section 7.3 second day central limit theorem. quick review
DESCRIPTION
PCFS ProportionsMeans p = the proportion of _____ who … Parameterµ = the mean … SRS np≥10 and n(1-p)≥10 Population ≥ 10n Conditions Random Normality Independence SRS Population ~ Normally Population ≥ 10n Formula Sentence The problem is that most populations AREN’T normally distributed.TRANSCRIPT
Section 7.3 Second DayCentral Limit Theorem
Quick Review
PCFSProportions Meansp = the proportion of _____ who …
Parameter µ = the mean …
SRSnp≥10 and n(1-p)≥10Population ≥ 10n
ConditionsRandom NormalityIndependence
SRSPopulation ~ NormallyPopulation ≥ 10n
Formula
Sentence
.x meanzstd dev
.x meanzstd dev
The problem is that most populations AREN’T normally distributed.
Houston,… We have a problem…Most population distributions
are not NormalEx. - Household Incomes
So when a population distribution is not Normal, what is the shape of the sampling distribution of x-bar?
Consider the strange population distribution from the Rice University sampling distribution applet.
Describe the shape of the sampling distributions as n increases. What do you notice?
Sample M
eans
The Central Limit Theorem If the population is normal, the sampling
distribution of x-bar is also normal. THIS IS TRUE NO MATTER HOW SMALL n IS.
As long as n is big enough, and there is a finite population standard deviation, the sampling distribution of x-bar will be normal even if the population is not distributed normally. This is called the Central Limit Theorem.
Memorize it.How big is “big enough?” We will use n > 30 and say, “since n > 30, the CLT says the sampling distribution of x-bar is Normal.”
PCFSProportions Meansp = the proportion of _____ who …
Parameter µ = the mean …
SRS
np≥10 and n(1-p)≥10
Population ≥ 10n
Conditions Random
Normality
Independence
SRS
Population ~ Normally OR “since n> 30, the CLT says the sampling distribution of x-bar is Normal.”Population ≥ 10n
Formula
Sentence
.x meanzstd dev
.
x meanzstd dev
Example: Servicing Air Conditioners
Based on service records from the past year, the time (in hours) that a technician requires to complete preventative maintenance on an air conditioner follows the distribution that is strongly right-skewed, and whose most likely outcomes are close to 0. The mean time is µ = 1 hour and the standard deviation is σ = 1
Your company will service an SRS of 70 air conditioners. You have budgeted 1.1 hours per unit. Will this be enough?