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help on stats 3

in a large city the distribution of incomes per family has a standard deviation of £5200. for a random sample of 400 families whats the probability that the sample mean income per family is within £500 of the actual mean income per family?
Original post by penelopecrux
in a large city the distribution of incomes per family has a standard deviation of £5200. for a random sample of 400 families whats the probability that the sample mean income per family is within £500 of the actual mean income per family?


You are essentially looking to find P(μ−500<xˉ<μ+500)=1−2P(xˉ<μ−500)P(\mu-500<\bar{x}<\mu+500) = 1-2P(\bar{x} < \mu - 500)
(edited 6 years ago)
Original post by RDKGames
You are essentially looking to find P(μ−500<xˉ<μ+500)=1−2P(xˉ<μ−500)P(\mu-500<\bar{x}<\mu+500) = 1-2P(\bar{x} < \mu - 500)


yeah but then how would you proceed from this step
Original post by penelopecrux
yeah but then how would you proceed from this step


Note that nn is large so we can approximate the distribution of xˉ\bar{x} by N(μ,52002400)N(\mu, \frac{5200^2}{400}), so (xˉ−μ)∼N(0,52002400)(\bar{x}-\mu) \sim N(0, \frac{5200^2}{400})


The probability above can be rewritten as 2P(xˉ−μ<500)−12P(\bar{x}-\mu < 500)-1.

Also might be better to use xˉ↦Xˉ\bar{x} \mapsto \bar{X} for better notation.
(edited 6 years ago)

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