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# Sampling distribution

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Name                                       Shakeel Nouman
Religion                                  Christian
Domicile                            Punjab (Lahore)
Contact #             0332-4462527. 0321-9898767
E.Mail                                sn_gcu@yahoo.com
sn_gcu@hotmail.com

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### Sampling distribution

1. 1. Sampling Distribution Slide 1 Shakeel Nouman M.Phil Statistics Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
2. 2. Sampling Distributions 6.1 6.2 Slide 2 The Sampling Distribution of the Sample Mean The Sampling Distribution of the Sample Proportion Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
3. 3. Sampling Distribution of the Sample Mean Slide 3 The sampling distribution of the sample mean is the probability distribution of the population of the sample means obtainable from all possible samples of size n from a population of size N Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
4. 4. Example: Sampling Annual % Return on 6 Stocks #1 Slide 4 • Population of the percent returns from six stocks – In order, the values of % return are: 10%, 20%, 30%, 40%, 50%, and 60% » Label each stock A, B, C, …, F in order of increasing % return » The mean rate of return is 35% with a standard deviation of 17.078% – Any one stock of these stocks is as likely to be picked as any other of the six » Uniform distribution with N = 6 » Each stock has a probability of being picked of 1/6 = 0.1667 Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
5. 5. Example: Sampling Annual % Return Slide 5 on 6 Stocks #2 Stock Stock A Stock B Stock C Stock D Stock E Stock F Total % Return 10 20 30 40 50 60 Frequency 1 1 1 1 1 1 6 Relative Frequency 1/6 1/6 1/6 1/6 1/6 1/6 1 Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
6. 6. Example: Sampling Annual % Return on 6 Stocks #3 Slide 6 • Now, select all possible samples of size n = 2 from this population of stocks of size N = 6 – Now select all possible pairs of stocks • How to select? – Sample randomly – Sample without replacement – Sample without regard to order Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
7. 7. Example: Sampling Annual % Return on 6 Stocks #4 Slide 7 • Result: There are 15 possible samples of size n = 2 • Calculate the sample mean of each and every sample Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
8. 8. Example: Sampling Annual % Return on 6 Stocks #5 Sample Mean 15 20 25 30 35 40 45 50 55 Slide 8 Relative Frequency Frequency 1 1/15 1 1/15 2 1/15 2 1/15 3 1/15 2 1/15 2 1/15 1 1/15 1 1/15 Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
9. 9. Observations • • Slide 9 Although the population of N = 6 stock returns has a uniform distribution, … … the histogram of n = 15 sample mean returns: 1. Seem to be centered over the sample mean return of 35%, and 2. Appears to be bell-shaped and less spread out than the histogram of individual returns Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
10. 10. General Conclusions Slide 10 • If the population of individual items is normal, then the population of all sample means is also normal • Even if the population of individual items is not normal, there are circumstances when the population of all sample means is normal (Central Limit Theorem)
11. 11. General Conclusions Continued 11 Slide • The mean of all possible sample means equals the population mean – That is, m = mx • The standard deviation sx of all sample means is less than the standard deviation of the population – That is, sx < s » Each sample mean averages out the high and the low measurements, and so are closer to m than many of the individual population measurements
12. 12. And the Empirical Rule Slide 12 • The empirical rule holds for the sampling distribution of the sample mean – 68.26% of all possible sample means are within (plus or minus) one standard deviation sx of m – 95.44% of all possible observed values of x are within (plus or minus) two sx of m – 99.73% of all possible observed values of x are within (plus or minus) three sx of m
13. 13. Properties of the Sampling Distribution of the Sample Mean #1 Slide 13 • If the population being sampled is normal, then so is the sampling distribution of the sample mean, x • The mean sx of the sampling distribution of x is mx = m • That is, the mean of all possible sample means is the same as the population mean Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
14. 14. Properties of the Sampling Distribution of the Sample Mean #2 Slide 14 • The variance s 2 of the sampling distribution of x is x s  2 x s2 n  That is, the variance of the sampling distribution x of is  directly proportional to the variance of the population, and  inversely proportional to the sample size Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
15. 15. Properties of the Sampling Distribution of the Sample Mean #3 Slide 15 • The standard deviation sx of the sampling distribution of x is sx  s n  That is, the standard deviation of the sampling distribution of x is  directly proportional to the standard deviation of the population, and  inversely proportional to the square root of the sample size Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
16. 16. Notes Slide 16 • The formulas for s2x and sx hold if the sampled population is infinite • The formulas hold approximately if the sampled population is finite but if N is much larger (at least 20 times larger) than the n (N/n ≥ 20) – x is the point estimate of m, and the larger the sample size n, the more accurate the estimate – Because as n increases, sx decreases as 1/√n » Additionally, as n increases, the more representative is the sample of the population • So, to reduce sx, take bigger samples!
