• Point Estimation- a single value used to estimate population parameter.
• Interval Estimation- A range of values (Confidence Interval) where the parameter is expected to be in.

Summary of dataset used to explain Point and Interval Estimation:

summary(USArrests)

##      Murder          Assault         UrbanPop          Rape      
##  Min.   : 0.800   Min.   : 45.0   Min.   :32.00   Min.   : 7.30  
##  1st Qu.: 4.075   1st Qu.:109.0   1st Qu.:54.50   1st Qu.:15.07  
##  Median : 7.250   Median :159.0   Median :66.00   Median :20.10  
##  Mean   : 7.788   Mean   :170.8   Mean   :65.54   Mean   :21.23  
##  3rd Qu.:11.250   3rd Qu.:249.0   3rd Qu.:77.75   3rd Qu.:26.18  
##  Max.   :17.400   Max.   :337.0   Max.   :91.00   Max.   :46.00

Formula:

\[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \]

Point Estimation’s formula is already implemented in R’s coding language through mean(). Using our data set, USArrests, we can find the point estimation (average) of murders through the following line.

mean(USArrests$Murder)

## [1] 7.788

Formula:

\[ \text{CI} = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}} \] Similarly to Point Estimation, Interval Estimation can also be found in r using t.test()$conf.int. Using our data set, we can find the confidence interval with the following code:

t.test(USArrests$Murder)$conf.int

## [1] 6.550178 9.025822
## attr(,"conf.level")
## [1] 0.95

Here is a 3D visual of the comparisons between the amount of murders, assaults, and rapes that were documented in this data set.

The bar plot shows the point estimation (average) of each category

For the visualization on the previous slide, here is the code that was written for it to see the average crime rates across the categories.

crime_means = colMeans(USArrests)
ggplot(data.frame(Crime = names(crime_means), Rate = crime_means), 
       aes(x = Crime, y = Rate)) + 
  geom_bar(stat = "identity", fill = "steelblue") + 
  theme_minimal() + 
  labs(title = "Average Crime Rates", 
       x = "Type of Crime", 
       y = "Rate")

This visualization compares the Urban Population to the Assault Rates of each state with the highlighted area being the confidence interval.

• Point Estimation provides the simplicity of a average number
• Interval Estimation provides precision and confidence of a possible range a number may fall between
• These two methods offer clarity and confident predictions by offering a best guess and a range of possible values.