Statistics for Managers 5 th Edition

1. Statistics for Managers 5th Edition Chapter 7 Sampling Distributions and Sampling

2. Chapter Topics • Sampling Distributions • Sampling Distribution of the Mean • Sampling Distribution of the Proportion • The Situation ofFinite Populations

3. Sampling Distribution - Definition • A sampling distribution is a probability distribution that shows the relationship between all of the possible values that a sample statistic can assume and the corresponding probabilities. • The sampling distribution of the mean shows all of the possible values that the sample mean can assume and the corresponding probabilities.

4. (X) Sampling Distribution of X for n = 2 Samples X X P ( X ) P ( X ) 1, 2 1.5 1/6 1, 3 2.0 1/6 1, 4 2.5 1/6 2, 3 2.5 1/6 2, 4 3.0 1/6 3, 4 3.5 1/6 6/6 Sampling Distribution of the mean Population = 1, 2, 3, 4 so N = 4 1.5 1/6 or 2.0 1/6 2.5 2/6 3.0 1/6 3.5 1/6

5. ) ( S · _ X X = P mX 1.5(1/6 ) + 2.0 (1/6 ) + 2.5 (2/6 ) + 3.0 (1/6 ) + 3.5 (1/6 ) = = 2.5 = · 2 _ S ( ) ( ) sX _ X - mX P X (1.5 - 2.5)2 · 1/6 + (2.0 - 2.5)2 · 1/6 + (2.5 - 2.5)2 · 2/6+ (3.0 - 2.5)2 · 1/6 + (3.5 - 2.5)2 · 1/6 = = .645 Summary Measures For Sampling Distribution

6. Properties of Summary Measures m = m x • Population Mean Equal to • Sampling Mean • The Standard Error (standard deviation) of the Sampling distribution is Less than Population Standard Deviation • Formula:   =  Asnincrease,decrease. x x

7. Properties of the Mean • Unbiasedness • Mean of sampling distribution equals population mean • Efficiency • Sample mean comescloser to population mean than any other unbiased estimator • Consistency • Assample size increases, variation of sample mean from population mean decreases

8. Unbiasedness P(X) Unbiased Biased  X

9. Efficiency Sampling Distribution of Median P(X) Sampling Distribution of Mean X 

10. Consistency Larger sample size P(X) B Smaller sample size A X 

11. When the Population is Normal Population Distribution Central Tendency   _ = x Variation  Sampling Distributions  _ = x n n = 4X = 5 n =16X = 2.5 Sampling with Replacement

12. Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal regardless of shape of population

13. Central Limit Theorem For a population with a mean u and a standard deviation s , the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normallydistributed assuming that the sample size is sufficiently large (n ≥ 30) regardless of the shape of the distribution of the population.

14. When The Population is Not Normal m = m x Population Distribution Central Tendency  = 10 Variation  = 50 s X s = Sampling Distributions x n n =30X = 1.8 Sampling with Replacement n = 4X = 5

15. Example: Central Limit Theorem • The number of seconds it takes to complete a task is normally distributed with a mean of 150 seconds and a standard deviation of 10 seconds. • What is the probability that a particular task takes more than 152 seconds? • If a random sample of 100 tasks is drawn, what is the probability that the sample mean number of seconds it takes to complete them is more than 152?

16. Population Sampling Distribution of X P ( X > 152 when n = 100) = .0228 .0793 .4772 .4207 .0228 0 150 .20 152 Z X 0 150 152 2.00 X Z X - m 152 - 150 X - m 152 - 150 Z = = = 2.00 Z = = = .20 s 10 100 s 10 n Example: Central Limit Theorem m = 150 s = 10 P ( X > 152 ) = .4207

17. X number of successes = = P n sample size Population Proportions • Categorical variable (e.g., gender) • % population having a characteristic • If two outcomes, binomial distribution Possess or don’t possess characteristic • Sample proportion (ps)

18. Sampling Distribution of the Proportion • The sampling distribution of the proportion shows the relationship between all of the possible values the sample proportion can assume and the corresponding probabilities.

19. Population Proportions π = the proportion of the population having some characteristic • Sample proportion( p ) provides an estimate of π: • 0 ≤ p ≤ 1 • p has a binomial distribution

20. Sampling Distribution of p • Approximated by anormal distribution if n ≥ 50 where and Sampling Distribution P(p) .3 .2 .1 0 p 0 . 2 .4 .6 8 1 (where π = population proportion)

21. Sampling Distribution of Proportion Example • 40% of all patients experience dry mouth when taking a specific drug. You select a random sample of 200 patients. What is the probability that the sample proportion of patients with dry mouth would be between 40% & 43% ?

22. Example: Sampling Distribution of Proportion ´ - - . 40 ( 1 . 40 ) π ( 1 π ) 200 n  p - π .43 - .40 Z  = = .87 Sampling Distribution Standardized - Normal Distribution p = .0346  = 1 ..3078 p Z p = .40 .43  = 0 .87

23. Sampling from Finite Sample • Modify standard error if sample size (n) is large relative to population size (N ) • Use finite population correction factor (fpc) • Standard error with FPC

24. Sampling Distribution of Proportion from Finite Population Example • 40% of all 1000 patients experience dry mouth when taking a specific drug. You select a random sample of 200 patients. What is the probability that the sample proportion of patients with dry mouth would be between 40% & 43% ?

25. Solution* P(.40 £P£ .43) Sampling Distribution Standardized Distribution s = .0310 s = 1 P Z .3340 Z P .97 .43

26. Chapter Summary • Introduced sampling distributions • Described the sampling distribution of the mean • For normal populations • Using the Central Limit Theorem • Described the sampling distribution of a proportion • Calculated probabilities using sampling distributions • Described the use of the Finite Population Correction