Hypothesis Testing Steps : 1. Review Data : –Sample size. –Type of data. –Measurement of data. –The parameter ( ,  2,P) you want to test. 2. Assumption :.For example: assumption about – the normality of the population distribution. – equality of variance. – independence of samples. 3. Hypothesis : Cases of hypothesis : Case 1 : H0 :  =  0 vs Ha :    0 (two-tailed test) Case 2 : H0 :    0 vs Ha :    0 (right-tailed test) Case 3 : H0 :    0 vs Ha :    0 (left-tailed test)

Hypothesis, Hypothesis and Testing HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing. HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.

Null and Alternate Hypothesis NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence. ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false.

Test Statistic versus Critical Value TEST STATISTIC A value, determined from sample information, used to determine whether to reject the null hypothesis. CRITICAL VALUE The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. Example: z, t, F,  2

Important Things to Remember about H 0 and H 1 H 0 : null hypothesis and H 1 : alternate hypothesis H 0 and H 1 are mutually exclusive and complementary H 0 is always presumed to be true H 1 has the burden of proof A random sample (n) is used to “reject H 0 ” If we conclude 'do not reject H 0 ', this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H 0 ; rejecting the null hypothesis then, suggests that the alternative hypothesis may be true. Equality is always part of H 0 (e.g. “=”, “≥”, “≤”). “≠” “ ” always part of H 1

How to Set Up a Claim as Hypothesis In actual practice, the status quo is set up as H 0 In problem solving, look for key words and convert them into symbols. Some key words include: “improved, better than, as effective as, different from, has changed, etc.” Keywords Inequality Symbol Part of: Larger (or more) than>H1H1 Smaller (or less)<H1H1 No more than  H0H0 At least≥H0H0 Has increased>H1H1 Is there difference?≠H1H1 Has not changed=H0H0 Has “improved”, “is better than”. “is more effective” See left textH1H1

H0 4. & 5. Test Statistic and it’s Distribution : For example : x -  0 z =   N(0,1)  /  n 6. Decision Rule : The rule that determines reject H0 or not reject it. –Determine the rejection region by using the level of significance (  ) –Use p-value to make the decision. 7. Calculation of test statistic: z cal = 1.06

Statistical decision & conclusion P-value   Or zcal falls in rejection region do not reject H0 H0 may be True reject H0 Ha is True 8. & 9. Statistical decision and conclusion : yesNo

P-VALUE Why do we use the p-value ? For solvent the decisions difference problem. reject H 0 do not reject H 0

Other names for the p-value: – Level of significance. – Observed significance level. – Significance probability. What is the p-value ? It is the smallest value possible where H 0 is rejected. Definition: P-value is the probability of obtaining a value as extreme or more extreme than the observed value of test statistic given H0 is true. 0  P-value  1

Steps for Calculating the P-value for a Test of Hypothesis: 1.Compute the value of the test statistic T. 2. If the test : Test statistic = - Z cal left-tailed test (Ha :  <  0 ) Test statistic = Z cal right-tailed test (Ha :  >  0 ) I. left-tailed (Ha :  <  0 ) P-value = P( T < t c ) II. right-tailed (Ha :    0 ) P-value = P( T > t c )

III. If the test two-tailed (Ha :    0): a. if the distribution of the test statistic is symmetric. For example: z cal & t cal P-value = 2 P( T < - |tc| ) Test statistic = - Z cal Test statistic Z cal negative Test statistic = Z cal Test statistic Z cal positive

test two-tailed (Ha :  2   2 0 ): b. if the distribution of the test statistic is not symmetric. For example: (  2 cal & F cal ) : P-value = 2 minimum (P L, P U ) PL = P( T < t c ) PU = P( T > t c ) F cal PUPU PLPL P-value = 2 P U PUPU PLPL P-value = 2 P L F cal

Note: The p-value dose not depend in calculation on . It only depends on the calculated value (t c ) and the sampling distribution of T.

Reporting Test Result as p-value: How to Decide Whether to Reject H0 –If the P-value is small, then the sample data does not support H0 and the decision is reject H0. – If the P-value is large, then the sample data support H0 and the decision is not reject H0. General Rule: –If the P-value  , then reject H0. –If the P-value > , then do not reject H0