Random sampling ensures that the sample is representative of the population, reducing bias and allowing for valid inferences about the population.

Increasing the sample size decreases the standard deviation of the sampling distribution, making the sample statistics more precise estimates of the population parameter.

The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (n ≥ 30), regardless of the shape of the population distribution. This allows us to use normal distribution methods for inference.

Checking conditions ensures that the sampling distribution is valid and that the statistical inferences based on the sample are reliable and accurate.

The 10% condition ensures independence of observations when sampling without replacement. It states that the sample size should be no more than 10% of the population size.

Sampling Distribution of Proportions: Deals with categorical data, uses sample proportions, and checks the large counts condition. | Sampling Distribution of Means: Deals with numerical data, uses sample means, and relies on the Central Limit Theorem or a normally distributed population.

One-Sample: Involves making inferences about a single population. | Two-Sample: Involves comparing two populations and requires checking conditions for both samples.

Population Standard Deviation: Measures the variability within the entire population. | Standard Deviation of Sampling Distribution: Measures the variability of sample statistics (like means or proportions) across different samples.

Large Counts: Used for sample proportions, requires at least 10 expected successes and failures. | Central Limit Theorem: Used for sample means, states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (n ≥ 30).

Conditions for Proportions: Random, Independence (10% condition), Normality (Large Counts condition). | Conditions for Means: Random, Independence (10% condition), Normality (Population is normal or n ≥ 30).

$\sqrt{\frac{p(1-p)}{n}}$, where p is the population proportion and n is the sample size.

$\frac{\sigma}{\sqrt{n}}$, where σ is the population standard deviation and n is the sample size.

$np \geq 10$ and $n(1-p) \geq 10$

$\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$

$\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$