Given a distribution X ~ N(60, 5). Suppose you form random samples of 16 from the distribution. Let be the random variable of averages and Ex be the random variable of sums. - What is the mean and the standard deviation of? What is the mean and the standard deviation of Ex? Find the probability P(= < 55). Find the probability P(E x > 1040).
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a.
The sampling distribution of the sample mean, X̄, based on a sample of size n, taken from a population with expectation μ and standard deviation σ, has expectation μX̄ = μ and standard deviation σX̄ = σ/√n.
If the sample size is large (n ≥ 30), or the population distribution is normal, then by the central limit theorem, the sampling distribution of the sample mean is normal, with parameters μX̄ and σX̄.
By convention, we write X ~ N (μ, σ2).
Thus, X has a normal distribution with parameters, μ = 60, σ2 = 5, so that σ = √5.
The sample size is, n = 16.
The mean of the sampling distribution the sample mean is:
μX̄
= μ
= 60.
The standard deviation of the sampling distribution the sample mean is:
σX̄
= σ/√n
= √5/√16
≈ 0.5590.
Hence, the distribution of X̄ is normal with mean μX̄ = 60 and standard deviation σX̄ = 0.5590.
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