ACJC 2026 Hypothesis Testing Summary

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Hypothesis Testing / Summary / Page 1 of 2 SUMMARY: Hypothesis Testing Six-Step Process for Hypothesis Testing (Z-test) 1. State the null and alternative hypotheses. To test 0H : 0= (a fixed no.) Against 1H : one of the 3 types 0 0 0or or      depending on the question. 2. State the level of significance at which the test is conducted. at % level of significance (usually given in question) 3. State distribution of X and Z under H0. Distribution of X Population Variance, 2 Under 0H , Test Statistic normal known X ~ 2 0N, n    ( )0 ~ N 0,1XZ n   −= normal unknown Since n is large ( 30n ), X ~ 2 0N, s n   approx. ( )0 ~ N 0,1 approx.XZ s n −= not normal known Since n is large ( 30n ), X ~ 2 0N, n    approx. by Central Limit Theorem ( )0 ~ N 0,1XZ n   −= approx. by Central Limit Theorem not normal unknown Since n is large ( 30n ), X ~ 2 0N, s n   approx. by Central Limit Theorem ( )0 ~ N 0,1XZ s n −= approx. by Central Limit Theorem 4. State value of test statistic and the value of p. Value of test statistic, 0xz n   −= = ___________ 0or xz s n  − =  p-value, p= _________ 5. Compare value of p with value of 100  . If p < 100  , state ‘We reject 0H at % level of significance.’ If p > 100  , state ‘We DO NOT reject 0H at % level of significance.’ 6. Conclusion in the context of the problem. In the case of 0rejection of H , “There is sufficient evidence at the ___% level of significance that 1H (in the context of the question) is true.” In the case of 0non-rejection of H , “There is insufficient evidence at the ___% level of significance that 1H (in the context of the question) is true.” The values of 0 ,  and n must be filled in.

Anglo-Chinese Junior College 2026 H2 Mathematics 9758: Hypothesis Testing / Summary / Page 2 of 2 Example A The mean lifetime of the fluorescent light bulbs produced by a company is  and the standard deviation is 120 hours. If the mean lifetime of a random sample of 100 fluorescent light bulbs produced by the company is found to be 1570 hours, use a 1% level of significance to test if the mean lifetime of the fluorescent light bulbs is less than 1600 hours. Example B The mean lifetime of the fluorescent light bulbs produced by a company is  and the standard deviation is 120 hours. If the mean lifetime of a random sample of 100 fluorescent light bulbs produced by the company is found to be 1623 hours, use a 1% level of significance to test if the mean lifetime of the fluorescent light bulbs is more than 1600 hours. Solution A: Solution B: To test 0H :  = 1600 Against 1H :   1600 at 1% level of significance To test 0H :  = 1600 Against 1H :   1600 at 1% level of significance Under 0H , since n = 100 is large, 2120~ N 1600, 100X   approx. by Central Limit Theorem Test statistic, ( )1600 ~ N 0,1120 100 XZ −= z = − 2.5 (from GC) p-value = 0.00621 (from GC) z = 1.92 (from GC) p-value = 0.0276 (from GC) Since p-value  0.01, reject 0H . Since p-value  0.01, do not reject 0H . There is sufficient evidence at the 1% level of significance that the mean lifetime of the fluorescent light bulbs is less than 1600 hours. There is insufficient evidence at the 1% level of significance that the mean lifetime of the fluorescent light bulbs is more than 1600 hours. Step 1 State the null and alternative hypotheses. Step 2 State the level of significance at which the test is conducted. Step 3 State distribution of X and Z under 0H . Step 6 Conclusion in the context of the problem. Step 5 Compare value of p with value of 100  . Step 4 State value of test statistic and the value of p.