exercise:2efc106914: Difference between revisions

Latest revision as of 21:12, 14 June 2024

Let [math]X[/math] be the outcome of a chance experiment with [math]E(X) =\mu[/math] and [math]V(X) = \sigma^2[/math]. When [math]\mu[/math] and [math]\sigma^2[/math] are unknown, the statistician often estimates them by repeating the experiment [math]n[/math] times with outcomes [math]x_1[/math], [math]x_2, \ldots, x_n[/math], estimating [math]\mu[/math] by the sample mean

[[math]] \bar{x} = \frac 1n \sum_{i = 1}^n x_i\ , [[/math]]

and [math]\sigma^2[/math] by the sample variance

[[math]] s^2 = \frac 1n \sum_{i = 1}^n (x_i - \bar x)^2\ . [[/math]]

Then [math]s[/math] is the sample standard deviation. These formulas should remind the reader of the definitions of the theoretical mean and variance. (Many statisticians define the sample variance with the coefficient [math]1/n[/math] replaced by [math]1/(n-1)[/math]. If this alternative definition is used, the expected value of [math]s^2[/math] is equal to [math]\sigma^2[/math]. (See Exercise, part (d).) Write a computer program that will roll a die [math]n[/math] times and compute the sample mean and sample variance. Repeat this experiment several times for [math]n = 10[/math] and [math]n =1000[/math]. How well do the sample mean and sample variance estimate the true mean 7/2 and variance 35/12?

@@ Line 1: / Line 1: @@
-<div class="d-none"><math>
+Let <math>X</math> be the outcome of a chance experiment with <math>E(X) =\mu</math> and <math>V(X) = \sigma^2</math>.  When <math>\mu</math> and <math>\sigma^2</math> are unknown, the statistician often estimates them by repeating the experiment <math>n</math> times with outcomes <math>x_1</math>, <math>x_2, \ldots, x_n</math>, estimating <math>\mu</math> by the  ''sample mean''
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mat}[1]{{\bf#1}}
-\newcommand{\exref}[1]{\ref{##1}}
-\newcommand{\secstoprocess}{\all}
-\newcommand{\NA}{{\rm NA}}
-\newcommand{\mathds}{\mathbb}</math></div> Let <math>X</math> be the outcome of a chance experiment with <math>E(X) =
-\mu</math> and <math>V(X) = \sigma^2</math>.  When <math>\mu</math> and <math>\sigma^2</math> are unknown, the statistician
-often estimates them by repeating the experiment <math>n</math> times with outcomes
-<math>x_1</math>, <math>x_2</math>, \dots, <math>x_n</math>, estimating <math>\mu</math> by the  ''sample mean''
 <math display="block">
@@ Line 21: / Line 12: @@
 should remind the reader of the definitions of the theoretical mean and variance.  (Many
 statisticians define the sample variance with the coefficient <math>1/n</math> replaced by
-<math>1/(n-1)</math>.  If this alternative definition is used, the expected value of <math>s^2</math> is equal to <math>\sigma^2</math>. See
+<math>1/(n-1)</math>.  If this alternative definition is used, the expected value of <math>s^2</math> is equal to <math>\sigma^2</math>. (See
-Exercise \ref{exer 6.2.19}, part (d).)
+[[exercise:974197a009|Exercise]], part (d).) Write a computer program that will roll a die <math>n</math> times and compute the sample mean and sample variance.  Repeat this experiment several times for <math>n = 10</math> and <math>n =1000</math>.  How well do the sample mean and sample variance estimate the true mean 7/2 and variance 35/12?
-Write a computer program that will roll a die <math>n</math> times and compute the sample mean
-and sample variance.  Repeat this experiment several times for <math>n = 10</math> and <math>n =
-</math>.  How well do the sample mean and sample variance estimate the true mean 7/2
-and variance 35/12?