Section 7.3 Second Day Central Limit Theorem

Quick Review

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.

Houston,… We have a problem…  Most population distributions are not Normal  Ex. - 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 Means

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.”

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 OR “since n> 30, the CLT says the sampling distribution of x-bar is Normal.” Population ≥ 10n Formula Sentence

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?