Final answer:
The student is asking about the Central Limit Theorem and how it relates to the distribution of sample means and the construction of confidence intervals in statistics. The theorem ensures that with a large enough sample size, the distribution of the sample means will be normally distributed, allowing for the calculation of confidence intervals around the population mean.
Step-by-step explanation:
The student's question pertains to the distribution of sample means (ℓ) and the use of confidence intervals in statistics. Specifically, this involves an understanding of the Central Limit Theorem which states that the distribution of sample means will tend to be normally distributed as the sample size increases, regardless of the population's distribution.
The standard deviation of a distribution of sample means (σ_{ℓ}) is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n), leading to the notation ℓ~N(μ_{ℓ}, σ_{ℓ}). This provides the framework to calculate confidence intervals for predicted sample means. When the size of the sample is large enough, as indicated by the empirical rule and the central limit theorem, the confidence interval can be constructed using the known mean and standard deviation values to approximate where a certain percentage of sample means will fall around the population mean (μ).
For instance, the empirical rule states that with a bell-shaped distribution, approximately 95% of the samples (the sample mean, ℓ) will fall within two standard deviations (σ) of the population mean (μ).
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