/// <summary>
/// Estimate the normal distribution of a sample proportion (for a categorical variable with two values { "SUCCESS", "FAILURE" })
///
/// The Centrl Limit Theorem (CLT) for proportions:
/// The distribution of sample proportions is nearly normal, centered at the population proportion, and with a standard error inversely proportional to the sample size.
///
/// Conditions for the CLT for proportions:
/// 1. Independence: Sampled observations must be independent.
/// > random sample/assignment
/// > if sampling without replacement, n < 10% population
/// 2. Sample size / skew: There should be at least 10 successes and 10 failures in the sample: np >= 10 and n(1-p) >= 10
/// </summary>
/// <param name="p"></param>
/// <param name="sampleSize"></param>
/// <returns></returns>
public static Gaussian EstimateSampleProportionDistribution(double p, int sampleSize)
{
double SE = StandardError.GetStandardErrorForProportion(p, sampleSize);
return(new Gaussian(p, SE));
}
/// <summary>
/// Estimate the normal distribution of a sample mean (for a continuous variable)
///
/// The Central Limit Theorem (CLT) states that:
/// The distribution of sample statistics (e.g., sample mean) is nearly normal, centered at the population mean, and with a standard deviation equal to the population standard deviation
/// divided by square root of the sample size.
///
/// With CTL, we can estimate the the normal distribution of a sample, given its estimated mean and stddev as well as the sample size.
///
/// For the CTL to hold true for a sample, the following conditions must be met:
/// 1. Independence: Sample observations must be independent.
/// > random sample/assignment
/// > if sampling without replacement, the sample size < 10% of the population
/// 2. Sample size/skew: Either the population distribution is normal, or if the population distribution is skewed, the sample size is large (rule of thumb: sample size > 30)
/// </summary>
/// <param name="sampleMean">point estimate of sample mean</param>
/// <param name="sampleStdDev">standard deviation of a random sample</param>
/// <param name="sampleSize">the size of the random sample</param>
/// <returns>The normal distribution of the sample means for a random sample drawn from the population</returns>
public static Gaussian EstimateSampleMeanDistribution(double sampleMean, double sampleStdDev, int sampleSize)
{
double SE = StandardError.GetStandardError(sampleStdDev, sampleSize);
return(new Gaussian(sampleMean, SE));
}
