Probability a sample mean of 25 reaches 165

Question

Draw $25$ independent observations from a normal population with mean $160$ and variance $900$ . Find, to four decimal places, the probability that their average is at least $165$ .

Short answer

$0.2023$ . The sample mean is normal with mean $160$ and standard deviation $30/\sqrt{25}=6$ , so the probability is $\mathbb{P}(Z\ge 5/6)\approx0.2023$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Populations, Samples, and Sampling Distributions
Sample Spaces, Events, and the Probability Axioms

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