Question

A sample of small bottles and their contents has the following weights in grams; 4, 2, 5, 4, 5, 2, and 6. What is the sample variance of bottle weight?

A. 6.92

B. 4.80

C. 1.96

D. 2.33

Answer 1

Concepts and reason

Mean: Mean is obtained by adding the observations in a set of data and then dividing the total value by the total number of observations.

Variance: The sample variance is defined as how much each observation in the data set deviates from a central point (average).

Fundamentals

The formula to obtain the sample mean is,

${\rm{Mean}}\left( {\bar x} \right) = \frac{{\sum x }}{n}$

The formula for variance is,

${\rm{Variance}} = \frac{1}{{n - 1}}\left\{ {\sum\limits_{i = 1}^n {{{\left( {{x_i} - \bar x} \right)}^2}} } \right\}$

The mean of bottle weight is obtained below:

From the information given, a sample of small bottles and their contents has the following weights in grams 4, 2, 5, 4, 5, 2, and 6.

The mean is,

$\begin{array}{c}\\{\rm{Mean}}\left( {\bar x} \right) = \frac{{4 + 2 + 5 + 4 + 5 + 2 + 6}}{7}\\\\ = \frac{{28}}{7}\\\\ = 4\\\end{array}$

The sample variance of bottle weight is obtained below:

The variance is,

$\begin{array}{c}\\{\rm{Variance}} = \frac{{\left[ \begin{array}{l}\\{\left( {4 - 4} \right)^2} + {\left( {2 - 4} \right)^2} + \\\\{\left( {5 - 4} \right)^2} + {\left( {4 - 4} \right)^2} + \\\\{\left( {5 - 4} \right)^2} + {\left( {2 - 4} \right)^2} + \\\\{\left( {6 - 4} \right)^2}\\\end{array} \right]}}{{\left( {7 - 1} \right)}}\\\\ = \frac{{\left[ {0 + 4 + 1 + 0 + 1 + 4 + 4} \right]}}{6}\\\\ = \frac{{14}}{6}\\\\ = 2.33\\\end{array}$

Ans:

The sample variance of bottle weight is 2.33.