2.1.1 - One Categorical Variable

Data concerning one categorical variable can be summarized using a proportion.

Proportion
\(Proportion=\dfrac{Number\;in\;the\;category}{Total\;number}\)

The symbol for a sample proportion is \(\widehat{p}\) and is read as "p-hat." The symbol for a population proportion is \(p\). 

The formula for a sample proportion may also be written as \(\widehat p = \frac{x}{n}\) where \(x\) is the number in the sample with the trait of interest and \(n\) is the sample size.

A proportion must be between 0 and 1.00.

Example: Black Cards Section

A standard 52-card deck contains \(26\) red cards and \(26\) black cards. What proportion of cards are black?

\(p=\dfrac{26}{52}=0.50\)

The symbol \(p\) was used because this is the proportion of all cards (i.e., the population) that are black.

Example: World Campus Undergraduate Students Section

In the Fall 2014 semester, there were \(82,382\) undergraduate students enrolled in Penn State. Of those, \(6,245\) were World Campus students. What proportion of all Penn State undergraduate students were World Campus students?

\(p=\dfrac{6245}{82382}=0.076\)

The symbol \(p\) was used because this is the proportion of all Penn State undergraduate students (i.e., the population) that are World Campus students.

Example: Broken Cookies Section

In a sample of \(30\) randomly selected packages of chocolate chip cookies, \(18\) contained broken cookies. What proportion of these selected packages had broken cookies?

\(\widehat{p}=\dfrac{18}{30}=0.60\)

These data were collected from a sample so the symbol \(\widehat{p}\) was used to denote a sample proportion.