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# First find the distribution of sample mean x bar x

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First find the distribution of sample mean X-bar X-bar has a normal distribution with mean = 50, std = 5/sqrt(100)=0.5 P(X-bar>51) = P(Z > (51-50)/0.5) = P(Z>2) = 1-P(Z<2) = 1- 0.9772 = 0.0228 5. Did you need to use the Central Limit theorem to answer #4? Why or why not? No, because the population is normal. For a normal population, the sampling distribution of X-bar is normal for all values of n. The annual rainfall in Cleveland, Ohio has an unknown distribution with mean 40.2 inches and standard deviation 8.4 inches

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6. What is the probability that next year's rainfall will be less than 40 inches? Can’t get the probability from the problem information because the population distribution is unknown here. 7. What is the probability that the average yearly rainfall over 30 years (selected at random) will be less than 40 inches? By CLT, since n=30 big enough, the sampling distribution of X-bar would be approximately normal with mean = 40.2 and std= 8.4/sqrt(30)= 1.53 P(X-bar<40 )= P(Z<(40-40.2)/1.53) = P(Z < -0.13) = 0.4483 8. Did you need to use the Central Limit Theorem for #7? Why or why not? Yes. Because the population distribution is unknown here, so we need CLT to get the
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First find the distribution of sample mean X bar X bar has...

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