17. 17. Reasoning from the Sampling Distribution Slide 17 • Recall from Chapter 2 mileage example, x = 31.5531 mpg for a sample of size n=49 – With s = 0.7992 • Does this give statistical evidence that the population mean m is greater than 31 mpg – That is, does the sample mean give evidence that m is at least 31 mpg • Calculate the probability of observing a sample mean that is greater than or equal to 31.5531 mpg if m = 31 mpg – Want P(x > 31.5531 if m = 31)
18. 18. Reasoning from the Sampling Distribution Continued Slide 18 • Use s as the point estimate for s so that sx  s n  0.7992  0.1143 49 • Then  31.5531 m x   Px  31.5531if m  31  P z    sx   31.5531 31    P z   0.1143    Pz  4.84 • But z = 4.84 is off the standard normal table • The largest z value in the table is 3.09, which has a right hand tail area of 0.001 Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
19. 19. Reasoning from the Sampling Distribution #3 Slide 19 • z = 4.84 > 3.09, so P(z ≥ 4.84) < 0.001 • That is, if m = 31 mpg, then fewer than 1 in 1,000 of all possible samples have a mean at least as large as observed • Have either of the following explanations: – If m is actually 31 mpg, then very unlucky in picking this sample OR – Not unlucky, but m is not 31 mpg, but is really larger • Difficult to believe such a small chance would occur, so conclude that there is strong evidence that m does not equal 31 mpg – Also, m is, in fact, actually larger than 31 mpg Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
20. 20. Central Limit Theorem Slide 20 • Now consider sampling a non-normal population • Still have: m x  m and sx  s n – Exactly correct if infinite population – Approximately correct if population size N finite but much larger than sample size n » Especially if N ≥ 20  n • But if population is non-normal, what is the shape of the sampling distribution of the sample mean? – Is it normal, like it is if the population is normal? – Yes, the sampling distribution is approximately normal if the sample is large enough, even if the population is non-normal » By the “Central Limit Theorem” Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
21. 21. The Central Limit Theorem #2 21 Slide • No matter what is the probability distribution that describes the population, if the sample size n is large enough, then the population of all possible sample means is approximately normal with mean m x  m and standard deviation s x  s n • Further, the larger the sample size n, the closer the sampling distribution of the sample mean is to being normal – In other words, the larger n, the better the approximation Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
22. 22. How Large? Slide 22 • How large is “large enough?” • If the sample size is at least 30, then for most sampled populations, the sampling distribution of sample means is approximately normal – Here, if n is at least 30, it will be assumed that the sampling distribution of x is approximately normal » If the population is normal, then the sampling distribution of x is normal no regardless of the sample size
23. 23. Unbiased Estimates Slide 23 • A sample statistic is an unbiased point estimate of a population parameter if the mean of all possible values of the sample statistic equals the population parameter • x is an unbiased estimate of m because mx=m – In general, the sample mean is always an unbiased estimate of m – The sample median is often an unbiased estimate of m » But not always Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
24. 24. Unbiased Estimates Continued 24 Slide • The sample variance s2 is an unbiased estimate of s2 – That is why s2 has a divisor of n – 1 and not n • However, s is not an unbiased estimate of s – Even so, the usual practice is to use s as an estimate of s Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
25. 25. Minimum Variance Estimates 25 Slide • Want the sample statistic to have a small standard deviation – All values of the sample statistic should be clustered around the population parameter » Then, the statistic from any sample should be close to the population parameter » Given a choice between unbiased estimates, choose one with smallest standard deviation » The sample mean and the sample median are both unbiased estimates of m » The sampling distribution of sample means generally has a smaller standard deviation than that of sample medians Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
26. 26. Finite Populations Slide 26 • If a finite population of size N is sampled randomly and without replacement, must use the “finite population correction” to calculate the correct standard deviation of the sampling distribution of the sample mean – If N is less than 20 times the sample size, that is, if N < 20  n – Otherwise sx  s n but instead s x  s n Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
27. 27. Finite Populations ContinuedSlide 27 • The finite population correction is N n N 1 • and the standard error is sx  s n N n N 1 Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
28. 28. Sampling Distribution of the Sample Proportion Slide 28 The probability distribution of all possible sample proportions is the sampling distribution of the sample proportion ˆ p If a random sample of size n is taken from a population then the sampling distribution of is ˆ p  approximately normal, if n is large  has mean m ˆ  p p p1  p   has standard deviation s ˆp  n p where p is the population proportion andˆ is a sampled proportion
29. 29. Slide 29 Name Religion Domicile Contact # E.Mail M.Phil (Statistics) Shakeel Nouman Christian Punjab (Lahore) 0332-4462527. 0321-9898767 sn_gcu@yahoo.com sn_gcu@hotmail.com GC University, . (Degree awarded by GC University) M.Sc (Statistics) Statitical Officer (BS-17) (Economics & Marketing Division) GC University, . (Degree awarded by GC University) Livestock Production Research Institute Bahadurnagar (Okara), Livestock & Dairy Development Department, Govt. of Punjab Sampling Distribution By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer