2.2 Notes
... If the points lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Will not be tested in Normal plots on the AP exam; however, students are welcome to use them ...
... If the points lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Will not be tested in Normal plots on the AP exam; however, students are welcome to use them ...
Estimation I 1 Introduction
... Closure under summation The sum of two independent normal distributions is normal. This follows immediately from above, since the product of two normal characteristic functions gives: ...
... Closure under summation The sum of two independent normal distributions is normal. This follows immediately from above, since the product of two normal characteristic functions gives: ...
S1 Normal Distribution June 13 Replacement June 13 Jan 13 Jan
... (a) Find the probability that a randomly selected employee has a journey to work of more than 20 km. ...
... (a) Find the probability that a randomly selected employee has a journey to work of more than 20 km. ...
Sampling Distribution Models AP Stats Chapter 18 Notes part
... E. Example: Suppose that about 13% of the population is left-handed. A 200-seat auditorium has been built with 15 “lefty seats,” seats that have the built-in desk on the left rather than the right arm of the chair. In a class of 90 students, what’s the probability that there will not be enough seat ...
... E. Example: Suppose that about 13% of the population is left-handed. A 200-seat auditorium has been built with 15 “lefty seats,” seats that have the built-in desk on the left rather than the right arm of the chair. In a class of 90 students, what’s the probability that there will not be enough seat ...
Full text
... All the equations (i)-(vi) involve contradictions. Of these, perhaps (ii) is the least obvious. Let us therefore examine (ii), which is true for m = 2 (even) leading to c2 = 1, cx = 2 from (ii) and (8). Now c2 = 1 = a2 - h2 implies that a2 = 2 (b2 = 1) or a2 - 1 (b2 = 0), i.e., a2 ^ 0, which contrad ...
... All the equations (i)-(vi) involve contradictions. Of these, perhaps (ii) is the least obvious. Let us therefore examine (ii), which is true for m = 2 (even) leading to c2 = 1, cx = 2 from (ii) and (8). Now c2 = 1 = a2 - h2 implies that a2 = 2 (b2 = 1) or a2 - 1 (b2 = 0), i.e., a2 ^ 0, which contrad ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.