Sampling distributions,

Point Estimation

Week 3: Lectures 3

•Sampling Distributions •Central limit theorem-sample mean•Point estimators-bias,efficiency•Random sampling-random # table

TEXT : CH.6, 7, 8 (p209-211)

Sampling Distributions•From a sample of values of a R.V.

(X),we calculate a statistic(e.g.x,sX

2) to estimate a population parameter (e.g. ,X

2).•Different samples -> different values of the statistic.•Many samples -> distribution of values•What are the properties of these distributions (shape,mean & variance) for (e.g.)sample mean, sample variance?

Sampling Distribution of the Sample Mean : meanThe mean of all possible sample means is

the population mean .i.e. if a random variable has mean then then the sampling distribution of X also has mean i.e. : E(X) = (proof is in textbook)

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Sampling distributionof mean: Variance ,SD

• The standard deviation of the sample mean is smaller than the population standard deviation, & the variance of X is smaller than 2. In fact:

s.d.( X ) X n

, or Var( X ) X 2

2

n

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Shape: central limit theorem• The CLT says that the average of

many identically distributed r.v.’s has an approx.normal distribution.X is such an average, so it is approx.normal for large samples ,even if X is not. Also,if X is normal, then X is always normal. i.e. : X ~ N ,

2

n

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Calculating Probabilities• so for large samples

(X-) /(/¦n) ~ N (0,1) (approx.) • i.e. the transform of X to a “Z” variable (standardized) has an approximately standard normal distribution, so if we know ,X we can calculate probabilities for X, using normal distribution tables

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Statistical Inference :

Estimation• Statistical Inference draws conclusions about populations from sample data

• sample statistics (x, s) are used in the estimation (point , interval estimates) of corresponding population parameters ().

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Point estimate : Example• Assume a sample of the selling

prices of houses on the Gold Coast was taken from the Gold Coast Bulletin on a single day.

• A point estimate of the average selling price of all such houses (population) would be given by the sample mean.

• Another newspaper on another day might give a different sample mean.

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Properties of estimators• So, different samples give different estimates of the population mean

• How can I be confident of making a good, useful, reliable point estimate, from my sample of observations?

• I must use an estimation procedure for which my point estimate is unbiased, efficient, and consistent

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Unbiasedness• Definition: a sample statistic is

an unbiased estimator() of a parameter (), if the expectation of the estimator equals the parameter i.e. : E()=

• e.g. It can be proved that E(X)=, so the sample mean is an unbiased estimator of the population mean.

• also s2 is an unbiased estimator

of 2.

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Efficiency• Estimators should be unbiased &

have minimum variance relative to all other estimators : 1 is more efficient than 2 , if Var(1) < Var(2)

• e.g. Var(sample median) = 1.57 times Var(sample mean)-->> sample mean is more efficient than the sample median

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Choice of point estimator• A minimum variance unbiased estimator is the most efficient of all unbiased estimators e.g. x,s2 where X ~ N

• for biased estimators, calculate : amount of bias & the mean square error (MSE)

• It is not always possible to find a min. var. unbiased estimator

• some biased estimators have smaller MSE’s than unbiased ones

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Population & Sample• Sampling Frame—a list of all

subjects from which the sample will be drawn

• Target Population—the population from which we hope we are drawing the sample

• E.g. target population could be all Bond students, but sampling frame is the Statistics class list

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Probability sampling• Simple random sampling: each individual or unit in the population has the same chance of selection

• Can be done with replacement or without replacement

• Sample should be representative of the population

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Random sampling methods• Use:a table,physical device,or

computer generated (pseudo) random values

• Using Excel, you can generate random numbers within the range [0,1]

• Convert these to random integers within an integer range such as 1–200 by multiplying and rounding

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Example : using a table • Method: assign values to all

units in sampling frame a random number table. Start anywhere & read off as many #’s in sequence as needed. Select appropriate digits from each number & match these #’s to values in the sampling frame to choose the units in the sample.

Midterm Exam preview

Week 6, Lecture 3

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Midterm Exam • Next Week on Wednesday

• 8:00 a.m. -->> 9:30 a.m.• 10 minutes perusal • 1hr 20 minutes writing time • 5 questions • approx. 16 minutes each

question

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Test Details• Test is in your usual Wednesday

lecture theatre at the usual lecture time.

• Be seated five minutes before the test start time. Those arriving late may be delayed in starting the test.

• Perusal start time: on the hour

Test start time : 10 min. past the hour Finish time : 30 min. past the 2nd hour

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Open book test

• You are expected to bring a calculator

• No computers may be used in the test

• You must bring your student card

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Material to be examined• The exam covers all the

material from weeks 1 to 5 inclusive

• You are also expected to remember Excel procedures which you have used in the lab, including any “Tools,Data Analysis” procedures, and any “functions” you have used