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Probability and Statistics Exam: Questions and Answers
Question
Probability and Statistics Exam: Questions and Answers
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To calculate the probability that the sample mean will be greater than 52, we need to know the distribution of the population and the sample size.

Assuming that the population is normally distributed and the sample size is sufficiently large (i.e., greater than 30), we can use the central limit theorem to approximate the distribution of the sample mean as normal with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.

Let's say we know the population mean is 50 and the population standard deviation is 10. If we take a sample of size 100, then the distribution of the sample mean can be approximated as normal with mean 50 and standard deviation 1 (i.e., 10 divided by the square root of 100).

To calculate the probability that the sample mean will be greater than 52, we need to standardize this value using the formula:

z = (x - μ) / (σ / sqrt(n))