OpenStax - Ch. 7 The Central Limit Theorem

Question 1

Average

Answer 1

a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

Question 2

Central Limit Theorem

Answer 2

Given a random variable (RV) with known mean μ and known standard deviation, σ, we are sampling with size n, and we are interested in two new RVs: the sample mean, X¯¯¯X¯, and the sample sum, ΣΧ. If the size (n) of the sample is sufficiently large, then X¯¯¯X¯ ~ N(μ, σn√σn) and ΣΧ ~ N(nμ, (n−−√n)(σ)). If the size (n) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distributions regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means, σn√σn, is called the standard error of the mean.

Question 3

Exponential Distribution

Answer 3

a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital, notation: X ~ Exp(m). The mean is μ = 1m1m and the standard deviation is σ = 1m1m. The probability density function is f(x) = me–mx, x ≥ 0 and the cumulative distribution function is P(X ≤ x) = 1 – e–mx.