【正文】 Abcot Example (con’t) At x = , z = (x m) / ? = ( 5) / = At x = , z = (x m) / ? = ( 5) / = From standardised normal distribution table, at z = P(?) = 1 = at z = P(?) = Hence, P( ? x ? ) = = (approx equal to by binomial distribution) ： ； :； ： Abcot Introduction to Sampling What is population in statistic ? A population in statistic refers to all items that have been chosen for study. What is a sample in statistic ? A sample in statistic refers to a portion chosen from a population, by which the data obtain can be used to infer on the actual performance of the population Population Sample 2 Sample 6 Sample 8 Sample 1 Sample 3 Sample 7 Sample 4 Sample 5 Abcot Symbols for population 和 sample For population, Population mean = m Population size = N Population 標(biāo)準(zhǔn)偏差 = ? For sample, Sample mean = X Sample size = n Sample 標(biāo)準(zhǔn)偏差 = S Sampling distribution a distribution of sample means If you take 10 samples out of the same populations, you will most likely end up with 10 different sample means 和 sample 標(biāo)準(zhǔn)偏差 s. A Sampling distribution describes the probability of all possible means of the samples taken from the same population. Population mean, m = 150 Sampling distribution mean, X Sample 2, mean X2 Sample 1, mean X1 Sample 3, mean X3 Sample n, mean Xn Collection of sample means Abcot m = 150 Population distribution with ? = 25 When sample size increases, the standard error (or the std deviation of sampling distribution) will get smaller. Sample distribution with std error ?x much less than 25 (when n = 30) Sample distribution with std error ?x 25 (when n = 5) Sampling distribution (con’t) Like all normal curve, the sampling distribution can be described by its mean, x 和 its 標(biāo)準(zhǔn)偏差 ?x (which is also known as standard error of the mean). As such, the standard error measures the extent to which we expect the means from different samples to vary due to chance error in sampling 流程 . Standard error = Population 標(biāo)準(zhǔn)偏差 / square root of sample size ?x = ? / n Abcot Central Limit Theorem 1. The mean x of the sampling distribution will approximately equal to the population mean regardless of the sample size. The larger the sample size, the closer the sample mean is towards the population mean. 2. The sampling distribution of the mean will approach normality regardless of the actual population distribution. 3. It assures us that the sampling distribution of the mean approaches normal as the sample size increases. It allow us to use sample statistics to make inferences about population parameters without knowing anything about the actual population distribution, other than what we can obtain from the sample. m = 150 Population distribution x = 150 Sampling distribution (n = 5) x = 150 Sampling distribution (n = 20) Abcot Example of sampling distribution The population distribution of annual ine of engineers is skewed negatively. This distribution has a mean of $48,000, 和 a 標(biāo)準(zhǔn)偏差 of $5000. If we draw a sample of 100 engineers, what is the probability that their average annual ine is $48700 和 more. m = $48K Population distribution ? = $5000 x = $48K Sampling distribution (n = 100) ?x = ? $ Since population mean is equal to sampling distribution mean (. central limit theorem), hence X = m = 48000 Sampling distribution mean = Standard error = ?x = ? / n = 5000 / 100 = 5000 / 10 = 500 Abcot Example of sampling distribution (con’t) x = $48K Sampling distribution (n = 100) ?x = 500 $ Therefore, mean = 48000 sigma = 500 X = 50000 Z = (48700 48000) / 500 = 700 / 500 = From the standardized normal distribution table, P(X ? $48700) = Therefore。 P(X ? $48700) = 1 = Thus, we have determined that it has only % chance for the average annual ine of 100 engineers to be more than $48700. Abcot Central Limit Theorem Exercise Break into 4 groups as below: Group 1: The population group. Group 2 to 4: The sample subgroup The population group will have 3 of their members throwing a single dices 60 times each. A total of 180 throws will be recorded 和 this data will be the population data. Each sample subgroup will have 3 of their members throwing 5 dices at one time, 和 collect the sum 和 average value of the particular throw. Each member is to conduct 20 throws 和obtain the sample mean of each throw. At the end of the exercise, a total of 180 sample means will be collected from the 3 subgroups. From the arrived data, plot the histogram 和 ment on the distribution of both the population 和 the samplings. Abcot The finite population multiplier Previously we say that: n x ?? ??e rro r St a n d a rdThis equation however applies only when the population is infinite or relatively large when pare to the sample size. In the case when the population is finite or relatively small when pare to the sample size, standard error is calculated as: 1n x ?????NnN??e rro r St a n d a rdFinite Population Multiplier Finite population multiplier with respect to population 和 sample size Rule of thumbs The finite population multiplier need only be included if population size to sample size ratio is less than 25. Abcot Confidence Interval “ Point estimates” A point estimate is a single number that is used to estimate an unknown population parameter. “實例 of point estimates” Sample mean, x as the estimator of the population mean, m. nxx ??Sample 標(biāo)準(zhǔn)偏差 , S as the estimator of the population 標(biāo)準(zhǔn)偏差 , ?. ? ?1nx x S ?? 2