Confidence Intervals: A Detailed Tutorial

In statistics, a **confidence interval** (CI) provides an estimated range of values that is likely to contain an unknown population parameter. The interval is associated with a confidence level, which quantifies the degree of certainty that the parameter lies within the interval.

This tutorial explains how to compute confidence intervals, interpret them, and understand their significance in statistical analysis using the mtcars dataset in R. We will walk through examples for estimating means, regression coefficients, and proportions, and also provide the results from the R code for these calculations.

1. What is a Confidence Interval?

A confidence interval is a range of values used to estimate the true population parameter from a sample. The **confidence level** tells us how confident we are that the true parameter lies within this interval. For instance, a 95% confidence interval means we can be 95% confident that the true parameter lies within this range.

For example, if we compute a 95% confidence interval for the mean weight of cars in the mtcars dataset, the interpretation would be: "We are 95% confident that the true mean weight of all cars in the population lies within this interval."

2. Confidence Intervals for Means

One of the most common applications of confidence intervals is estimating the mean of a population from a sample. In this case, we will estimate the average weight of cars in the mtcars dataset.

The formula for a confidence interval for the population mean is:

Where:

• 
  : Sample mean
• 
  : t-value corresponding to the desired confidence level (e.g., for 95%, this value corresponds to the 97.5th percentile of the t-distribution)
• 
  : Standard deviation of the sample
• 
  : Sample size

To calculate the 95% confidence interval for the mean weight of cars in the mtcars dataset, we use the following R code:

# Load the mtcars dataset
data(mtcars)

# Calculate the sample mean and standard deviation for the 'wt' (weight) variable
mean_weight <- mean(mtcars$wt)
sd_weight <- sd(mtcars$wt)

# Calculate the sample size
n <- length(mtcars$wt)

# Calculate the t-value for a 95% confidence interval (df = n - 1)
t_value <- qt(0.975, df = n - 1)

# Calculate the margin of error
margin_of_error <- t_value * (sd_weight / sqrt(n))

# Calculate the confidence interval
ci_lower <- mean_weight - margin_of_error
ci_upper <- mean_weight + margin_of_error

# Print the confidence interval
ci_lower
ci_upper

The output of the code gives us the following confidence interval:

Interpretation: We are 95% confident that the true mean weight of all cars in the population lies between 2.86 and 3.57 units (thousand pounds). This range provides an estimate of the population mean based on the sample data.

3. Confidence Intervals for Regression Coefficients

Confidence intervals are also used in regression analysis to assess the uncertainty around the estimated regression coefficients. Let's fit a simple linear regression model predicting car weight from horsepower in the mtcars dataset, and calculate the confidence intervals for the intercept and the slope.

The formula for a confidence interval for a regression coefficient is:

Where:

• 
  : Estimated regression coefficient
• 
  : Standard error of the regression coefficient
• 
  : Number of observations in the regression model

We fit the following linear regression model:

# Fit a linear regression model predicting weight from horsepower
model <- lm(wt ~ hp, data = mtcars)

# Summary of the model
summary(model)

The model summary shows the following results:

Call:
lm(formula = wt ~ hp, data = mtcars)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.41757 -0.53122 -0.02038  0.42536  1.56455 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.838247   0.316520   5.808 2.39e-06 ***
hp          0.009401   0.001960   4.796 4.15e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7483 on 30 degrees of freedom
Multiple R-squared:  0.4339,	Adjusted R-squared:  0.4151 
F-statistic:    23 on 1 and 30 DF,  p-value: 4.146e-05

The coefficients table shows the following values for the intercept and the slope:

• Intercept estimate: 1.838247
• Slope estimate (hp): 0.009401
• Both coefficients have very low p-values, indicating statistical significance at the 0.05 level (marked by ‘***’).

Now, we calculate the 95% confidence intervals for the intercept and the slope:

# Extract the regression coefficients and their standard errors
coef <- summary(model)$coefficients

# Extract the t-value for a 95% confidence interval
t_value <- qt(0.975, df = n - 2)

# Calculate the margin of error for the intercept and slope
intercept_margin <- t_value * coef[1, 2]
slope_margin <- t_value * coef[2, 2]

# Calculate the confidence intervals
intercept_ci <- coef[1, 1] + c(-intercept_margin, intercept_margin)
slope_ci <- coef[2, 1] + c(-slope_margin, slope_margin)

# Print the confidence intervals
intercept_ci
slope_ci

The output gives us the following confidence intervals for the intercept and slope:

Interpretation:

• We are 95% confident that the true intercept (when horsepower is zero) lies between 1.19 and 2.48.
• We are 95% confident that the true slope (change in weight per unit increase in horsepower) lies between 0.0054 and 0.0134.

4. Confidence Intervals for Proportions

Finally, we will calculate a confidence interval for a population proportion. Let's say we want to estimate the proportion of cars in the mtcars dataset that have horsepower greater than 100. We will calculate the 95% confidence interval for this proportion.

# Calculate the proportion of cars with horsepower greater than 100
prop <- mean(mtcars$hp > 100)

# Calculate the margin of error using the formula for proportions
z_value <- qnorm(0.975)  # z-value for 95% confidence
margin_of_error <- z_value * sqrt((prop * (1 - prop)) / n)

# Calculate the confidence interval for the proportion
ci_lower <- prop - margin_of_error
ci_upper <- prop + margin_of_error

# Print the confidence interval
ci_lower
ci_upper

The output of the code gives us the following confidence interval:

Interpretation: We are 95% confident that the true proportion of cars with horsepower greater than 100 lies between 0.56 and 0.87.

5. Conclusion

In this tutorial, we have covered the concept of confidence intervals and how to calculate them for different statistical scenarios, including means, regression coefficients, and proportions. Confidence intervals provide important information about the uncertainty of our estimates, and they help us make more informed decisions based on sample data. We also demonstrated the use of R code to compute confidence intervals for various types of data and models, providing real output and explanations for each case.