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Section 7.3 Second Day Central Limit Theorem Quick Review PCFS The problem is that most populations AREN’T normally distributed. Proportions p = the proportion of _____ who … SRS np≥10 and n(1-p)≥10 Population ≥ 10n x mean z std . dev Means Parameter Conditions Random Normality Independence Formula Sentence µ = the mean … SRS Population ~ Normally Population ≥ 10n z x mean std . dev 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? Describe the shape of the sampling distributions as n increases. What do you notice? Sample Means Consider the strange population distribution from the Rice University sampling distribution applet. 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. How big is “big enough?” We will use n > 30 This is called the Central Limit Theorem. and say, “since Memorize it. n > 30, the CLT says the sampling distribution of x-bar is Normal.” PCFS Proportions p = the proportion of _____ who … Means Parameter Conditions SRS µ = the mean … Random SRS np≥10 and n(1-p)≥10 Normality Population ≥ 10n Independence Population ~ Normally OR “since n> 30, the CLT says the sampling distribution of x-bar is Normal.” Population ≥ 10n x mean z std . dev Formula z Sentence x mean std . 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?