Ri Sampling Estimation HypoTesting CorNReg Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 15 (Summary and Tutorial) Topic: Sampling and Estimations, Hypothesis Testing, Correlation&Regression Summary for Sampling and Estimations Definitions - A population is the entire collection of data (persons or items or individuals) that we want to study e.g. apples produced by a farm. - A sample is random if every element in the population has an equal chance of being selected, and the selection of an element is independent of another. e.g. ‘Every biscuit bar has an equal chance of being selected, and the selection of one biscuit bar is not affected or influenced by the selection of another biscuit bar’. [Note that it is not sufficient to say ‘each biscuit bar has an equal chance of being selected’] - A sample is non-random if each element in the population does not have an equal chance of being selected, resulting in certain segments of the population being over-represented, as some members are “systematically or deliberately excluded” from the study and the sample being biased. The Sample Mean, X as a Random Variable Let 1 2 3, , , , nX X X X be a random sample of size n taken from an infinite population (or finite population if sampling is done with replacement) with mean  and variance 2 . Then the sample mean X, defined by 1 2 31 nX X X XX Xn n      , is a random variable with E( )X  and 2 Var( )X n  .

The Distribution of the Sample Mean Let 1 2 3, , , , nX X X X be a random sample of size n taken from a normal population with mean  and variance 2 . Then (1) (2) 2 1 2 3 ~ N( , )nX X X X n n      Let 1 2 3, , , , nX X X X be a large random sample of size n taken from a non-normal population with mean  and variance 2 . Then since sample size n is large, (1) approximately by Central Limit Theorem (2) 2 1 2 3 ~ N( , )nX X X X n n      approximately by Central Limit Theorem General Tips 1. To identify questions on Central Limit Theorem, look out when there is no mention of ‘normally distributed’ or ‘normal population’ e.g. “The random variable X is thought to have a mean of 50 but it is known that the standard deviation is 14.5.” and the question asks for ‘find the probability that the sample mean / average value of X / sum …’ , ‘by using a suitable approximation’ or ‘estimate the probability’. 2. Do not make the mistake of writing  2~ N ,X   at the first sighting of mean and variance/standard deviation in the question. Mean and variance applies to any population, not just normal, so look out for the phrases in point 1 above. 3. Also, writing 2~ N ,X   approximately by Central Limit Theorem is wrong as the theorem applies to X. 4. When the population variance 2 is unknown, the unbiased estimate of population variance 2s will be used and the notation will change accordingly. This is especially important when we write the distribution of 2 2 ~ N , sX n     approximately under hypothesis testing. 2 ~ N ,X n     2 ~ N ,X n    

Unbiased Estimates of Population Mean and Population Variance. In statistics, an estimate is considered to be “good” if it’s unbiased, i.e. the average value of the sample statistic (used to estimate the population parameter) for all possible samples gives the true value of the population parameter. In particular, the average value of Xgives the true value of , that is, E .X  Population Estimate from sample Unbiased estimate? How?  Sample mean x Yes, sample mean is an unbiased estimate of population mean.  or x x ax x an n     2 Sample variance No, sample variance is NOT an unbiased estimate of population variance. We will have to calculate the unbiased estimate of population variance 2s by using one of the following:  2 2(1) 1 x xns n n             2 2 2 1(2) 1 xs x n n          2(3) sample variance1 ns n      2 22 1(4) 1 x as x an n          

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: Let X be the IQ of a student in ABC University. Step 1: To test 0H : 118 vs 1H : 118 Step 2: Perform a 1-tail / 2-tail test at 5% level of significance. Step 3: (Sample from a Normal population of known variance) Under 0H , 2 0~ N ,X n     , where 0 118  and 12 . From the sample, 121x , 50n . OR Step 3 (Large sample from a Normal population of unknown variance): Under 0H , 2 0~ N ,sX n    approximately, with 0 118  . From the sample, 121x , 97.454s OR Step 3 (Large sample from a non-Normal population of unknown variance): Remember to always define X if they are not defined in the question. This is an example. Read question carefully to decide >, < or ≠.

Under 0H , since 60n is large, 2 0~ N ,sX n    approximately, by Central Limit Theorem, with 0 118  . From the sample, 121x , 97.454s Step 4: Using a z-test, -valuep P( 121)X  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. OR (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 5% 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 two phrases in bold and the end of the sentence is to describe the alternative hypothesis H1.] General Tips - Take note of the nature of the “claim” in the question and the respective conclusion e.g. if the claim is “the mean IQ is equal to 118 or at least 118 or at most 118”, then when 0H : 118 is rejected, there is “sufficient evidence at … to conclude that the claim is invalid”; whereas if the claim is “the mean IQ is higher or lower than 118”, then when 0H : 118 is rejected, there is “sufficient evidence at … to conclude that the claim is valid”. - If 0H is rejected at 5% level of significance for a certain p-value, 0H will also be rejected at 10% level of significance or any level higher than 5%. On the contrary, 0H may or may not be rejected at 1% or any level lower than 5%.

Definitions 1. The level of significance (or significance level) of a hypothesis test, denoted by , is defined as the probability of rejecting 0H when 0H is true e.g. “5% level of significance” means “there is a 0.05 probability of wrongly concluding that the mean IQ of the students is more than 118 when in fact it is 118”. 2. The -valuep P( 121)X  in this context refers to the probability of getting a sample with average IQ at least 121. This is also the value we will put down on our script if question asked for the smallest level of significance at which 0H can be rejected in favour of 1H . Important Cases where conclusion is given and we are asked to find: Case 1: Level of significance % Carry on as if we are performing a test. After finding the p-value: If given 0H is rejected, p-value 100  ; if given 0H is not rejected, p-value 100  . Case 2: Sample mean x(No standardizat