Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data is collected on a population to draw conclusions, or make statistical inferences , about the population. Types of statistical inference: 1) parameter estimation (e.g. estimating μ) - with a certain level of confidence 2) hypothesis testing (e.g. H 0 : μ = 50) 1
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Chapter 7: SAMPLING DISTRIBUTIONS& POINT ESTIMATIONOF PARAMETERS
Part 1: IntroductionSampling Distributions &the Central Limit Theorem
Point Estimation & EstimatorsSections 7-1 to 7-2
Sample data is collected on a population to drawconclusions, or make statistical inferences, aboutthe population.
Types of statistical inference:
1) parameter estimation (e.g. estimating µ)- with a certain level of confidence
2) hypothesis testing (e.g. H0 : µ = 50)
1
Example of parameter estimation(or point estimation):
We’re interested in the value of µ.
We collected data and we use theobserved x̄ as a point estimate for µ.
µ is the unknown parameter being estimated.
NOTATION: µ̂ = X̄X̄ is the estimator.
{We often show an estimator as a ‘hat’
over its respective parameter.}
The observed x̄ estimate is a single value,or a point estimate.
Prior to data collection, X̄ is random variableand it is the statistic of interest from the data.
2
Sample-to-sample variability
The value we get for X̄ (the sample mean) de-pends on the specific sample chosen.
Sample
Population
This means, X̄ is a random variable!
The distribution of the random variable X̄is called the sampling distribution of X̄ .
We expect X̄ to be close to µ (we ARE us-ing it to estimate µ) but there is variability inX̄ before it is observed because we use randomsampling to choose our sample of size n.
3
The Sampling Distribution of X̄...
• Tells us what kind of values are likely to occurfor X̄ .
• Puts a probability distribution over the pos-sible values for X̄ .HINT: It’s distribution will be normal when conditions are met.
In a simple random sample of n observationsfrom a population,
E(X̄) = µ
⇒ X̄ is an unbiased estimator of µ.
This gives us a measure of center for the sam-pling distribution for X̄ , but what about thevariability of the X̄ random variable?
4
Sampling distribution of X̄
Case 1 Original population is normally dis-tributed.
x
f(x)
The x̄ I observe depends on the sample (theparticular n observations) I chose from thisnormal distribution.
Let’s look at the distribution of x̄ values if Ichoose a sample of size n and compute x̄ forthat sample, and I repeat this process 1000times...
5
x
f(x)
1) Choose a sample of size n from a normaldistribution
2) Compute x̄
3) Plot the x̄ on our frequency histogram
4) Do steps 1-3 1000 times
See applet at:http://onlinestatbook.com/stat sim/sampling dist/index.html
6
SKETCH THE PLOTS:
Distribution of X̄ for n=2 when original pop-ulation is normal.
Distribution of X̄ for n=25 when originalpopulation is normal.
7
Turns out, in this case, the random variableX̄ is normally distributed.
This normal distribution is centered at µ (themean of the original population we were sam-pling from).
The variability of X̄ depends on the samplesize n, and the variability in the original pop-ulation.
SPECIFICALLY:
When X ∼ N(µ, σ2),
X̄ ∼ N(µ,σ2
n)
NOTE: the distribution for X̄ is less vari-able than the distribution for X .
8
X̄ ∼ N(µ,σ2
n)
NOTE: X̄ from n = 25 is less variable thanX̄ from n = 2.
More data (larger n) gives us a better esti-mate of µ from X̄ .
The distribution of our estimator X̄ is squishedcloser, or is tighter, around the thing we’retrying to estimate. Which is beneficial whenestimating something.
9
Sampling distribution of X̄
Case 2 Original population is NOT normallydistributed.
x
f(x)
x
f(x)
x
f(x)
Or anything else...
10
What does the distribution of X̄ look like?
1) Choose a sample of size nfrom the distribution
2) Compute x̄
3) Plot the x̄ on our frequency histogram
4) Do steps 1-3 1000 times
———————————————————–
Right-skewed with n = 10.
11
Really non-normal (mass out at the ends)with n = 2.
Really non-normal (mass out at the ends)with n = 25.
12
Turns out the random variable X̄ is normallydistributed no matter what your original dis-tribution was IF n is large enough...
What’s large enough?Rule of thumb is n ≥ 30
So, what have we learned...
if X is normally distributed, thenX̄ ∼ N(µ, σ2/n) for any n.
if X is NOT normally distributed, thenX̄ ∼ N(µ, σ2/n) for n ≥ 30.
if X is not severely non-normal, thenX̄ ∼ N(µ, σ2/n) is close to true for n < 30.
