Instructor Nahid Farnaz Nhn North South University Statistics

Instructor: Nahid Farnaz (Nhn) North South University Statistics for Business and Economics 6 th Edition Sampling and Sampling Distributions Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -1

Populations and Samples § A Population is the set of all items or individuals of interest § Examples: All likely voters in the next election All parts produced today All sales receipts for November § A Sample is a subset of the population § Examples: 1000 voters selected at random for interview A few parts selected for destructive testing Random receipts selected for audit Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -2

Why Sample? § Less time consuming than a census § Less costly to administer than a census § It is possible to obtain statistical results of a sufficiently high precision based on samples. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -3

Simple Random Samples § Every object in the population has an equal chance of being selected § Objects are selected independently § Samples can be obtained from a table of random numbers or computer random number generators § A simple random sample is the ideal against which other sample methods are compared Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -4

Sampling Distributions § A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -5

Chapter Outline Sampling Distributions Sampling Distribution of Sample Mean Sampling Distribution of Sample Proportion Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Sampling Distribution of Sample Variance Chap 7 -6

Developing a Sampling Distribution § Assume there is a population … § Population size N=4 A B C D § Random variable, X, is age of individuals § Values of X: 18, 20, 22, 24 (years) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -7

Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: P(x). 25 0 18 20 22 24 A B C D x Uniform Distribution Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -8

Developing a Sampling Distribution (continued) Now consider all possible samples of size n = 2 16 Sample Means 16 possible samples (sampling with replacement) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -9

Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means Distribution 16 Sample Means _ P(X). 3. 2. 1 0 18 19 20 21 22 23 (no longer uniform) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 24 _ X Chap 7 -10

Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -11

Expected Value of Sample Mean § Let X 1, X 2, . . . Xn represent a random sample from a population § The sample mean value of these observations is defined as Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -12

Standard Error of the Mean § Different samples of the same size from the same population will yield different sample means § A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: § Note that the standard error of the mean decreases as the sample size increases Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -13

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 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -14

Z-value for Sampling Distribution of the Mean § Z-value for the sampling distribution of where: : = sample mean = population standard deviation n = sample size Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -15

Sampling Distribution Properties Normal Population Distribution § (i. e. is unbiased ) Normal Sampling Distribution (has the same mean) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -16

Sampling Distribution Properties (continued) § For sampling with replacement: As n increases, decreases Larger sample size Smaller sample size Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -17

Example 7. 2: “Executive Salary Distributions” § Suppose that the annual percentage salary increases for the chief executive officers of all midsize corporations are normally distributed with mean 12. 2% and standard deviation 3. 6%. § A random sample of nine observations is obtained from this population and the sample mean computed. § What is the probability that the sample mean will be less than 10%? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -18

Example 7. 3 “Spark Plug Life” § A spark plug manufacturer claims that the lives of its plugs are normally distributed with mean 36, 000 miles and standard deviation 4000 miles. § A random sample of 16 plugs had an average life of 34, 500 miles. § If the manufacturer’s claim is correct, what is the probability of finding a sample mean of 34, 500 or less? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -19

If the Population is not Normal § We can apply the Central Limit Theorem: § 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. Properties of the sampling distribution: and Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -20

Central Limit Theorem As the sample size gets large enough… n↑ Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. the sampling distribution becomes almost normal regardless of shape of population Chap 7 -21

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 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Larger sample size Chap 7 -22

Example § 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? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -23

Example (continued) Solution: § Even if the population is not normally distributed, the central limit theorem can be used (n > 25) § … so the sampling distribution of approximately normal § … with mean is = 8 § …and standard deviation Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -24

Example (continued) Solution (continued): Population Distribution ? ? ? Sampling Distribution Standard Normal Distribution Sample ? X . 1915 +. 1915 Standardize 7. 8 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 8. 2 -0. 5 Z Chap 7 -25

Acceptance Intervals § Goal: determine a range within which sample means are likely to occur, given a population mean and variance § By the Central Limit Theorem, we know that the distribution of X is approximately normal if n is large enough, with mean μ and standard deviation § Let zα/2 be the z-value that leaves area α/2 in the upper tail of the normal distribution (i. e. , the interval - zα/2 to zα/2 encloses probability 1 – α) § Then is the interval that includes X with probability 1 – α Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -26

Sampling Distributions of Sample Proportions Sampling Distribution of Sample Mean Sampling Distribution of Sample Proportion Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Sampling Distribution of Sample Variance Chap 7 -27

