151 Notes 20, Page 13 One-Tailed Test on Mean  H 0 :  =  0, H 1 :  >  0  X 1, …, X n ~ i.i.d. normal with variance  2 if the true mean is  0, it is unlikely for

IEEM 320IEEM151 Notes 20, Page 14 One-Tailed Test on Mean 1–   00 Critical regions (reject H 0 if here) Acceptance region (accept H 0 if here) Critical values

IEEM 320IEEM151 Notes 20, Page 15 One-Tailed Test on Mean  H 0 :  =  0, H 1 :  <  0  X 1, …, X n ~ i.i.d. normal with variance  2 if the true mean is  0, it is unlikely for

IEEM 320IEEM151 Notes 20, Page 16 One-Tailed Test on Mean 1–  00 Critical regions (reject H 0 if here) Acceptance region (accept H 0 if here) Critical values 

IEEM 320IEEM151 Notes 20, Page 17 Type I  and Type II Error   type I error: Rejecting H 0 when it is true ? type II error: Accepting H 0 when it is false Correct decisionType I errorReject H 0 Type II errorCorrect decisionAccept H 0 H 0 is falseH 0 is true 11  1–   /2 00 Probability of committing a type I error. Probability of committing a type II error if  =  1 Type II error: change with a given  1

IEEM 320IEEM151 Notes 20, Page 18 Effect of Sample Size on Type I Error Solution: follows a normal distribution with  = 68 and  =3.6/6 = 0.6 Example: Find the type 1 error. H 0 :  = 68, H 1 :   68. given  = 3.6, n = 36; critical regions: Hen ce

IEEM 320IEEM151 Notes 20, Page 19 Effect of Sample Size on Type I Solution: follows a normal distribution with  = 68 and  =3.6/8 = 0.45 Hen ce Example: Find the type 1 error. H 0 :  = 68, H 1 :   68. given  = 3.6, n = 64; critical regions:

IEEM 320IEEM151 Notes 20, Page 20 p-Value  A p-value is the lowest level (of significance) at which the observed value of the test statistic is significant.  Calculate the p-value and compare it with a preset significance level . If the p-value is smaller than , we reject the null hypothesis.

IEEM 320IEEM151 Notes 20, Page 21 Type II Error  accepting when H 0 is false  type II error: a function of the true value of parameter  Find type II error. H 0 :  = 68,H 1 :   68.  = 3.6; n = 64, critical regions: the true  = 70 Example on page 292

IEEM 320IEEM151 Notes 20, Page 22 Important Properties  Relationships among ,  and sample size ? type I error   type II error  ; type I error   type II error  ? type I error changes with the critical value(s) ? n    and  ?  if the difference between the true value and the hypothesized value increases

IEEM 320IEEM151 Notes 20, Page 23 Examples ? A random sample of 100 recorded deaths in U.S> during the past year showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean life span today is greater than 70 years? Use a 0.05 level of significance.

IEEM 320IEEM151 Notes 20, Page 24 Examples ? A manufacturer of sports equipment has developed a new fishing line that claims has a mean breaking strength of 8 kilograms with s standard deviation of 0.5 kilogram. Test the hypothesis that u=8 kilograms again H1 that u is not equal to 8 if a random of sample of 50 lines is tested and found to have a mean breaking strength of 7.8 kilograms. Use a 0.01 level significance.

IEEM 320IEEM151 Notes 20, Page 25 Examples ? Some company has published figures on the annual number of kilowatt-hours expended by various home appliances. It is claimed that a vacuum cleaner expends an average of 46 kilowatt-hours per hour. If a random sample of 12 homes included in a planned study indicates that vacuum cleaners expended an average of 42 kilowatt-hours per year with a standard deviation of 11.9 kilowatt-hours, does this suggest at the 0.05 level of significance that vacuum cleaners expend, on average, less than 46 kilowatt-hours annually? Assume the population of kilowatt-hours to be normal.