What is a confidence interval?

Confidence is another term for probability in statistics. A confidence interval (CI) is the mean of your estimate plus and minus the variation in that estimate.

The confidence interval for data which follows a standard normal distribution is:

\(CI = \bar{X} \pm \text{t*} \frac{\sigma}{\sqrt{n}}\)

Where:

• \(CI\) = the confidence interval
• \(\bar{X}\) = the population mean
• \(\text{t*}\) = the critical value of the t distribution
• \(\sigma\) = the population standard deviation
• \(\sqrt{n}\) = the square root of the population size

In reality, the population values are usually unknown, so the population values need to be replaced with sample values. As a result, the formula is changed to:

\(CI = \hat{x} \pm \text{t*} \frac{s}{\sqrt{n}}\)

Where:

• \(\hat{x}\) = the sample mean
• \(s\) = the sample standard deviation

deaths_df <- tibble (
  date = seq(from = as.Date("1973-01-01"),
             to = as.Date("1978-12-01"),
             by = "month"),
  deaths = as.vector(USAccDeaths)
  )
deaths_summary <- summary(USAccDeaths)

• Time period: January 1973 to December 1978
• Observations: 72 months
• Minimum: 6892 deaths
• Maximum: 1.1317^{4} deaths
• Mean: 8789

Goal: Estimate the pop. mean monthly accidental deaths with a 95% CI.

alpha <- 0.05  # 1-alpha = 95% confidence
n <- length(USAccDeaths)
sample_mean <- mean(USAccDeaths)
sample_sd <- sd(USAccDeaths)

# Calc margin of error using t-dist
margin_error <- qt(1-alpha/2, df=n-1) * sample_sd/sqrt(n)

# Calc confidence interval
lower_ci <- sample_mean - margin_error
upper_ci <- sample_mean + margin_error

• Sample mean: 8789 deaths per month
• 95% Confidence Interval: [8564, 9014]