【正文】 occurence a = 1 – Shaded area Applicable when: ? Sample size 30 ? 標(biāo)準(zhǔn)偏差 is unknown ? Population distribution is at least approximately normally distributed 0 . 7 5 0 . 9 0 0 . 9 5 0 . 9 7 5 0 . 9 9 0 . 9 9 5 0 . 9 9 9 51 1 . 0 0 0 0 3 . 0 7 7 7 6 . 3 1 3 7 1 2 . 7 0 6 2 3 1 . 8 2 1 0 6 3 . 6 5 5 9 6 3 6 . 5 7 7 62 0 . 8 1 6 5 1 . 8 8 5 6 2 . 9 2 0 0 4 . 3 0 2 7 6 . 9 6 4 5 9 . 9 2 5 0 3 1 . 5 9 9 83 0 . 7 6 4 9 1 . 6 3 7 7 2 . 3 5 3 4 3 . 1 8 2 4 4 . 5 4 0 7 5 . 8 4 0 8 1 2 . 9 2 4 44 0 . 7 4 0 7 1 . 5 3 3 2 2 . 1 3 1 8 2 . 7 7 6 5 3 . 7 4 6 9 4 . 6 0 4 1 8 . 6 1 0 15 0 . 7 2 6 7 1 . 4 7 5 9 2 . 0 1 5 0 2 . 5 7 0 6 3 . 3 6 4 9 4 . 0 3 2 1 6 . 8 6 8 56 0 . 7 1 7 6 1 . 4 3 9 8 1 . 9 4 3 2 2 . 4 4 6 9 3 . 1 4 2 7 3 . 7 0 7 4 5 . 9 5 8 77 0 . 7 1 1 1 1 . 4 1 4 9 1 . 8 9 4 6 2 . 3 6 4 6 2 . 9 9 7 9 3 . 4 9 9 5 5 . 4 0 8 18 0 . 7 0 6 4 1 . 3 9 6 8 1 . 8 5 9 5 2 . 3 0 6 0 2 . 8 9 6 5 3 . 3 5 5 4 5 . 0 4 1 49 0 . 7 0 2 7 1 . 3 8 3 0 1 . 8 3 3 1 2 . 2 6 2 2 2 . 8 2 1 4 3 . 2 4 9 8 4 . 7 8 0 910 0 . 6 9 9 8 1 . 3 7 2 2 1 . 8 1 2 5 2 . 2 2 8 1 2 . 7 6 3 8 3 . 1 6 9 3 4 . 5 8 6 811 0 . 6 9 7 4 1 . 3 6 3 4 1 . 7 9 5 9 2 . 2 0 1 0 2 . 7 1 8 1 3 . 1 0 5 8 4 . 4 3 6 912 0 . 6 9 5 5 1 . 3 5 6 2 1 . 7 8 2 3 2 . 1 7 8 8 2 . 6 8 1 0 3 . 0 5 4 5 4 . 3 1 7 813 0 . 6 9 3 8 1 . 3 5 0 2 1 . 7 7 0 9 2 . 1 6 0 4 2 . 6 5 0 3 3 . 0 1 2 3 4 . 2 2 0 914 0 . 6 9 2 4 1 . 3 4 5 0 1 . 7 6 1 3 2 . 1 4 4 8 2 . 6 2 4 5 2 . 9 7 6 8 4 . 1 4 0 315 0 . 6 9 1 2 1 . 3 4 0 6 1 . 7 5 3 1 2 . 1 3 1 5 2 . 6 0 2 5 2 . 9 4 6 7 4 . 0 7 2 816 0 . 6 9 0 1 1 . 3 3 6 8 1 . 7 4 5 9 2 . 1 1 9 9 2 . 5 8 3 5 2 . 9 2 0 8 4 . 0 1 4 917 0 . 6 8 9 2 1 . 3 3 3 4 1 . 7 3 9 6 2 . 1 0 9 8 2 . 5 6 6 9 2 . 8 9 8 2 3 . 9 6 5 118 0 . 6 8 8 4 1 . 3 3 0 4 1 . 7 3 4 1 2 . 1 0 0 9 2 . 5 5 2 4 2 . 8 7 8 4 3 . 9 2 1 719 0 . 6 8 7 6 1 . 3 2 7 7 1 . 7 2 9 1 2 . 0 9 3 0 2 . 5 3 9 5 2 . 8 6 0 9 3 . 8 8 3 320 0 . 6 8 7 0 1 . 3 2 5 3 1 . 7 2 4 7 2 . 0 8 6 0 2 . 5 2 8 0 2 . 8 4 5 3 3 . 8 4 9 621 0 . 6 8 6 4 1 . 3 2 3 2 1 . 7 2 0 7 2 . 