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 notatio