Business Statistics:

Transcription

1 Chapter 7 Student Lecture Notes 7-1 Department of Quantitative Methods & Information Systems Business Statistics: Chapter 7 Introduction to Sampling Distributions QMIS 220 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should be able to: Define the concept of sampling error Determine the mean and standard deviation _ for the sampling distribution of the sample mean, Determine the mean and standard deviation for the _ sampling distribution of the sample proportion, p Describe the Central Limit Theorem and its importance Apply sampling distributions for both and p Chap 7-2

2 Chapter 7 Student Lecture Notes 7-2 Review: Inferential Statistics Inferential statistics Drawing conclusions and/or making decisions concerning a population based only on sample data Consists of methods that use sample results to help make decisions or predictions about a population. Elections Chap 7-3 Review: Inferential Statistics Sample statistics Population parameters (known) Inference (unknown, but can be estimated from sample evidence) Sample Population Chap 7-4

3 Chapter 7 Student Lecture Notes 7-3 Review: Inferential Statistics Drawing conclusions and/or making decisions concerning a population based on sample results. Estimation e.g., Estimate the population mean weight using the sample mean weight Hypothesis Testing e.g., Use sample evidence to test the claim that the population mean weight is 120 pounds Chap 7-5 Review: Key Definitions A population is the entire collection of things under consideration A parameter is a summary measure computed to describe a characteristic of the population A sample is a portion of the population selected for analysis A statistic is a summary measure computed to describe a characteristic of the sample Chap 7-6

4 Chapter 7 Student Lecture Notes 7-4 Review: Population vs. Sample Population a b c d ef gh i jk l m n o p q rs t u v w y z Sample b c g i n o r u y Chap 7-7 Review: 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. Chap 7-8

5 Chapter 7 Student Lecture Notes 7-5 Review: Sampling Techniques Sampling Techniques Nonstatistical Sampling Convenience Judgment Simple Random Statistical Sampling Stratified Systematic Cluster Chap 7-9 Review: Statistical Sampling Items of the sample are chosen based on known or calculable probabilities Statistical Sampling (Probability Sampling) Simple Random Stratified Systematic Cluster Chap 7-10

6 Chapter 7 Student Lecture Notes 7-6 Simple Random Sampling Every possible sample of a given size has an equal chance of being selected Selection may be with replacement or without replacement The sample can be obtained using a table of random numbers or computer random number generator Chap 7-11 Stratified Random Sampling Divide population into subgroups (called strata) according to some common characteristic Select a simple random sample from each subgroup Combine samples from subgroups into one Population Divided into 4 strata Sample Chap 7-12

7 Chapter 7 Student Lecture Notes 7-7 Systematic Random Sampling Decide on sample size: n Divide frame of N individuals into groups of k individuals: k=n/n Randomly select one individual from the 1 st group Select every k th individual thereafter N = 64 n = 8 k = 8 First Group Chap 7-13 Cluster Sampling Divide population into several clusters, each representative of the population Select a simple random sample of clusters All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique Population divided into 16 clusters. Randomly selected clusters for sample Chap 7-14

8 Chapter 7 Student Lecture Notes 7-8 Eamples of poor samplings The technique of sampling has been widely used, both properly and improperly, in the area of politics. During the 1936 presidential race where the Literary Digest predicted Alf Landon to win the election over Franklin D. Roosevelt. Chap 7-15 Sampling Error So far, we have stressed the benefits of drawing a sample from a population. However, in statistics, as in life, there's no such thing as a free lunch. By sampling, we epose ourselves to errors that can lead to inaccurate conclusions about the population. The type of error that a statistician is most concerned about is called sampling error. Chap 7-16

9 Chapter 7 Student Lecture Notes 7-9 Sampling Error Sample Statistics are used to estimate Population Parameters Problems: e: X is an estimate of the population mean, μ Different samples provide different estimates of the population parameter Sample results have potential variability, thus sampling error eits Chap 7-17 Sampling Error As the entire population is rarely measured, the sampling error cannot be directly calculated. With inferential statistics, we'll be able to assign probabilities to certain amounts of sampling error later. It occurs when we select a sample that is not a perfect match to the entire population. Sampling errors are a small price to pay to avoid measuring an entire population. Chap 7-18

10 Chapter 7 Student Lecture Notes 7-10 Sampling Error One way to reduce the sampling error of a statistical study is to increase the size of the sample. In general, the larger the sample size, the smaller the sampling error. If you increase the sample size until it reaches the size of the population, then the sampling error will be reduced to 0. But in doing so, we lose the benefits of sampling. Chap 7-19 Calculating Sampling Error Sampling Error: The difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population Eample: (for the mean) SamplingError - μ where: sample mean μ population mean Chap 7-20

