Distribution of X

• Random variable X is a variable whose possible values are determined by a random phenomenon.
• The probability distribution of X is a rule that assigns a probability to each possible value of X.
• In simple terms, it describes the chances of occurrence of different possible values.
• The mean (mu) and standard deviation (sigma) are important characteristics that describe a distribution. Other key features include the mode and median.
• The type of distribution dictates the shape of the distribution graph. Common types include the uniform distribution, normal distributions, and the binomial distribution.
• Understanding the distribution of X is essential in the calculation of probability, statistical inference, and hypothesis tests.

Central Limit Theorem (CLT)

• The Central Limit Theorem is a fundamental concept in statistical and probability theory.
• It states that as the sample size (n) becomes very large, the shape of the sample means approximates a normal distribution, regardless of the shape of the population distribution.
• The mean of the sample means equals to the population mean (mu).
• The standard deviation of the sample means (also known as the standard error) is equal to the population standard deviation (sigma) divided by the square root of the sample size (n). This is represented as: `Standard Error = sigma / sqrt(n)`.
• The CLT provides a justification for making inferences about a population from a sample.
• Practical application of the CLT includes construction of confidence intervals and hypothesis testing.

Application to Hypothesis Testing and Confidence Intervals

• Understanding the distribution of X and the CLT is key to apply hypothesis testing and constructing confidence intervals.
• In hypothesis testing, we use sample data to test a claim about a population parameter, such as the population mean (mu). The underlying distribution is important when deciding the type of test (Z-test, T-test, chi-square test, etc).
• Confidence intervals provide an estimation range for a population parameter with a certain level of ‘confidence’. This is computed based on the sample mean and the standard error. Again CLT comes into role when we assume the sample means to follow a normal distribution and apply a Z-score or T-score for computation.
• Mastery of these concepts not only helps to tackle specific questions on these topics, but also lays the foundation for more advanced statistical studies in university and beyond.