Lecture Probability & statistics: Chapter 7 - Bùi Dương Hải

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Lecture "Probability & statistics - Chapter 7: Sampling" has contents: Sampling, sampling distribution, point estimate, acceptance interval, random sample, sampling distribution, acceptance interval.

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Nội dung Text: Lecture Probability & statistics: Chapter 7 - Bùi Dương Hải

Lecture 7. SAMPLING  Sampling  Sampling distribution  Point Estimate  Acceptance Interval  [1] Chapter 7. pp. 298 - 335 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1
Inferential Statistics  Inferential Statistics: Deduce information of Population from Sample data.  Population size: , = ( , ,…, )  Parameters: Population Mean Population proportion Proportion variance PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2
Example  Population: 4 8 5 7 6 7 6 2 5 8  Parameter: = 5.8 = 3.16  Proportion of “odd value” = = 0.4  Sample 1: 4 8 5 ̅ = 5.67, = 4.33, ̅ = = 0.33  Sample 2: 8 6 7 8 ̅ = 7.25, = 0.917, ̅ = =0 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3
7.1. Random Sample Population data: from census  Exactly  Maybe impossible  Difficult to gather  Costly, much time Sample data: from surveys  Possilbe to gather  Easier than census  Less cost and time PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4
Random Sample  Random Sample: Sample drawn from population that every elements are selected with equal probability, and independently.  For random sample  =⋯= =  =⋯= = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5
Sampling Method  Stratified sampling  Cluster sampling  Systematic sampling  Convenience sampling  Judgment sampling  [1] p.331 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6
7.2. Sampling Distribution  A sampling distribution is a distribution of all of the possible values of a statistic for a given sample selected from a population  Distribution of Sample Mean  Distribution of Sample Proportion  Distribution of Sample Variance PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7
Distribution of Sample Mean  Sample Mean (Random Sample) ∑ ̅=  ̅ is random variable  ̅ = ̅=  ̅ = ̅ =  ̅ = = Standard Error (S.E)  Sample mean has same expectation with , but smaller variance. PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8
Correction Factor  In case of large sample or finite population, n is relatively large in comparison to N  The correction factor is  ̅ = ×  ̅ = ×  In lectures: sample is not large PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9
Normal Distribution  If Population is Normal distributed: ~ ( , ) or Non-normal distributed but > 30 then:  ̅~ ( ̅, ̅ )  ̅~ , Ex. Population ~ (20,4 )  Sample = 16  ̅ ~ 20, 1 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10
Example Example 7.1. Workers’ salary is Normal distributed with Mean of 300 $ and Standard deviation of 20 $. (a) What is the probability that salary of a worker chosen randomly exceeds 305? (b) Random choose 10 workers, what is the probability that sample mean exceeds 305? (c) What is the probability that sample mean of 100 workers exceeds 305? (d) With the probability of 0.67, what is the maximum of sample mean of 10 workers? (e) With the probability of 0.67, what is the maximum of sample mean of 100 workers? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11
Distribution of Sample Proportion  Population Proportion = Probability =  Sample proportion = ̅ ( )  ̅ = ; ̅ =  With ≥ 100 1− ̅~ , Example 7.2. Probability that candidate pass the exam is 0.4. Find the probability that proportion of pass in 200 candidate is greater than 45% PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12
7.3. Acceptance Interval  Assume that Population is known  Parameter , , are known  Deduce for statistics in sample  With probability of 95%, 90%, or (1 − ) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13
Sample mean  With probability of 95% −1.96 < < 1.96 = 0.95 ̅− −1.96 < < 1.96 = 0.95 /  Acceptance interval 95% of sample mean − 1.96 < ̅< + 1.96  Or: ± 1.96 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14
Sample mean  In general, probability of 1 −  The acceptance interval for sample mean: − / < < + / or ± / Example 7.3. Worker income ($) is normal distribution with mean of 300 and variance of 400. What is the interval that average income of 25 workers falls into, with probability of 95%, 90%, 80% ? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15
Sample Proportion  If population parameter is known  is population proportion or probability  Acceptance interval of sample proportion ̅ ( − ) ( − ) − / < < + /  Or ( − ) ± / PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16
Sample Proportion Example 7.4. Probability that a visitor buying at least one item in the shopping mall is 0.3. (a) Find the probability that in 200 visitors, there are at least 65 customers. (b) At probability level of 95%, in 200 visitors, what are acceptance interval of relative frequency of number of buyers, and acceptance interval of number of buyers? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17
Key Concepts  Random Sample  Sampling Distribution  Acceptance Interval Exercise [1] Chapter 7  (309) 13, 16,  (320) 21, 27, 29  (326) 37, 38 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18