Question

Wendy counted the number of words per page in her new book. Number of words Number of pages 28 2 33 1 69 1 78 3 84 1 98 2 X is the number of words that a randomly chosen page has. What is the expected value of X? Write your answer as a decimal.

Answer 1

The variable of interest is:

X: the number of words that a randomly chosen page has

The recorded data is displayed on a frequency table, where the first column shows the possible values of the variable (xi) and the second column shows the corresponding observed frequencies (fi).

You can read the information as follows:

For example, for the first row, xi=28 and fi=2, this means that two pages had 28 words.

To determine the expected value of a data set expressed using a frequency table, you have to use the following formula:

$E(X)=\frac{\Sigma x_if_i}{n}$

Where

E(X) is the expected value

xi represents the possible values of the variable

fi represents the observed frequencies for each value of X

n represents the sample size

Before calculating the expected value, you have to determine the sample size. To do so, add all observed frequencies:

$\begin{gathered} n=\Sigma f_i \\ n=2+1+1+3+1+2 \\ n=10 \end{gathered}$

Expected value:

Calculate the product of each value of x by its corresponding frequency, add them and divide the result by the sample size:

$\begin{gathered} E(X)=\frac{(28\cdot2)+(33\cdot1)+(69\cdot1)+(78\cdot3)+(84\cdot1)+(98\cdot2)}{10} \\ E(X)=\frac{56+33+69+234+84+196}{10} \\ E(X)=\frac{672}{10} \\ E(X)=67.2 \end{gathered}$

The expected value is E(X)=67.2 words/page