CONCEPTS OF HYPOTHESIS TESTING As the previous section indicated, we can use sample data to estimate the value of an known parameter. Another way of drawing inferences about a population is hypothesis testing. The objective of this from of statistical inference is to determine whether or not the sample data support some belief or hypothesis about the population. STRUCTURE OF HYPOTHESIS TESTING. The tests of hypotheses that the present in this section are called parametric test, because they test the value of a population parameter.

These tests consist of the following four components : 1 These tests consist of the following four components : 1. Null hypothesis 2. Alternative hypothesis 3. Test statistic 4. Rejection region

STRUCTURE OF HYPOTHESIS TESTING TEST STATISTIC The test statistic is the sample statistic upon which we base our decision to either reject or not reject the null hypothesis REJECTION REGION The rejection region is a range of values such that, if the test statistic falls into that range, we decide to reject the null hypothesis STRUCTURE OF HYPOTHESIS TESTING ALTERNATIVE HYPOTHESIS/* The alternative hypothesis denoted H1 or HA ; answers the question by specifying that the parameter is one of the following Greater than the value shown in the null hypothesis. Less than the value shown in the null hypothesis Different from the value shown in the null hypothesis NULL HYPOTHESIS The null hypothesis, which is denoted Ho, must specify that the parameter is equal to a single value * = RESEARCH HYPOTHESIS

RESULTS OF A TEST OF THE NULL HYPOTHESIS STATE OF NATURE DECISION Ho is true Ho is false Reject Ho Type I error P(type I error) =  Correct Decision Do not reject Ho Type II error P(type II error) =   is also known as the level of statistical significance or the significance level of the test Note : When hypothesis testing is viewed as a problem in decision making, two alternative actions can be taken : “accept Ho” or “reject Ho”. The two alternatives, truth or falsity of hypothesis Ho, are viewed as “states of nature” or “states of the world”

Example : Suppose we can tolerate a size of type I error up to 0,06 when testing the null hypothesis : Ho :  = 10 versus H1 :  > 10 for the assembly time problem. Assume the distribution of the time required to assemble a unit is normal with standard deviation=1,4 minutes. Say we observe the assembly time of 25 randomly selected units and chose the sample mean as the test statistic. In particular, we want to compare the following three critical regions: to determine which one satisfies the size of type I error that can be tolerated and which has the smallest  among the three. SOLUTION : discuss in class-room

EXERCISES 1. Suppose you wish to test the hypothesis Ho :  = 5 against the alternative H1 :  = 8 by means of a single observed value of a random variable with probability density function If the maximum size of type I error that can be tolerated is 0,15, which of the following tests is best for choosing between the two hypotheses? a. Reject Ho if X  9 b. Reject Ho if X  10 c. Reject Ho if X  11

2. Suppose a manufacturer of memory chips observes that the probability of chip failure is p = 0,05. A new procedure is introduced to improve the design of chips. To test this new procedure and tested. Let random variable X denote the number of these 200 chips that fail. We set the test rule that we would accept the new procedure, if X  5. Let Ho : p = 0,05, versus H1 : p < 0,05 Find the probability of a type I error

3. Let be the mean of a random sample of size n = 36 from NOR (, 9) 3. Let be the mean of a random sample of size n = 36 from NOR (, 9). Our decision rule is to reject Ho :  = 50 and to accept H1 :  > 50 if Determine the OC () curve and evaluate it at  = 50,0 ; 50,5; 51,0; 51,5. What is the significance level of the test? 4. Consider a NOR (, 2=40) distribution. To test Ho :  = 32 against H1 :  > 32, we reject Ho if the sample mean Find the sample size n and the constant c such that OC (=32) = 0,90 and OC(=35) = 0,15

5. Let Y have binomial distribution with parameter n and p 5. Let Y have binomial distribution with parameter n and p. In a test of Ho : p = 0,25, against H1 : p < 0,25 we reject Ho if Y  c. Find n and c if OC (p=0,25)=0,90 and OC (p=0,20)=0,05. state your assumptions