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Sampling Distributions andthe Central Limit Theorem
Section 7-2
Sample data is collected on a population to drawconclusions, or make statistical inferences, aboutthe population.
NOTATION:
− A large letter like X̄ represents the randomvariable X̄ , and X̄ can take on many values.
− A small letter like x̄ represents an actual ob-served x̄ from a sample, and it is a fixedquanitity once observed.
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•Random Sample
The random variables X1, X2, . . . , Xn are arandom sample of size n if...
a) theXi’s are independent random variables,and
b) every Xi has the same sample probabilitydistribution (i.e. they are drawn from thesame population).
NOTE: the observed data x1, x2, . . . , xn isalso referred to as a random sample.
15
• Statistic– A statistic is any function of the observa-
tions in a random sample.
∗ Example:The mean X̄ is a function of the obser-vations (specifically, a linear combina-tion of the observations).
X̄ =
∑ni=1Xin
=1
nX1+
1
nX2+· · ·+1
nXn
– A statistic is a random variable,and it has a probability distribution
– The distribution of a statistic is called thesampling distribution of the statis-tic because is depends on the sample cho-sen.
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– The sampling distribution of the meanis very important.
What is the expected value of the samplemean X̄ in a random sample?
E(X̄) = E(1
nX1 +
1
nX2 + · · · + 1
nXn)
=1
n
∑E(Xi)
=1
n
∑µ =
nµ
n= µ = µX̄
Notation: E(X̄) = µX̄ = µ
where µ is the population mean.
(µ is also the expected valueof a single Xi)
17
What is the variance of the sample meanX̄ in a random sample?
(Xi’s in a random sample are independent.)
V (X̄) = V (1
nX1 +
1
nX2 + · · · + 1
nXn)
=
(1
n
)2∑V (Xi)
=
(1
n
)2∑σ2
=
(1
n
)2
nσ2 =σ2
n
Notation: V (X̄) = σ2X̄
= σ2
n
where σ2 is the population variance.
(σ2 is also the variance of a single Xi)
18
As we have described earlier, for n ≥ 30
X̄ ∼ N(µ,σ2
n)
(and this is also true for n < 30 if eachXi comesfrom a normal population).
Using this fact, and what we know about stan-dardizing variables, leads to...
• The Central Limit Theorem
If X1, X2, . . . , Xn is a random sample of sizen taken from a population with mean µ andvariance σ2, the limiting form of the distri-bution of
Z =X̄ − µσ/√n
as n → ∞ is the standard normal distribu-tion, or N(0, 1).
19
The approximation of
X̄ − µσ/√n∼ N(0, 1)
depends on the size of n.
Satisfactory approximation for n ≥ 30 forany population.
Satisfactory approximation for n < 30 fornear normal populations.
————————————————————
The next graphic shows 3 different original pop-ulations (one nearly normal, two that are not),and the sampling distribution for X̄ based on asample of size n = 5 and size n = 30.
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The three original distributions are on the farleft (one that is nearly symmetric and bell-shaped,one that is right skewed, and one that is highlyright skewed).
As shown in: Navidi, W. ‘Statistics for Engineers and Scientists’, McGraw Hill, 2006
21
Things to notice from the previous graphic:
• The variability of X̄ decreases as n increases
Recall: V (X̄) = σ2
n .
• If the original population has a shape that’scloser to normal, smaller n is sufficient for X̄to be normal.
• The normal approximation gets better withlarger n when you’re starting with a non-normal population.
• Even when X has a very non-normal distri-bution, X̄ still has a normal distribution witha large enough n.
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• Example: Flaws in a copper wire.
Let X denote the number of flaws in a 1 inchlength of copper wire. The probability massfunction of X is presented in the followingtable:
x P (X = x)0 0.481 0.392 0.123 0.01
Suppose n = 100 wires are sampled from thispopulation. What is the probability that theaverage number of flaws per wire in the sam-ple is less than 0.5?
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ANS:
P (X̄ < 0.5) =?
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Some Notation: Sampling distributionfor sample mean (X̄)
Suppose we have a random sample of size ndrawn from a parent (original) population withan expected value µ and variance σ2. Then,
X̄ ∼ N(µ,σ2
n)
is true for sample size n > 30 no matter whatthe distribution of the parent population, butalso true for smaller n when the parent popula-tion is normal or near-normal.
Notation:
E(X̄) = µX̄ = E(X) = µ
V (X̄) = σ2X̄
=V (X)n = σ2
n
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Terminology:
The term standard deviation refers to thepopulation standard deviation, or
√V (X) = σ,
and...
Z =X − µσ
The term standard error is a value related toX̄ and is also more fully stated as the standarderror of the sample mean and it is the squareroot of the variance of X̄ , or√