What allows us to use the standard Normal distribution to compute probabilities on x¯ when we don’t know the shape of the population?
the central limit theorem
The Central Limit Theorem (CLT) is what allows us to use the standard Normal distribution to compute probabilities on x̄ when we don’t know the shape of the population. The CLT states that for a sufficiently large sample size (30 or greater), the sampling distribution of the sample means will approximate a normal distribution, regardless of the shape of the population.
In other words, if we take enough random samples of a certain size from any population, the distribution of the means of those samples will resemble a normal distribution, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
Using the standard Normal distribution to compute probabilities on x̄ is possible because the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population, as long as the sample size is large enough. The mean of the sampling distribution will be equal to the population mean, and the standard deviation of the sampling distribution will be equal to the population standard deviation divided by the square root of the sample size. This allows us to use the standard Normal distribution to compute probabilities on x̄ even when we don’t know the shape of the population.
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