11 Chapter 7 Student Lecture Notes 7-11 Review Population mean: Sample Mean: μ N i n i where: μ = Population mean = sample mean i = Values in the population or sample N = Population size n = sample size Chap 7-21 Eample If the population mean is μ = 98.6 degrees and a sample of n = 5 temperatures yields a sample mean of = 99.2 degrees, then the sampling error is Chap 7-22

12 Chapter 7 Student Lecture Notes 7-12 Sampling Errors Different samples will yield different sampling errors The sampling error may be positive or negative ( may be greater than or less than μ) The epected sampling error decreases as the sample size increases Chap 7-23 Sampling Distribution A sampling distribution is a distribution of the possible values of a statistic for a given size sample selected from a population Chap 7-24

13 Chapter 7 Student Lecture Notes 7-13 Developing a Sampling Distribution Assume there is a population Population size N=4 Random variable,, is age of individuals Values of : 18, 20, 22, 24 (years) A B C D Chap 7-25 Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: Chap 7-26

14 Chapter 7 Student Lecture Notes 7-14 Developing a Sampling Distribution Now consider all possible samples of size n=2 (continued) Chap 7-27 Developing a Sampling Distribution Sampling Distribution of All Sample Means (continued) Chap 7-28

15 Chapter 7 Student Lecture Notes 7-15 Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: Chap 7-29 Comparing the Population with its Sampling Distribution Chap 7-30

16 Chapter 7 Student Lecture Notes 7-16 Properties of a Sampling Distribution For any population, the average value of all possible sample means computed from all possible random samples of a given size from the population is equal to the population mean: μ μ The standard deviation of the possible sample means computed from all random samples of size n is equal to the population standard deviation divided by the square root of the sample size: σ σ n Theorem 1 Theorem 2 Chap 7-31 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 μ μ and σ σ n Theorem 3 Chap 7-32

17 Chapter 7 Student Lecture Notes 7-17 z-value for Sampling Distribution of Z-value for the sampling distribution of : z ( μ) σ n where: μ σ = sample mean = population mean = population standard deviation n = sample size Chap 7-33 Finite Population Correction Apply the Finite Population Correction if: the sample is large relative to the population (n is greater than 5% of N) and Sampling is without replacement Then ( μ) z σ N n n N 1 Chap 7-34

18 Chapter 7 Student Lecture Notes 7-18 Sampling Distribution Properties The sample mean is an unbiased estimator Normal Population Distribution μ μ Normal Sampling Distribution (has the same mean) μ Chap 7-35 μ Sampling Distribution Properties The sample mean is a consistent estimator (the value of becomes closer to μ as n increases): (continued) As n increases, σ σ/ n decreases Small sample size Population Larger sample size Chap 7-36 μ

19 Chapter 7 Student Lecture Notes 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 approimately normal as long as the sample size is large enough and the sampling distribution will have μ μ and σ σ n Theorem 4 Chap 7-37 Central Limit Theorem As the sample size gets large enough n the sampling distribution becomes almost normal regardless of shape of population Chap 7-38

20 Chapter 7 Student Lecture Notes 7-20 If the Population is not Normal Sampling distribution properties: Central Tendency Variation μ μ σ σ n (Sampling with replacement) Population Distribution Sampling Distribution (becomes normal as n increases) Smaller sample size (continued) Larger sample size Chap 7-39 μ μ How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 is sufficient For normal population distributions, the sampling distribution of the mean is always normally distributed Chap 7-40

21 Chapter 7 Student Lecture Notes 7-21 Eample 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? Chap 7-41 Solution: Eample (continued) Chap 7-42

22 Chapter 7 Student Lecture Notes 7-22 Eample Solution (continued) -- find z-scores: (continued) Chap 7-43 Population Proportions, π π = the proportion of the population having some characteristic Sample proportion ( p ) provides an estimate of π : p n number of successesinthe sample samplesize If two outcomes, p has a binomial distribution Chap 7-44

23 Chapter 7 Student Lecture Notes 7-23 Sampling Distribution of p Approimated by a normal distribution if: nπ 5 n(1 π) 5 Sampling Distribution P( p ) p where μ p π and σ p π(1 π) n (where π = population proportion) Chap 7-45 z-value for Proportions Standardize p to a z value with the formula: p π z σ p p π π(1 π) n If sampling is without replacement and n is greater than 5% of the population size, then σ p must use the finite population correction factor: σ p π(1 π) n N n N 1 Chap 7-46

24 Chapter 7 Student Lecture Notes 7-24 Eample 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)? Chap 7-47 Eample if π =.4 and n = 200, what is P(.40 p.45)? (continued) Chap 7-48

25 Chapter 7 Student Lecture Notes 7-25 Eample if π =.4 and n = 200, what is P(.40 p.45)? (continued) Chap 7-49 Chapter Summary Discussed sampling error 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 Discussed sampling from finite populations Chap 7-50

26 Chapter 7 Student Lecture Notes 7-26 Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc. Chap 7-51