Statistics Lecture 8 Point and interval estimates •David Bartl •Statistics •INM/BASTA Outline of the lecture •Central Limit Theorem: the Lindenberg-Lévy Theorem •Sampling and survey data collection — sampling with replacement — sampling without replacement •Point estimates — point estimates for the population mean and for the population variance •Interval estimates — interval estimates for the population mean and for the population variance Normal distribution anima2 1 z=0 z=0 p=0,5 p=0,5 CLT: Lindeberg-Lévy Theorem CLT: Lindeberg-Lévy Theorem in brief CLT: Lindeberg-Lévy Theorem in brief CLT: Lindeberg-Lévy Theorem in in other words Example: Invoices Example: Invoices Example: Invoices •The table of the frequencies for the sample mean of the 50-element samples: Interval of the sample mean Frequency (number) Relative frequency 3800 ≤ x¯ < 4000   1   0.2 % 4000 ≤ x¯ < 4200   2   0.4 % 4200 ≤ x¯ < 4400  17   3.4 % 4400 ≤ x¯ < 4600  39   7.8 % 4600 ≤ x¯ < 4800  52  10.4 % 4800 ≤ x¯ < 5000  85  17.0 % 5000 ≤ x¯ < 5200 110  22.0 % 5200 ≤ x¯ < 5400  77  15.4 % 5400 ≤ x¯ < 5600  64  12.8 % 5600 ≤ x < 5800  37   7.4 % 5800 ≤ x¯ < 6000  10   2.0 % 6000 ≤ x¯ < 6200   4   0.8 % 6200 ≤ x¯ < 6400   2   0.4 % TOTAL: 500 100.0 % Example: Invoices •Histogram – the relative frequencies of the sample mean – 500-element sample: 5 % 15 % 10 % 20 % 0 % 4000 4200 4400 Amount of money 25 % 4600 Relative frequency 4800 5000 5200 5400 5600 5800 3800 5400 5600 5800 0.2 0.4 3.4 7.8 10.4 17.0 22.0 15.4 2.0 12.8 7.4 0.8 0.4 Example: Invoices •Histogram – the relative frequencies of the sample mean – 5000-element sample: Sampling and survey data collection •The population is often very large, or the population may not be clearly specified in practice. •Sampling is a method to obtain a relevant sample of the entire population. •The sample should be representative, i.e. its structure should follow the structure of the entire population. •Sampling plan (random sample, etc.) – see below. •The methods of collecting the data: • — [opinion] poll, survey • — questionnaire • — on-line / telephone / mail / face-to-face / … Sampling and survey data collection Sampling and survey data collection •Simple random sampling – each element has equal chance of being selected. •Systematic sampling. •Stratified sampling. •Cluster sampling. •Accidental sampling. •… Sampling with and without replacement •Replacement of selected units: • •sampling without replacement: — an element can appear no more than once in the sample • •sampling with replacement: — an element can appear several times in the sample • • •Note that sampling without replacement is often the case in practice. Point estimates & Interval estimates Point estimates Point estimates Point estimates Point estimates (always) (if the sampled variables are normal) Point estimates An illustrative example • • Interval estimates Interval estimates An illustrative example An illustrative example An illustrative example An illustrative example Theorem Theorem Theorem – Corollary Theorem – Corollary – Proof: An illustrative example An illustrative example An illustrative example An illustrative example The sample size for the confidence interval The sample size for the confidence interval The sample size for the confidence interval The sample size for the confidence interval • • Interval estimate for the variance Theorem An illustrative example An illustrative example An illustrative example An illustrative example An illustrative example An illustrative example The variance of the sample variance •The variance of the sample variance for normal distribution • The variance of the sample variance The variance of the sample variance