- Simple Random Sampling:
Definition: Simple random sampling (SRS) is a method of selecting a subset (sample) from a larger population in such a way that every individual or element in the population has an equal and independent chance of being included in the sample.
Process: In simple random sampling, each element in the population is assigned a unique identification number. Then, a random number generator or a random sampling technique is used to select elements for the sample without any bias.
Advantages: Simple random sampling is unbiased and ensures that every part of the population has a chance of being represented in the sample, making it a reliable method for drawing conclusions about the population.
- Sampling Bias:
Definition: Sampling bias occurs when the method of selecting a sample systematically favors certain outcomes or subgroups of the population over others, leading to a non-representative sample.
Types of Bias:
Selection Bias: Occurs when certain elements in the population are more likely to be included in the sample than others. Non-Response Bias: Arises when individuals selected for the sample do not respond, and the non-respondents differ from the respondents in important ways.
Undercoverage Bias: Occurs when certain groups or segments of the population are not adequately represented in the sample.
Impact: Sampling bias can lead to erroneous conclusions and generalizations about the population because the sample does not accurately reflect the true characteristics of the population.
- Central Limit Theorem:
Definition: The Central Limit Theorem (CLT) is a fundamental theorem in statistics that states that the sampling distribution of the sample mean (or other sample statistics) approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
Key Points:
The CLT is essential because it allows us to make inferences about a population using sample statistics, assuming that the sample size is sufficiently large.
It states that even if the population distribution is not normal, the distribution of sample means will tend to follow a normal distribution as long as the sample size is reasonably large (usually n ≥ 30 is considered sufficient).
Applications: The CLT is widely used in hypothesis testing, confidence intervals, and other statistical analyses to make inferences about population parameters.
- Sampling Distributions (e.g., t-distribution):
Definition: A sampling distribution is a probability distribution that describes the behavior of a sample statistic (e.g., sample mean or sample proportion) across all possible random samples of a given size from a population.
Types of Sampling Distributions:
Sampling Distribution of the Sample Mean: Describes the distribution of sample means from different random samples of the same size.
Sampling Distribution of the Sample Proportion: Describes the distribution of sample proportions from different random samples.
t-Distribution: The t-distribution, also known as the Student’s t-distribution, is a probability distribution used in hypothesis testing when the population standard deviation is unknown. It is similar in shape to the normal distribution but has heavier tails.
Applications: Sampling distributions are used to calculate confidence intervals and conduct hypothesis tests. The t-distribution is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown.