# Random Sampling

When taking a survey of a population, a random sample is a randomly chosen subset which represents the entire population.

If the size of the sample and the size of the entire population is known, then predictions can be made about the entire population using proportions .

Example:

In a school with $4400$ students, $66$ students are chosen at random and asked what their favorite subject is. Eighteen of them say mathematics. About how many students at the school would name mathematics as their favorite subject?

Write the proportion.

$\frac{18}{66}=\frac{x}{4400}$

Take cross products .

$\begin{array}{l}18\left(4400\right)=66x\\ 79200=66x\\ x=1200\end{array}$

So, about $1200$ students in the school would name mathematics as their favorite subject.

Some care must be taken to ensure that the sample is actually random with respect to the question being asked.

For example, if the survey questions concern income, calling $100$ randomly chosen from the phone book during the day may not yield a good random sample; the people contacted will be those who are home during the day, and may not be highest earners. On the other hand, if the survey question asks the color of one's house or apartment building, calling $100$ random people from the phone book is probably fine.