Chapter 7 Sampling and Sampling Distributions Yandell Econ

Chapter 7 Sampling and Sampling Distributions Yandell – Econ 216 Chap 7 -1

Chapter Goals After completing this chapter, you should be able to: _ for the n Determine the mean and standard deviation sampling distribution of the sample mean, X n Determine the mean and standard deviation for the sampling distribution of the sample proportion, p n Describe the Central Limit Theorem and its importance n Apply sampling distributions for both X and p Yandell – Econ 216 _ Chap 7 -2

Sampling Distribution n Yandell – Econ 216 A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population Chap 7 -3

Developing a Sampling Distribution n Assume there is a population … n Population size N=4 n n A B C D Random variable, x, is age of individuals Values of x: 18, 20, 22, 24 (years) Yandell – Econ 216 Chap 7 -4

Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: P(x). 3. 2. 1 0 18 19 20 21 22 23 24 A Yandell – Econ 216 B C Equally likely x D Chap 7 -5

Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 16 Sample Means 16 possible samples (sampling with replacement) Yandell – Econ 216 Chap 7 -6

Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Distribution 16 Sample Means _ P(X). 3. 2. 1 0 Yandell – Econ 216 listen 18 19 20 21 22 23 (no longer uniform) 24 _ X Chap 7 -7

Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Yandell – Econ 216 Chap 7 -8

Comparing the Population with its Sampling Distribution Population N=4 Sample Means Distribution n=2 _ P(X). 3 . 2 . 1 0 18 20 22 24 A B C D Yandell – Econ 216 X 0 18 19 20 21 22 23 24 _ X Chap 7 -9

If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and Yandell – Econ 216 Chap 7 -10

Z-value for Sampling Distribution of X n Z-value for the sampling distribution of where: Yandell – Econ 216 : = sample mean = population standard deviation n = sample size Chap 7 -11

Finite Population Correction n Apply the Finite Population Correction if: n the sample is large relative to the population (n is greater than 5% of N) and… n Sampling is without replacement Then Yandell – Econ 216 Chap 7 -12

Sampling Distribution Properties Normal Population Distribution n (i. e. is unbiased ) Normal Sampling Distribution (has the same mean) listen Yandell – Econ 216 Chap 7 -13

Sampling Distribution Properties (continued) n For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size Yandell – Econ 216 Chap 7 -14

If the Population is not Normal n We can apply the Central Limit Theorem: n n n Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough …and the sampling distribution will have and Yandell – Econ 216 Chap 7 -15

Central Limit Theorem As the sample size gets large enough… n↑ the sampling distribution becomes almost normal regardless of shape of population listen Yandell – Econ 216 Chap 7 -16

If the Population is not Normal (continued) Sampling distribution properties: Population Distribution Central Tendency Variation Sampling Distribution (becomes normal as n increases) Smaller sample size Larger sample size (Sampling with replacement) Yandell – Econ 216 Chap 7 -17

How Large is Large Enough? n n n Yandell – Econ 216 For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 For normal population distributions, the sampling distribution of the mean is always normally distributed Chap 7 -18

Example n n Suppose a population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7. 8 and 8. 2? Yandell – Econ 216 Chap 7 -19

Example (continued) Solution: n n Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of approximately normal n … with mean n …and standard deviation Yandell – Econ 216 is = 8 Chap 7 -20

Example (continued) Solution (continued): Population Distribution ? ? ? Sampling Distribution Sample ? X Yandell – Econ 216 Standard Normal Distribution . 1554 +. 1554 Standardize 7. 8 8. 2 -0. 4 Z Chap 7 -21

Population Proportions, π π = the proportion of population having some characteristic n n Sample proportion ( p ) provides an estimate of π : If two outcomes, p has a binomial distribution Yandell – Econ 216 Chap 7 -22

Sampling Distribution of p n Approximated by a normal distribution if: n P( p ). 3. 2. 1 0 0 Sampling Distribution . 2 . 4 . 6 8 1 p where and listen (where π = population proportion) Yandell – Econ 216 Chap 7 -23

Z-Value for Proportions Standardize p to a Z value with the formula: n If sampling is without replacement and n is greater than 5% of the population size, then must use the finite population correction factor: Yandell – Econ 216 Chap 7 -24

Example n n If the true proportion of voters who support Proposition A is π =. 4, what is the probability that a sample of size 200 yields a sample proportion between. 40 and. 45? i. e. : if π =. 4 and n = 200, what is P(. 40 ≤ p ≤. 45) ? Yandell – Econ 216 Chap 7 -25

Example (continued) if π =. 4 and n = 200, what is P(. 40 ≤ p ≤. 45) ? n Find : Convert to standard normal: Yandell – Econ 216 Chap 7 -26

Example (continued) n if π =. 4 and n = 200, what is P(. 40 ≤ p ≤. 45) ? Use standard normal table: P(0 ≤ Z ≤ 1. 44) =. 4251 Standardized Normal Distribution Sampling Distribution . 4251 Standardize . 40 Yandell – Econ 216 . 45 p 0 1. 44 Z Chap 7 -27

Chapter Summary n n Introduced sampling distributions Described the sampling distribution of the mean n n For normal populations Using the Central Limit Theorem Described the sampling distribution of a proportion Calculated probabilities using sampling distributions Discussed sampling from finite populations Yandell – Econ 216 Chap 7 -28

Demonstration of Central Limit Theorem n Visit this website to see a Normal Distribution Demonstration (watch a live electronic quincunx demo) http: //www. mathsisfun. com/data/quincunx. html Yandell – Econ 216 Chap 7 -29