Sta247h1f fall sampling distribution of the sample

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STA247H1FFall 201411 / 16
Sampling Distribution of The Sample MeanNotes on Central Limit TheoremTheCentral Limit Theorem (CLT)states that the sample mean fromany probability distribution (as long as they have a mean and variance)will have an approximate normal distribution, if the sample is sufficientlylarge.“Largen” meansn30in general, but in some cases may even bemuch less.The larger the sample size, the more nearly normally distributed is thepopulation of all possible sample means.For fairly symmetric distributions,n >15will be sufficient.STA247H1FFall 201412 / 16
Sampling Distribution of The Sample MeanSummaryForn30,X≈ N(μ, σ2/n).It follows that:P(Xb) =PX-μσ/nb-μσ/n=PZb-μσ/n.Forn30,Sn≈ N(nμ, nσ2).It follows that:P(Snb) =PSn-n σb-n σ=PZb-n σ.STA247H1FFall 201413 / 16
Sampling Distribution of The Sample MeanExampleAn electrical firm manufactures light bulbs that have a length of life thatis approximately normally distributed, with mean equal to 800 hours and astandard deviation of 40 hours. Find the probability that a random sampleof 16 bulbs will have an average life of less than 775 hours?