Sampling Distributions: Understanding the Variability of Sample Means - Prof. Murali S. Sh, Study notes of Business Statistics

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Chapter Goals

To use information from the sample to make inference about the population.

Formulate Theories

Collect Data

Summarize Results

Interpret Results/Make Decisions

Statistical Sampling

Definition Parameters are numerical descriptive measures of populations. Definition Statistics are numerical descriptive measures of samples (more generally, a quantity computed from the observations in a sample). Definition Estimators are sample statistics that are used to estimate the unknown population/process parameter. E.g., the sample mean X is a point estimator of the population mean μ. Note that X is also a random variable.

Question? How close is our sample statistic to the true, but unknown population parameter?

Notations

Parameters Statistics μ X , Md , mode, μ σ^2 s^2 , σ  p  p

Sampling Distribution: An Example

B

D

A

C

Population Characteristics

Population consists of N = 4 items A , B , C , and D , with values 1, 2, 3, and 4 respectively.

Summary measures :

μ =

∑ i = 1

(^4) X i 4 =^

1 + 2 + 3 + 4 4 =^ 2. 5

σ^2 =

∑ i^ N =^ 1  X^ i − μ ^2

N =^

 1 −2.5^2 + 2 −2.5^2 + 3 −2.5^2 + 4 −2.5^2 4 =^ 1. 25

σ = σ^2 = 1. 25 = 1. 118

Sampling Distribution of the Sample

Mean

Now let us draw all possible samples of size n = 2.

16 possible samples 1 st^ Observation 2 nd^ Observation A B C D A AA AB AC AD B BA BB BC BD C CA CB CC CD D DA DB DC DD

16 corresponding sample means 1 st^ Observation 2 nd^ Observation A B C D A 1.0 1.5 2.0 2. B 1.5 2.0 2.5 3. C 2.0 2.5 3.0 3. D 2.5 3.0 3.5 4.

Comparison of Population and Sampling

Distribution

Population Sampling μ (Mean) 2.5 2. σ^2 (Variance) 1.25 0. σ (Standard Deviation) 1.118 0.

Question: What is the relationship between the variance in the population and sampling distributions?

Empirical Derivation of Sampling

Distribution of X

- Select a random sample of n observations from a given population or process. - Compute X. - Repeat steps (1) and (2) a “large” number of times. - Construct a relative frequency histogram of the resulting set of X ’s.

Example Let X the length of pregnancy be X ~ N 266, 256.

- What is the probability that a randomly selected pregnancy lasts more than 274 days. I.e., what is P  X > 274 ? - Suppose we have a random sample of n = 25 pregnant women. Is it less likely or more likely (as compared to the above question), that we might observe a sample mean pregnancy length of more than 274 days. I.e., what is P  X > 274 ?

Let’s do it! 9.

Exercise The model for breaking strength of steel bars is normal with a mean of 260 pounds per square inch and a variance of 400_. What is the probability that a randomly selected steel bar will have a breaking strength greater than_ 250 pounds per square inch?

Exercise A shipment of steel bars will be accepted if the mean breaking strength of a random sample of 10 steel bars is greater than 250 pounds per square inch. What is the probability that a shipment will be accepted?

Desirable Characteristics of Estimators

Definition An estimator θ^ is unbiased if the mean of its sampling distribution is equal to the population parameter θ to be estimated. That is, θ^ is an unbiased estimator of θ if E θ^ = θ. For example, E  X  = μ, therefore X is an unbiased estimator of μ. Definition An estimator is a consistent estimator of a population parameter θ if the larger the sample size, the more likely it is that the estimate will be closer to θ. Is X a consistent estimator? Definition The efficiency of an unbiased estimator is measured by the variance of its sampling distribution. If two estimators based on the same sample size are both unbiased, the one with the smaller variance is said to have greater relative efficiency than the other.