• Point Estimation: Using sample data to estimate population parameters.
• Confidence Intervals (CI): A range of values that likely contains the true population parameter.
• Example: Estimating the mean miles per gallon (mpg) of cars using the mtcars dataset.

data(mtcars)
mpg_mean <- mean(mtcars$mpg)
mpg_sd <- sd(mtcars$mpg)
n <- length(mtcars$mpg)

cat("Sample Mean:", mpg_mean, "\nSample Standard Deviation:", mpg_sd)

## Sample Mean: 20.09062 
## Sample Standard Deviation: 6.026948

• Explanation: This is to calculate the sample mean and standard deviation for mpg in the mtcars dataset. These values help us understand the central tendency and variability in the data.

\[ CI = \bar{x} \pm Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \]

Where: - \(\bar{x}\) = sample mean - \(s\) = sample standard deviation - \(n\) = sample size - \(Z_{\alpha/2}\) = critical value for confidence level

\[ MOE = Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \]

• Margin of Error (MOE) helps define the range of confidence.
• A larger sample size results in a smaller MOE, making the estimate more precise.

alpha <- 0.05
z_value <- qnorm(1 - alpha/2)
margin_of_error <- z_value * (mpg_sd / sqrt(n))

lower_bound <- mpg_mean - margin_of_error
upper_bound <- mpg_mean + margin_of_error

cat("95% Confidence Interval: (", lower_bound, ",", upper_bound, ")")

## 95% Confidence Interval: ( 18.00243 , 22.17882 )

• Explanation: This is to calculate a 95% confidence interval for the mpg dataset. This provides a range where the true population mean is likely to be.

- Explanation: A histogram and a boxplot of mpg values with confidence interval markers. A visual representation makes it easier to understand the spread and confidence range.

• Explanation: A 3D surface plot showing the probability distribution. This helps visualize the relationship between mpg values and probability density.

• Point Estimation provides a best guess of a population parameter.
• Confidence Intervals give a range of likely values.
• Larger samples result in narrower intervals, increasing accuracy.

• R Documentation: mtcars dataset
• ggplot2 & plotly for visualizations