Point estimation is a statistical concept, where we use sample data to estimate a populations Mean, Proportion and Variance. In the following presentation, we will explore the basics of point estimation and its application using the cars data set.
Point Estimation
The Population,
includes every observation, in this case 50. 
The Sample,
includes only a portion of the observations, in this case 25. 
The R Code for the Previous Graphs
Population: ggplot(cars, aes(x = speed, y = dist)) + geom_point() + labs(title = "Speed vs. Distance Using the Population") Sample: ggplot(sample, aes(x = speed, y = dist)) + geom_point() + labs(title = "Speed vs. Distance Using A Sample of the Population")
Sample Mean
The sample mean \({x}\) estimates the population mean \(\). It’s calculated using the formula:
\( {x} = _{i=1}^{n}x_i \)
Where \( n \) is the sample size and \( x_i \) are the individual observations in the sample.
Sample Proportion
The sample proportion (\( \)) estimates the population proportion (\( p \)). It is calculated using the formula:
\[ \hat{p} = \frac{x}{n} \]
Where \( x \) is the number of times the event of interest occurs in the sample and \( n \) is the number of observations in the sample.
Sample Variance
The sample variance \(s^2\) is an estimator of the population variance. It’s calculated as:
\[s^2 = _{i=1}^{n}(x_i - {x})^2 \]
Where \( {x} \) is the sample mean, \(x_i\) is the individual observation and \(n\) is the total number of observations.
Example
We can expand on the data we have to find the time of each observation in the sample
The R Code for the Previous Graphs
plot_ly(data = sample, x = ~speed, y = ~dist, z = ~dist/speed,
type = "scatter3d",
mode = "markers",
marker = list(size = 5)) %>%
layout(scene = list(xaxis = list(title = "Speed"),
yaxis = list(title = "Distance"),
zaxis = list(title = "Time")))
Example of Sample Proportion
\[ \begin{array}{c} \text{Sample} \\ \text{Proportion} \end{array} \ (\hat{p}) = \frac{\text{Observations in sample with a } time > 2s}{\text{Total number of cars in sample}} \]
\[ \hat{p} = \frac{\text{number of times } (time > 2 \text{ seconds})}{n} \]
\[ \hat{p} = \frac{15}{25} \]
\[ \hat{p} = 0.6 \]
Example of Sample Mean
\[ \begin{array}{c} \text{Sample} \\ \text{Mean} \end{array} \ (\bar{x}_{\text{time}}) = \frac{\text{Sum of times in the sample}}{\text{Total number of observations}} \]
\[ \bar{x}_{\text{time}} = \frac{61.9104094}{25} \]
\[ \bar{x}_{\text{time}} = 2.4764164 \]
Example of Sample Variance
\[ \begin{array}{c} \text{Sample} \\ \text{Variance} \end{array} \ (s^2_{\text{time}}) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
\[ s^2_{\text{time}} = 0.9814676 \]