Tema 4. Muestreo 4.1 Definición de muestreo. 4.1.1 Tipos de muestreo aleatorio, sistematizado, estratificado y conglomerado. Ejercicios resueltos pasos a paso. 4.2 Concepto de distribución de muestreo de la media. 4.2.1 Distribución muestral de la media con varianza conocida y desconocida. Ejercicios resueltos pasos a paso 4.2.2 Distribución muestral de la diferencia entre dos medias con varianza conocida y desconocida. 4.2.3 Distribución muestral de la proporción. 4.2.4 Distribución muestral de la diferencia de dos proporciones. 4.3 Teorema de límites central. 4.4 Tipos de estimaciones y características. 4.5 Determinación del tamaño de la muestra de una población. 4.6 Intervalos de confianza para la media, con el uso de la distribución.

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Sampling is the process of selecting a representative subset from a larger population for statistical analysis. There are four main types of sampling methods. Simple random sampling gives each element equal probability of selection. Systematic sampling selects elements at regular intervals. Stratified sampling divides the population into groups first. Cluster sampling uses natural groupings. Let's see a systematic sampling example with 100 elements where we need 20 samples, so we select every 5th element. A sampling distribution is the distribution of a statistic calculated from all possible samples of a given size. The sampling distribution of the mean has important properties: its mean equals the population mean, and its standard deviation, called standard error, equals population standard deviation divided by square root of sample size. When population variance is known, we use the Z-distribution. When unknown, we use the t-distribution. In our example with population mean 50, standard deviation 10, and sample size 25, the sampling distribution has mean 50 and standard error 2. The Central Limit Theorem states that as sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the population's original shape. Key components include sample size of at least 30, applicability to any population distribution, and standard error decreasing with larger samples. The theorem shows that sample means follow a normal distribution with population mean and variance divided by sample size. This animation demonstrates how distributions become more normal as sample size increases from 10 to 30 to 100. Two-sample distributions extend single-sample concepts to compare two groups. For differences between two means, the sampling distribution is normal with mean equal to the difference of population means and variance equal to the sum of individual variances divided by sample sizes. For proportions, we use similar logic with proportion-specific variance formulas. In our height comparison example with two groups having different sample sizes and standard deviations, we calculate the standard error as 2.24, creating a distribution of the difference centered at 5 centimeters. Estimation theory connects sampling distributions to practical inference. Point estimation provides a single value estimate, while interval estimation gives a range of plausible values. Confidence intervals use the formula: sample mean plus or minus critical value times standard error. For sample size determination, we solve for n using the desired margin of error. In our example, to estimate a population mean within 2 units with 95% confidence and known standard deviation of 10, we need approximately 97 samples. The visualization shows how confidence intervals capture the true parameter with specified probability.