0 7 9 6 2 . 5 1 7 6 2 . 8 3 1 4 3 . 8 1 9 322 0 . 6 8 5 8 1 . 3 2 1 2 1 . 7 1 7 1 2 . 0 7 3 9 2 . 5 0 8 3 2 . 8 1 8 8 3 . 7 9 2 223 0 . 6 8 5 3 1 . 3 1 9 5 1 . 7 1 3 9 2 . 0 6 8 7 2 . 4 9 9 9 2 . 8 0 7 3 3 . 7 6 7 624 0 . 6 8 4 8 1 . 3 1 7 8 1 . 7 1 0 9 2 . 0 6 3 9 2 . 4 9 2 2 2 . 7 9 7 0 3 . 7 4 5 425 0 . 6 8 4 4 1 . 3 1 6 3 1 . 7 0 8 1 2 . 0 5 9 5 2 . 4 8 5 1 2 . 7 8 7 4 3 . 7 2 5 126 0 . 6 8 4 0 1 . 3 1 5 0 1 . 7 0 5 6 2 . 0 5 5 5 2 . 4 7 8 6 2 . 7 7 8 7 3 . 7 0 6 727 0 . 6 8 3 7 1 . 3 1 3 7 1 . 7 0 3 3 2 . 0 5 1 8 2 . 4 7 2 7 2 . 7 7 0 7 3 . 6 8 9 528 0 . 6 8 3 4 1 . 3 1 2 5 1 . 7 0 1 1 2 . 0 4 8 4 2 . 4 6 7 1 2 . 7 6 3 3 3 . 6 7 3 929 0 . 6 8 3 0 1 . 3 1 1 4 1 . 6 9 9 1 2 . 0 4 5 2 2 . 4 6 2 0 2 . 7 5 6 4 3 . 6 5 9 530 0 . 6 8 2 8 1 . 3 1 0 4 1 . 6 9 7 3 2 . 0 4 2 3 2 . 4 5 7 3 2 . 7 5 0 0 3 . 6 4 6 0165。 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(shí)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 ?? 2Abcot What does 95% confidence intervals means ? It defines 95% of the time, the average value of a random sampling will fall within a value range which is +/ standard error from the sample mean. 為什么 standard error ? Referring to the standardized normal distribution table, when z = , the associated probability is (or %) as below: But this is a 2 tails interval (. +/ ), hence we need to minus off another % from the other end, giving a total coverage of 95%. % uncovered at one end + % uncovered at another end Abcot Equation for puting confidence intervals For continuous data: ? ?? ?s i z e S a m p l e n)c o n f i d e n c e 9 9 % ( f o r 2 . 5 8 )c o n f i d e n c e 9 5 % ( f o r 1 . 9 6 Za v e r a g e S a m p l e X E r r o r S t dE r r o r S t dZ x l i m i t c o n f i d e n c e L o w e rE r r o r S t dZ x l i m i t c o n f i d e n c e Up p e r????????u n k n o wn ) is ( i f So r ??nnFor discrete data: ? ?? ?s i z e S a m p l e nd a t a c o n t i n u o u s as s a m e Zp)1 ( i . e . f a i l u r e of P r o p o r t i o n qs u c c e s s of P r o p o r t i