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Class Notes – Chapter 11 – Sampling Distributions

A parameter is a number that describes the population. In practice when thepopulation is large, the value of a parameter is unnown because we cannot

e!amine the entire population. "he mean of a population is denoted b#  μ  $the

%ree letter mu&

A statistic is a number computed from sample data. In practice, we use a statistic

to estimate an unnown parameter. "he mean of a sample is denoted b# ´ x .

'!ample( )lorida has pla#ed a e# role in recent presidential elections. *oterregistration records show that +1 of )lorida voters are registered as Democratsand - as /epublicans. 02 randoml# chosen )loridians are contacted and ofthese -- are registered Democrats. 3hich of these numbers are parameters andwhich are statistics4

Statistical inference uses sample data to draw conclusions about the entirepopulation. 'ven though ´ x  depends on the sample, if we eep taing larger and

larger samples, the statistic ´ x  gets closer and closer to the parameter,  μ . "his

is called the law of large numbers. "he law of large numbers is the foundation ofgambling casinos and insurance companies.

Sampling distributions:

 "he population distribution of a variable is the distribution of values of thevariable among all the individuals in the population.

 "he sampling distribution of a statistic is the distribution of values taen b# thestatistic in all possible samples of the same si5e from the same population. "hisdescribes how the statistic varies in man# samples from the population.

 "he following sampling distribution shows that histogram of the samples means of1222 separate S/Ss of si5e 12 is close to a Normal distribution

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3hat can we sa# about the shape, center, and spread of this distribution4

• Shape: It loos Normal. Detailed e!amination con6rms that the distribution of from man# samples is ver# close to Normal.

• Center:  "he mean of the 1222 7s is 0+.8. "hat is, the distribution iscentered ver# close to the population mean  μ 9 0.

• Spread: "he standard deviation of the 1222 7s is 0.01, notabl# smallerthan the standard deviation σ  9 of the population of individual sub:ects.

Although these results describe :ust one simulation of a sampling distribution, the#re;ect facts that are true whenever we use random sampling.

The Sampling Distribution of :

Sample mean, : Suppose that is the mean of an S/S of si5e n drawn from a

large population with mean  μ and standard deviation σ . "hen the sampling

distribution of has mean  μ and standard deviation  σ /√ n . So the sampling

distribution is less spread out than the distribution of individual observations

because averages are less variable than individual observations.

If individual observations,  X  , have the N$ μ, σ & distribution, then the sample mean

of an S/S of si5e n has the N$ μ, σ /√ n & distribution.

CENTRAL L!T T"E#RE!

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Draw an S/S of si5e n from an# population with mean  μ and 6nite standard

deviation σ . "he central limit theorem sa#s that when n is large $accepted to be

n>30¿ , the sampling distribution of the sample mean  is appro!imatel# Normal(

 "he central limit theorem allows us to use Normal probabilit# calculations to answer

<uestions about sample means from man# observations even when the population 

distribution is not Normal. 

$$%$& "he =ureau of >abor Statistics announces that last month it interviewed allmembers of the labor force in a sample of ?2,222 households@ &%'( of the peopleinterviewed were unemplo#ed. "he boldface number is a

$a& sampling distribution.

$b& parameter.

$c& statistic.

$$%$) A stud# of voting chose ??- registered voters at random shortl# after anelection. f these, 0 said the# had voted in the election. 'lection records showthat onl# )*( of registered voters voted in the election. "he boldface number is a

$a& sampling distribution.

$b& parameter.

$c& statistic.

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$$%$* Annual returns on the more than 222 common stocs available to investorsvar# a lot. In a recent #ear, the mean return was B.- and the standard deviation of returns was 0B.. "he law of large numbers sa#s that

$a& #ou can get an average return higher than the mean B.- b# investing in a

large number of stocs.

$b& as #ou invest in more and more stocs chosen at random, #our average

return on these stocs gets ever closer to B.-.

$c& if #ou invest in a large number of stocs chosen at random, #our average

return will have appro!imatel# a Normal distribution.

$$%$+ Scores on the mathematics part of the SA" e!am in a recent #ear wereroughl# Normal with mean 1 and standard deviation 11+. ou choose an S/S of122 students and average their SA" math scores. If #ou do this man# times, themean of the average scores #ou get will be close to

$a& 1.

$b& 1122 9 .1.

$c& 1E100 9 .1.

$$%$ Scores on the mathematics part of the SA" e!am in a recent #ear wereroughl# Normal with mean 1 and standard deviation 11+. ou choose an S/S of122 students and average their SA" math scores. If #ou do this man# times, the

standard deviation of the average scores #ou get will be close to

$a& 11+.

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$b& 11+122 9 1.1+.

$c& 11+E100 9 11.+.

$$%$' A newborn bab# has e!tremel# low birth weight $'>=3& if it weighs less than1222 grams. A stud# of the health of such children in later #ears e!amined arandom sample of 018 children. "heir mean weight at birth was grams. "hissample mean is an unbiased estimator  of the mean weight μ in the population of all'>=3 babies. "his means that

$a& in man# samples from this population, the mean of the man# values of will

be e<ual to μ.

$b& as we tae larger and larger samples from this population, will get closer

and closer to μ.

$c& in man# samples from this population, the man# values of will have a

distribution that is close to Normal.

$$%-. "he number of hours a light bulb burns before failing varies from bulb tobulb. "he distribution of burnout times is strongl# sewed to the right. "he centrallimit theorem sa#s that

$a& as we loo at more and more bulbs, their average burnout time gets ever

closer to the mean μ for all bulbs of this t#pe.

$b& the average burnout time of a large number of bulbs has a distribution of the

same shape $strongl# sewed& as the distribution for individual bulbs.

$c& the average burnout time of a large number of bulbs has a distribution that is

close to Normal.

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$$%-$ "he length of human pregnancies from conception to birth varies accordingto a distribution that is appro!imatel# Normal with mean 0?? da#s and standarddeviation 1? da#s. "he probabilit# that the average pregnanc# length for ?randoml# chosen women e!ceeds 02 da#s is about

$a& 2.+2.

$b& 2.0.

$a& 2.2.