2.1: Proportion
Proportions are usually calculated when dealing with qualitative variables. Suppose that you want to know the proportion of time that a basketball player will make a free throw. You could look at how often the player tries to make the free throw, and how often they do make a free throw. Then you could divide the number made by the number attempted. This is how we find proportion. This is a sample statistic, since we cannot look at all of the attempts, because the player could attempt more in the future. If the player retires, and never wants to play basketball ever again, then we could find the population parameter for that player. Since there are rare cases where you can find this, then we will define both the population parameter and the sample statistic. Remember though, usually we use the sample statistic to estimate the population parameter.
Population Proportion:
\[p=\frac{r}{N} \label{population} \]
where \(r\) = number of successes observed
\(N\) = number of times the activity could be tried
\[\hat{p}=\frac{r}{n} \label{sample} \]
where \(r\) = number of successes observed and \(n\) = number of times the activity was tried
Example \(\PageIndex{1}\): Finding Proportion
Suppose that you ask 140 people if they prefer vanilla ice cream to other flavors, and 86 say yes. What is the proportion of people who prefer vanilla ice cream?
Solution
Since you only asked 140 people, and there are many more than 140 people in the world, then this is a sample and we use the sample proportion formula (Equation \ref{sample}.
\[\hat{p}=\frac{r}{n} \nonumber \]
with \(r\) = 86 and \(n\) = 140.
\[\hat{p}=\frac{86}{140} \approx 0.614=61.4 \% \nonumber \]
So 61.4% of the people in the sample like vanilla ice cream. This could mean that 61.4% of all people in the world like vanilla ice cream. We do not know for sure, but this is a good guess for the true proportion, p , as long as our sample was representative of the population. If you own an ice cream shop, then you probably want to make sure you order more vanilla ice cream than other flavors.