Interval estimation refers to the statistical technique used to estimate a population parameter by calculating an interval within which the parameter is expected to fall with the specified level of confidence. This interval is known as the confidence interval. The width of the interval reflects the precision of the estimate: narrower intervals indicate the more precise estimates while wider intervals suggest the greater uncertainty. A key factor in determining the width of the confidence interval is the margin of error, which represents the maximum expected difference between the sample estimate and the true population parameter. The margin of error directly influences interval estimation by accounting for variability in the data and the chosen confidence level. In the following slides, I will be using the air quality dataset to provide examples of interval estimation.

##  95% Confidence Interval for Ozone: [35.840, 48.358] 
##  95% Confidence Interval for Solar.R: [167.656, 201.948] 
##  95% Confidence Interval for Wind: [9.270, 10.609]

• Blue line represents the mean of the data.
• Red lines represent the bounds of the 95% CI.

• Blue line represents the mean of the data.
• Red lines represent the bounds of the 95% CI.

• Blue line represents the mean of the data.
• Red lines represent the bounds of the 95% CI.

\[ \textbf{Formula} \\ \bar{x} = \frac{\sum_{i=1}^{n}x_i}{n} \] \[ \textbf{Values} \\ \text {n = sample size} \]

\[ \textbf{Formula} \\ \text{ME} = t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}} \quad \]

\[ \textbf{Values} \\ \text {t$_{\alpha/2, n-1}$ = t-critical value} \\ \text {s = sample standard deviation} \\ \text {n = sample size} \]

\[ \textbf{Formula} \\ \text{CI} = (\bar{x} - \text{ME}, \bar{x} + \text{ME}) \]

\[ \textbf{Values} \\ \text {$\bar{x}$ = sample mean} \\ \text {ME = margin of error} \]

library(dplyr)
library(plotly)
library(ggplot2)

data(airquality)
air_quality <- na.omit(airquality)

ozone_c <- t.test(air_quality$Ozone, conf.level=0.95)
solar_c <- t.test(air_quality$Solar.R, conf.level=0.95)
wind_c <- t.test(air_quality$Wind, conf.level=0.95)

cat(
  sprintf(" 95%% Confidence Interval for Ozone: [%.3f, %.3f] \n",
    ozone_c$conf.int[1], ozone_c$conf.int[2]),
  sprintf("95%% Confidence Interval for Solar.R: [%.3f, %.3f] \n",
    solar_c$conf.int[1], solar_c$conf.int[2]),
  sprintf("95%% Confidence Interval for Wind: [%.3f, %.3f]",
    wind_c$conf.int[1], wind_c$conf.int[2])
)

##  95% Confidence Interval for Ozone: [35.840, 48.358] 
##  95% Confidence Interval for Solar.R: [167.656, 201.948] 
##  95% Confidence Interval for Wind: [9.270, 10.609]

# use the pan feature to move the plot around
# use the rotate features to view from different angles
plot_ly(data=air_quality, x = ~Ozone, y = ~Solar.R, z = ~Wind,
        type="scatter3d", mode="markers", 
        marker = list(color=~Temp, colorscale="Viridis", size=5))