Population Proportions, P P = the proportion of the population having some characteristic § Sample proportion ( ) provides an estimate of P: § 0≤ § ≤ 1 has a binomial distribution, but can be approximated by a normal distribution when n. P(1 – P) > 9 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -28

^ Sampling Distribution of P § Normal approximation: Sampling Distribution. 3. 2. 1 0 0 . 2 . 4 . 6 8 1 Properties: and (where P = population proportion) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -29

Z-Value for Proportions Standardize to a Z value with the formula: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -30

Example 7. 8 Business Course Selection (Prob. Of Sample Proportion) § It has been estimated that 43% of business graduates believe that a course in business ethics is important. § Find the probability that more than one-half of a random sample of 80 business graduates have this belief. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -31

Example (for Practice) § If the true proportion of voters who support Proposition A is P =. 4, what is the probability that a sample of size 200 yields a sample proportion between. 40 and. 45? § i. e. : if P =. 4 and n = 200, what is P(. 40 ≤ Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. ≤. 45) ? Chap 7 -32

Example (continued) § Find if P =. 4 and n = 200, what is P(. 40 ≤ ≤. 45) ? : Convert to standard normal: Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -33

Example (continued) § if p =. 4 and n = 200, what is P(. 40 ≤ ≤. 45) ? Use standard normal table: P(0 ≤ Z ≤ 1. 44) =. 4251 Standardized Normal Distribution Sampling Distribution . 4251 Standardize . 40 . 45 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 0 1. 44 Z Chap 7 -34

Sampling Distributions of Sample Proportions Sampling Distribution of Sample Mean Sampling Distribution of Sample Proportion Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Sampling Distribution of Sample Variance Chap 7 -35

Sample Variance § Let x 1, x 2, . . . , xn be a random sample from a population. The sample variance is § the square root of the sample variance is called the sample standard deviation § the sample variance is different for different random samples from the same population Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -36

Sampling Distribution of Sample Variances § The sampling distribution of s 2 has mean σ2 § If the population distribution is normal, then § If the population distribution is normal then has a 2 distribution with n – 1 degrees of freedom Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -37

The Chi-square Distribution § The chi-square distribution is a family of distributions, depending on degrees of freedom: § d. f. = n – 1 0 4 8 12 16 20 24 28 d. f. = 1 2 0 4 8 12 16 20 24 28 d. f. = 5 2 0 4 8 12 16 20 24 28 2 d. f. = 15 § Text Table 7 contains chi-square probabilities Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -38

Degrees of Freedom (df) Idea: Number of observations that are free to vary after sample mean has been calculated Example: Suppose the mean of 3 numbers is 8. 0 Let X 1 = 7 Let X 2 = 8 What is X 3? If the mean of these three values is 8. 0, then X 3 must be 9 (i. e. , X 3 is not free to vary) Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2 (2 values can be any numbers, but the third is not free to vary for a given mean) Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -39

Chi-square Example § A commercial freezer must hold a selected temperature with little variation. Specifications call for a standard deviation of no more than 4 degrees (a variance of 16 degrees 2). § A sample of 14 freezers is to be tested § What is the upper limit (K) for the sample variance such that the probability of exceeding this limit, given that the population standard deviation is 4, is less than 0. 05? Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -40

Finding the Chi-square Value Is chi-square distributed with (n – 1) = 13 degrees of freedom § Use the chi-square distribution with area 0. 05 in the upper tail: 213 = 22. 36 (α =. 05 and 14 – 1 = 13 d. f. ) probability α =. 05 2 213 = 22. 36 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -41

Chi-square Example (continued) 213 = 22. 36 (α =. 05 and 14 – 1 = 13 d. f. ) So: or (where n = 14) so If s 2 from the sample of size n = 14 is greater than 27. 52, there is strong evidence to suggest the population variance exceeds 16. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -42

Example 7. 9 George Sampson is responsible for quality assurance at Integrated Electronics. He has asked you to establish a quality monitoring process for the manufacturer of control device A. The variability of the electrical resistance, measured in ohms, is critical for this device. Manufacturing standards specify a standard deviation of 3. 6 and normal distribution. The monitoring process requires that a random sample of n=6 observations be obtained from the population of devices and the sample variance be computed. Determine an upper limit for the sample variance such that the probability of exceeding this limit, given a population S. D. of 3. 6, is less than 0. 05. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 7 -43