Dada una variable aleatoria normal X con media 20 y varianza...

Dada una variable aleatoria normal $X$ con media 20 y varianza 9 , y una muestra aleatoria de tamaño $n$ tomada de la distribución, ¿qué tamaño de la muestra $n$ se necesita para que
$$
P(19.9 \leq \bar{X} \leq 20.1)=0.95 \text { ? }
$$

Key Concepts

Sampling Distribution of the Mean

When taking samples from a normal distribution, the distribution of the sample mean is itself normally distributed. Its mean is the same as the population mean and its variance is the population variance divided by the sample size. This concept is key in assessing how sample means vary around the true mean.

Normal Distribution

This is a probability distribution characterized by a symmetric bell-shaped curve where the mean, median, and mode are equal. It is defined by its mean and variance, and many statistical methods assume that the data are normally distributed, which simplifies analysis and inference.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, commonly the sample mean. It is calculated as the population standard deviation divided by the square root of the sample size, and it quantifies the expected fluctuation of the sample mean from the true mean.

Confidence Interval

A confidence interval is a range of values derived from the sample data that is believed, with a certain level of confidence, to contain the true population parameter. For the normal distribution, it often involves using the standard error and the appropriate z-score corresponding to the desired confidence level.

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and a sample estimate of that parameter. It is determined by the product of the critical value (from the normal distribution for a given confidence level) and the standard error, establishing how close the sample mean is expected to be to the true mean.

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