RI 2025_Y6T2W9Mathfocus_Summary+on+Hypothesis+Testing

Page 1 of 4 RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 2 Revision (Summary) Topic: Hypothesis Testing Summary for Hypothesis Testing Performing a Hypothesis Test Step 1 Understand the given question and write down the null hypothesis 0H and the alternative hypothesis 1H Step 2 Write down the level of significance  (usually given in the question) Step 3 Decide on the test statistic to be used and determine its distribution Step 4 Use the GC to calculate the p-value Step 5 Reject 0H if p-value , OR Do not reject 0H if p-value Write down the conclusion in the context of the question Example: From records, the IQ of university students is normally distributed with a mean of 118. Students from ABC University claim that the mean IQ of the students in their university is greater than 118. A sample of 50 students was tested and the mean IQ was found to be 121. Test, at 5% significance level, whether the sample provides significant evidence to support the students’ claim, assuming that (a) the IQ of university students is normally distributed with standard deviation, , where 12 , (b) the IQ of university students is normally distributed and an unbiased estimate of the population variance, calculated based on the sample of 50 students, is 97.454, (c) an unbiased estimate of the population variance for the IQ of university students, calculated based on the sample of 50 students, is 97.454. Solution Let X be the IQ and be the mean IQ of a student in ABC University. Step 1: To test 0H : 118 vs 1H : 118 Remember to always define X and any notations used if they are not defined in the question. Read question carefully to decide >, < or ≠.

Page 2 of 4 Step 2: Perform a 1-tail test at 5% level of significance. For part a: Sample from a Normal population of known variance Step 3: Under 0H , 2 0~ N ,X n     , where 0 118  and 12 . From the sample, 121x , 50n . Step 4: Using a z-test, -valuep  P 121X  0.0385 (3 s.f.) Step 5: Since -valuep 0.0385 0.05, we reject 0H and conclude that there is sufficient evidence, at 5% level of significance, to support the claim that the mean IQ of students in ABC University is greater than 118. If test is performed at 1% level of significance, Step 5: Since -valuep 0.0385 0.01, we do not reject 0H and conclude that there is insufficient evidence, at 1% level of significance, to support the claim that the mean IQ of students in ABC University is greater than 118. [Notice the only difference are the phrases in bold, the end of the sentence is to describe the alternative hypothesis H1.] For part b: Large sample from a Normal population of unknown variance Step 3: Under 0H , 2 0~ N , sX n    approximately, with 0 118  . From the sample, 121x , 97.454s Step 4: Using a z-test, -valuep P 121X  0.0158 (3 s.f.)