(Basic Statistics for Citizen Data Scientist) Simulation It is often useful to create a model using simulation. Usually, this takes the form of generating a series of random observations (often based on a specific statistical distribution) and then studying the resulting observations using techniques described throughout the rest of this website. This approach is commonly called Monte …
(Basic Statistics for Citizen Data Scientist) Comparing two means when variances are known Theorem 1: Let x̄ and ȳ be the means of two samples of size nx and ny respectively. If x and y are normal or nx and ny are sufficiently large for the Central Limit Theorem to hold, then x̄ – ȳ has normal distribution with mean μx – μy and standard deviation Proof: Since the samples are random, x̄ and ȳ are normally and independently distributed. By the Central Limit …
(Basic Statistics for Citizen Data Scientist) Hypothesis Testing using the Central Limit Theorem Using the Central Limit Theorem we can extend the approach employed in Single Sample Hypothesis Testing for normally distributed populations to those that are not normally distributed. Suppose we take a sample of size n, where n is sufficiently large, and pose a null hypothesis that the …
(Basic Statistics for Citizen Data Scientist) Central Limit Theorem Theorem 1 – Central Limit Theorem: If x has a distribution with mean μ and standard deviation σ then for n sufficiently large, the variable has a distribution which is approximately the standard normal distribution. Observation: The larger the value of n the better the approximation will be. For practical purposes when n ≥ 30, then the approximation will be …
(Basic Statistics for Citizen Data Scientist) Confidence Intervals for Sampling Distributions Suppose we take a sample of size n from a normal population N(μ, σ) and ask whether the sample mean differs significantly from the overall population mean. As we have seen in Single Sample Hypothesis Testing The exact point of rejection (at the right tail), zcrit, has value And so …
(Basic Statistics for Citizen Data Scientist) Standardized Effect Size Definition 1: Cohen’s d, a statistic which is independent of the sample size and is defined as where m1 and m2 represent two means and σpooled is some combined value for the standard deviation. The effect size given by d is conventionally viewed as small, medium or large as follows: d = 0.20 – small effect d = …
(Basic Statistics for Citizen Data Scientist) Single Sample Hypothesis Testing Suppose we take a sample of size n from a normal population N(μ, σ) and ask whether the sample mean differs significantly from the overall population mean. This is equivalent to testing the following null hypothesis H0: We use a two-tailed hypothesis, although sometimes a one-tailed hypothesis is preferred …
(Basic Statistics for Citizen Data Scientist) Basic Concepts of Sampling Distributions Definition 1: Let x be a random variable with normal distribution N(μ, σ). Now consider a random sample {x1, x2,…, xn} from this population. The mean of the sample (called the sample mean) is x̄ can be considered to be a number representing the mean of the actual sample taken, …
(Basic Statistics for Citizen Data Scientist) Truncated Normal Distribution Definition 1: Let -∞ ≤ a < b ≤ ∞. Then the pdf of the truncated normal distribution with mean μ and variance σ2 constrained by is where φ is the pdf of the normal distribution and Φ is the cdf of the normal distribution. We assume that if x < a or x = -∞ then φ(x, µ, σ) = 0 and …
(Basic Statistics for Citizen Data Scientist) Log-normal Distribution Definition 1: A random variable x is log-normally distributed provided the natural log of x, ln x, is normally distributed. The probability density function (pdf) of the log-normal distribution is Observation: Some key statistical properties are: Observation: Sometimes it is useful to use a transformation of the population being studied. In particular, …