Question

an education researcher recorded the number of students at each of the schools in his county.
number of students
178 178 178 178 178 193 193 193 193 594 594 594 594 594 788 788 788 788 788 788
x is the number of students that a randomly - chosen school has. what is the expected value of x?
write your answer as a decimal.

Explanation:

Step1: Count the frequencies

There are 5 schools with 178 students, 4 schools with 193 students, 5 schools with 594 students and 6 schools with 788 students. The total number of schools is $5 + 4+5 + 6=20$.

Step2: Calculate the expected - value formula

The formula for the expected value $E(X)=\sum_{i}x_ip_i$, where $x_i$ is the value of the random - variable and $p_i$ is the probability of that value.
$p_1=\frac{5}{20}$, $x_1 = 178$; $p_2=\frac{4}{20}$, $x_2 = 193$; $p_3=\frac{5}{20}$, $x_3 = 594$; $p_4=\frac{6}{20}$, $x_4 = 788$.
$E(X)=178\times\frac{5}{20}+193\times\frac{4}{20}+594\times\frac{5}{20}+788\times\frac{6}{20}$.

Step3: Calculate each term

$178\times\frac{5}{20}=\frac{890}{20}=44.5$;
$193\times\frac{4}{20}=\frac{772}{20}=38.6$;
$594\times\frac{5}{20}=\frac{2970}{20}=148.5$;
$788\times\frac{6}{20}=\frac{4728}{20}=236.4$.

Step4: Sum up the terms

$E(X)=44.5 + 38.6+148.5+236.4$.
$E(X)=468$.

Answer:

$468.0$